1 20 111 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/274fe4b15080f353b6f7e68bf921ed25.pdf 8d1ec494b7e407030402e87c31182af9 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 22 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Broadcast Radio Description An account of the resource Transcript of a lecture given by tutor emeritus Howard Fisher on April 29, 2026 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "This talk will recount the paths by which the basic techniques of wireless signaling were adapted to make possible the wireless transmission of voice and sound—the technology we now call radio. Essential in this endeavor were additional capabilities of the vacuum tube: oscillation, which enabled production of electromagnetic waves having a defined frequency, and modulation, which varied the properties of electromagnetic waves. I will conclude by suggesting that technologies should not be viewed as being chiefly solutions to problems, but might also be seen as practical extensions of speculative thinking." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2026-04-29 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Radiotelephone Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Broadcast_Radio_2026-04-29 Relation A related resource <a href="https://digitalarchives.sjc.edu/items/show/8182">Broadcast Radio audio recording</a> https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a1c5f0ffcdc4279e8be3c202431eedcf.pdf 0495d614641a49be958fc447b3e122c9 PDF Text Text The Vacuum Tube 1 Howard J.Fisher Good afternoon, everyone, and thank you for coming to this second of three lectures on early radio technology. Our topic is the vacuum tube—I chose the two examples on display here for their large size, which allows us to see their construction. The vacuum tube made possible the age of radio—that is, wireless transmission and broadcast of sound—as opposed to radiotelegraphy, which was limited to sending wireless messages in the dots and dashes of Morse code. What did the vacuum tube do, that made it such a consequential device? And what are the principles of its operation? The vacuum tube finds its beginnings in Edison’s light bulb. Here is one of Edison’s later designs. The hair-like filament is heated to incandescence by a strong electric current, producing a very creditable glow, as you see here. But at temperatures high enough to achieve incandescence, almost any filament material would soon burn up in the presence of air; the glass bulb therefore had to be evacuated. Edison’s light bulb was by no means the first. Earlier researchers had crafted similar incandescent devices; but the limited capability of early vacuum pumps achieved only mediocre vacuums, in which the glowing elements would burn up quickly. Edison benefited from the development of highly effective pumps such as Sprengel’s and Geissler’s, which could achieve pressures as low as a tenth of a millimeter of mercury. 2 The ability to obtain such low pressures enabled Edison to develop by 1880 a carbonized bamboo filament capable of lasting hundreds of hours; but very soon another difficulty emerged. Early versions of this filament tended to throw off some of their carbon, which would then be deposited on the interior of the glass, gradually blackening it, which, of course, defeated the very purpose of a lighting device. This lecture was given 15 April 2026 at St. John’s College, Santa Fe. It is the second of a series of three talks on early radio technology. 1 2 Andrade, E. N. da C. (1953). "The history of the vacuum pump." Vacuum 9 (1): 41–47. �2 Interestingly, though, the pattern of carbon deposit was not uniform. Edison noted a peculiar “reverse shadow” of the filament—a line of less darkening, as you can see in the photograph on the left. 3 Curiously, this lighter line appeared only on the side of the bulb adjacent to the positive arm of the filament—the line labeled bb in the diagram on the right. For, indeed, there was a definite positive and a definite negative arm of the filament, as the diagram indicates. That is because Edison powered his lamp with direct current, for which the positive extremity of the source remains positive, and the negative remains negative, throughout the current’s duration. In contrast, the common household current of today is alternating current, in which the positive and negative polarities exchange places 60 times each second. Evidently, in Edison’s bulb, many of the carbon particles traveling in the direction from negative to positive were being blocked before reaching the glass, but particles traveling in the opposite direction were not so impeded. What could cause this asymmetry? Well, if the carbon particles escaping from the filament carried a negative charge, then at least some of the particles originating from the negative arm of the filament would be attracted by the positive arm and might be captured by it before reaching the glass, while those escaping from the positive arm would be repelled from the negative arm; they would be less susceptible to capture and would continue beyond it, and would at last deposit themselves on the glass. But did these roving carbon particles carry an electric charge? Edison investigated that question by mounting a metal plate between the filament arms, as shown in this photograph. On their way from the negative arm to the positive, some particles, at least, would presumably strike the plate and give up their charge to it. Then, if that plate were connected back to the filament, the circulating charge would constitute a 3 Left: from Tyne, Gerald F. J., Saga of the Vacuum Tube, Howard W. Sams & Co. (1977). Right: adapted from Fleming, J. A., The Thermionic Valve and its Developments in Radiotelegraphy and Telephony. London, The Wireless Press (1919). �3 current and should be detected by a galvanometer. Edison began a series of experiments with this special bulb in 1882. When he connected the plate to the positive end of the filament, a current did indeed pass; but no current passed when it was connected to the negative end. Let us see if we can understand this. Figure (1) is a diagram of Edison’s original light bulb. The righthand filament arm, AB, is connected to the positive terminal of the battery; the lefthand arm, BC, is connected to the negative terminal. Thus there is a potential difference between the two extremities of the filament; and since the filament is a conductor, a current will flow through it. But the same potential difference exists in the space between the two arms, since the filament is bent in a horseshoe shape. Thus there is an electric field between AB and BC, represented by the curved red arrows. If, then, electrically-charged particles are emitted from the filament, they will be urged in the direction of the arrows if they are positive, and in the opposite direction if they are negative. Now consider Figure (2). With the addition of plate D connected (through the meter) to the positive battery terminal, plate D and arm AB are effectively connected together. They are therefore at the same electric potential and no electric field exists between them. But the full battery voltage now obtains between plate D and arm BC, connected to the negative terminal. An electric field therefore exists between them, having direction from D to BC, as indicated by the little red arrows. If, then, as Edison suspected, negatively-charged particles of carbon were being emitted from the filament, they would be urged in the direction opposite to the electric field, that is, in the direction from BC to D; particles emitted from BC would thus carry negative electricity from filament to plate, producing the current that Edison witnessed. What about negative particles emitted from the other arm of the filament, AB? They are exposed to no field at all and therefore are not urged toward the plate; they would not reach it and so would play no part in the measured current. �4 But now consider Figure (3), where the plate is connected to the negative terminal of the battery. Here plate D and arm BC are effectively joined together; they must therefore have the same electric potential, and there will no longer be any electric field between them. Instead, a field exists between arm AB and plate D, in the direction from AB to D. Negative particles emitted from the filament now have nowhere to go! Particles emitted from BC are exposed to no field at all and so are not drawn to the plate, as I mentioned before. Particles emitted from AB are exposed to a field; but if they are negatively charged, they are urged in the opposite direction, that is, away from the plate. Negative particles thus cannot form a current no matter what part of the filament they originate from, which explains why Edison observed no current when the plate was connected to the negative battery terminal. This asymmetric current—a current that flows to one end of the filament but not to the other, came to be known as the “Edison Effect.” 4 Edison’s experiments thus confirmed that the carbon particles thrown off from his light bulb filaments were negatively-charged. But that knowledge proved of little value for preventing the original problem—the progressive blackening of his bulbs with use. 5 In fact that blackening could not be averted so long as carbon was the filament material; eventually carbon gave way to thoriated tungsten wire, which today is the material used almost universally for incandescent lamp filaments. 6 We can witness the Edison Effect for ourselves. 7 An automotive taillight bulb, such as the number 1157 bulb shown here, has two filaments: one for the taillight, and a brighter Preece, W. H., “On a Peculiar Behaviour of Glow Lamps when Raised to High Incandescence,” Proc. Roy. Soc. (1885) 38: 219–230. 4 5 Edison did, however, envision a different application for his specially-equipped bulb, its use as a kind of sensitive voltmeter. He took out a patent on the device in 1884. 6 Even bulbs with tungsten filaments exhibit a long-term blackening, though at far slower rates than do those with carbon filaments. The problem is a serious one only in lamps of very high wattage, such lamps control this by filling the bulb with a gas such as nitrogen, argon, krypton, or xenon, which do not readily react with tungsten, at a pressure great enough to reduce filament evaporation. This clever procedure is described in an article by ZhiQi Lin, Wei Zhang, and HuiJie Li: “Using car taillights as an experimental device to demonstrate the Edison effect” in The Physics Teacher, 56, pp. 410–411 (2018). 7 �one for the brake light. They have a terminal in common, as the diagram indicates; so they are not isolated from one another the way Edison’s plate was isolated from his filament. But if we deliberately burn out one filament, as this slide shows, one fragment of the 5 destroyed filament will be isolated from the filament that is still intact, and so should play the role of Edison’s plate. Let us try. The intact filament is being heated with 11 volts, and I connect the isolated element, through the milliammeter, to the positive end of the filament. The meter shows an Edison current of 0.5 microamperes. But when I connect it to the negative end, we find no current, just as Edison had observed. Now since currents require a closed circuit, whatever current I measure in the wire outside the bulb must also be flowing in the space within the bulb. What, then, is the nature of this internal current? It obviously does not consist of carbon particles, as did Edison’s; for our filament is tungsten. But neither is it likely to consist of charged tungsten particles, since the very advantages that make tungsten a superior filament material—lack of “blackening” and long lifetime—indicate that the filament is not losing tungsten at a sufficient rate to be carrying the Edison current. 8 Perhaps, then, the current within the bulb does not consist of charged material particles at all—perhaps it is a flow of electricity itself! But what does that even mean? We are accustomed to thinking that electricity requires a conductor in order to “flow”; and vacuum is not a conductor! The engineer Edwin Houston expressed this puzzlement in a paper he presented at the International Electrical Exhibition at Philadelphia in 1884. 9 Speaking of the Edison Effect, he wrote: The question is, what is the origin of this current? How is it produced? Since we have within the globe a nearly perfect vacuum, we cannot conceive the current as flowing across the vacuous space, as this is not in accordance with our preconceived ideas connected with high vacua. Houston went on to muse further: This inference was conclusively demonstrated by O. W. Richardson; see “The Emission of Electrons from Tungsten at High Temperatures,” Science 38, 967 (1913). 8 Houston, E. J.: “Notes on phenomena in incandescent lamps,” Transactions of the American Institute of Electrical Engineers, I, 1 (1884). 9 �6 It may be electricity flowing through empty space, which I don’t think probable. Even if we set aside the perplexity as to how electricity can possibly move through empty space, we are confronted anew with Edison’s question: whether the current is positive or negative. But we cannot answer it as definitively as he did. For in the absence of occurrences like “blackening” we no longer have physical evidence that something is traveling from filament to plate; and therefore we can no longer know whether we have a current of negative electricity passing from the filament to the plate or a current of positive electricity from the plate to the filament. One piece of evidence, at least, does certainly implicate the filament as the source; watch what happens when I start with a cold filament and gradually raise its temperature by increasing the heating voltage. As before, the meter will measure the Edison current. I started with no filament voltage at all, and of course no plate current either; but we’re still seeing no current, even as I continue to raise the voltage enough to make the filament glow... Ah! We’re getting tenth of a microampere with about 8.6 volts on the filament... As I continue to raise the filament voltage, the plate current rises... and we reach a full microampere when the voltage is a little over eleven volts. Now why would the current depend on the temperature of the filament? If the current is “electricity flowing through empty space”—the case Edwin Houston thought “not probable”—it is hard to see why there should be such a connection, since no theory of electricity implies any inherent relation between it and heat. But that dependency would make sense if the electricity traveled in the form of material particles, inasmuch as the temperature of a body indicates the kinetic energy of the particles which compose it; and the more agitated those particles are, the greater the number that may escape the forces which bind them to their host—just as a liquid will evaporate more quickly when it is hot than when it is cold. But wait! Did we not conclude earlier that the current could not consist of charged particles of tungsten—the material of the filament? If the charged particles are not tungsten, what possible material could they be? That is the very question seniors take up in the first semester lab when they study cathode rays, and the answer is startling: it appears that current in the evacuated tube is carried by charged particles whose material belongs to no known species of matter! 10 10 Thomson, J. J., "Cathode Rays," Philosophical Magazine. 5. 44 (269): 293 (7 August 1897); “On Bodies Smaller Than Atoms,” Popular Science Monthly (August 1901) �Shall we then affirm that the heated filament is the source of charged particles of some as-yet-unknown material, and that these constitute the Edison current? Unfortunately, the mere association of current strength with filament temperature overlooks a subtle but crucial factor. The problem is that our power supply does two things at once: it both heats the filament and establishes the electric field between the plate and the filament. This diagram shows the interior of our taillight bulb. The isolated fragment of the burned-out 7 filament is our “plate,” as I mentioned before. And, as before, both the “plate” and one end of the surviving filament are connected to the positive terminal of the power supply; the other end of the filament is connected to the negative terminal. The full potential difference of the supply therefore exists between the plate and the negative end of the filament. I have drawn Faradayan lines of force—they are the curved red arrows—to represent the electric field associated with this potential difference. But this field is what furnishes the very force that would propel charged particles along their paths—from plate to filament if they are positive, from filament to plate if they are negative. These charged particles, whatever their source, will be subject to a greater electric force if the voltage is raised; they will be accelerated more quickly and will attain a higher velocity. More charge, whether negative or positive, will be transferred in a shorter time; and this will result in a greater Edison current, wholly apart from any increase in filament temperature. You see, then, the weakness in our former reasoning. In raising the voltage of our power supply we both raise the filament temperature and increase the electric field; and a stronger electric field would tend to increase the rate of particle transfer, whether they are negative particles emitted from the filament or positive particles emitted from the plate. It is clear, then, that we cannot definitively identify the filament as the source of charged particles so long as we rely on a single electrical source. But if we were to use separate supplies for the plate and the filament, we could control their voltages independently, and so distinguish the influence of filament temperature from that of electrostatic attraction. That is what the British engineer John Ambrose Fleming 11 did in 1896. Using samples of Edison’s special bulbs that had been made specifically for him, Fleming connected separate batteries to the filament and the plate. A variable resistance, or rheostat, controlled the voltage at the filament, and hence the filament’s temperature. With a fixed voltage between the plate and filament, Fleming obtained a current through the plate that rose rapidly with 11 In addition to the researches to be described here, Fleming is known for having originated one of the many “right-hand rules” used in physics, and for designing the transmitter that sent the first transatlantic radiotelegraphy message. He studied under Maxwell at Cambridge in 1878 and 1879, and was by his own account one of only two or three students who attended Maxwell’s final lectures. See Fleming, Ambrose, “Some Memories of Professor James Clerk Maxwell” in James Clerk Maxwell, A Commemoration Volume 1831–1931, Cambridge University Press (1931). �8 increasing filament voltage. That rise is depicted in Fleming’s graph, on the right. 12 And since he kept the plate voltage constant while varying the filament voltage, the plate current’s dependence on the filament voltage means that we can now point confidently to the heated filament as the source of electrified particles. 13 The current must therefore consist of negatively-charged particles that were liberated from the filament by heat in the manner we described a moment ago. Once freed from the filament they will be accelerated towards the plate by the electric field between plate and filament, thereby constituting the Edison current. In 1904 Fleming constructed the device shown on the left; it was essentially Edison’s bulb, with a flat plate like his. The one on the right was a later design employing a cylindrical plate to surround the filament completely. Fleming’s “valve,” as he called it—we will soon see why that name is appropriate—launched decades of commercially manufactured devices having essentially the same design. One of them is this elegant creation, an Eimac model 250R, introduced in the late 1940s; you can see how like it is to 12 Fleming, J. A., The Thermionic Valve and its Developments in Radiotelegraphy and Telephony. London, The Wireless Press (1919). The graph records measurements he had made in 1896. 13 This conclusion is valid provided the filament voltage is small compared to the potential difference between plate and filament. That was true for Fleming’s devices and will be true in our adaptations of his experiments. �9 Fleming’s. And if you wonder what possible commercial application could have supported such a product as this one, 40 years after Fleming patented his apparatus, I invite you to take one of these handouts 14 after the talk; they deal with the topic of current rectification. We, however, will employ the Eimac tube to continue our own investigation of the role of the heated filament. In his initial work with the special Edison bulb, Fleming kept the plate voltage constant while varying the filament voltage. Later he examined the complementary case: varying the plate voltage while keeping the filament voltage constant. 15 We can replicate these experiments, using variable power supplies in place of Fleming’s batteries. Here is our setup. I am applying 2 volts to the filament. The plate voltage is zero; and as you see, the plate current is also zero. But keep your eye on the plate current as I apply more and more voltage to the plate. . . . You see how the plate current increased to one milliampere right away, but its growth seems to have stopped . . . No, it has grown to 2 milliamperes, but it is not increasing any further, even as I continue to raise the plate voltage . . . The plate current appears to have reached a maximum—at least for this degree of heating the filament. Let’s do the same thing using a higher filament voltage, and therefore a higher filament temperature. The filament voltage is now 2.3. As before, I am gradually applying voltage to the plate, and the plate current has risen to 2 milliamperes . . . and now 3 milliamperes . . . Now it has risen to 4 milliamperes . . . But it is definitely holding there . . . It is still 4 milliamperes, even at the maximum voltage of 118.7 volts. 14 15 The handout for this lecture begins on page 21 below. Fleming, J. A., Principles of Electric Wave Telegraphy and Telephony (1916), page 533. �10 Here is a graph of the measurements we just made. Just as in our first trial, when we raise the plate voltage, the plate current increases rapidly at first; but it once again levels off to a plateau. But notice that the plateau attained with 2.3 volts on the filament is higher than when the filament was being heated with only 2.0 volts. The plateau is now 4 milliamperes, while before it was only 2 milliamperes. I made several other sets of measurements using higher and higher filament voltages. Let us append those results to our initial graph and produce a composite presentation of all the trials: The interpretation of these curves is clear: once charged particles are emitted from the filament, they become subject to acceleration towards the plate by the electric field that exists between the filament and the plate. At low plate voltages, for which the electric field is weak, particles are emitted in greater numbers than they can be swept away. But as the plate voltage rises, more and more of the emitted particles are drawn to the plate. Finally the plate voltage reaches a value at which particles are being removed at the same rate as they are emitted; the measured current cannot then undergo any further increase, no matter how much more the plate voltage is raised. The value of this current maximum, or “plateau,” is therefore a measure of the rate at which particles are being emitted from the filament; and since the plateau rises when the filament voltage rises—and hence when the filament temperature rises—it should now be clear that the rate at which negative particles are emitted depends essentially on the filament temperature. Because these charged particles, or ions, 16 are released by the action of heat, Owen Richardson of Cambridge 16 Ion, from ιέναι, go, since charged particles undergo motion when placed in an electric field. Like so many other electrochemical terms, it was introduced by Faraday at the suggestion of William Whewell. �University called them “thermions,” and their travel to the plate he called the “thermionic current.” 17 11 With the aid of a vacuum tube essentially identical to Fleming’s we have finally, I think, been able to appreciate the role of the heated filament. But that was not the question which had excited Fleming’s interest. His chief concern was the asymmetric character of the conduction—the same asymmetry we noted in the Edison Effect. Here the power supply is connected so as to make the plate positive; and we are obtaining a generous 37 milliamperes of plate current. It is much higher than what we obtained a moment ago, because I am heating the filament with a much higher voltage—4.0 volts. But if I reverse the connections to make the plate negative... there is no current, even though I am applying a very strong voltage to the plate. To Fleming this meant that current could flow through the tube in only one direction—a feature that, he realized, would have immediate application to wireless telegraphy. To help us see why, let me revisit a topic I raised in February’s talk. At the time of Fleming’s researches, the standard device for detecting an incoming radiotelegraphy signal—that is, an electromagnetic wave—was still the enigmatic coherer, which we discussed last time; but as I noted then, that apparatus’s principle of operation was poorly understood; and it had many deficiencies, concerning which I quoted some choice words from several disgruntled engineers. Why not, then, simply send the current from the receiving antenna to one of the more ordinary current indicators: a galvanometer, a telephone receiver, or some other electromechanical device? The reason is that the current induced by an incoming 17 “Here we have two currents: the current used to heat the wire and the thermionic current away from the surface of the latter.” O. W. Richardson in London, Edinburgh & Dublin Philosophical Magazine 6th Series vol. 17, 814 (1909) �12 electromagnetic wave is oscillatory, or alternating. A graph of that current with respect to time would look like this: The current swings regularly between positive and negative; and the number of complete swings per second is the frequency of the current. But in wireless telegraphy the frequencies are so high that no ordinary mechanical responder—neither the needle of a galvanometer, nor the diaphragm of a telephone receiver—could possibly vibrate at those rates; the alternating pulses would simply end up nullifying one another and would produce no response at all. Now it had been known since the 1870s that certain crystals, when brought into electrical contact with a metal probe, could somehow make these radio-frequency alternating currents compatible with indicators like the telephone receiver. 18 In February’s talk I singled out the work of G. W. Pickard in this connection; his contact devices would soon become practical “crystal detectors” for radiotelegraphy and would remain in use well into the broadcast radio era. But at the time of Fleming’s work, these contrivances were temperamental and delicate; and because their operation was poorly understood, improvements in their design could be pursued only by trial and error, not by systematic analysis. In contrast, Fleming’s vacuum tube excelled in all these areas. It was mechanically sturdy, its electrical characteristics were stable and reliable; and, most importantly, its principle of operation—"one-way” conduction—was understood. 19 By conducting electric current in one direction only, Fleming’s vacuum tube could prevent the mutual nullification of positive and negative wave phases by blocking the negative phases. It would then send onward a pulsating current whose graph would look like this: To be sure, those pulses would still be arriving at a rate too high for either a galvanometer needle or a telephone receiver to reproduce; but they would be in the same direction! They would not nullify one another, as a train of alternating pulses must do. On the contrary, if sent to a galvanometer, they would reinforce one another to displace the needle by a clearly visible amount. If sent to a telephone receiver, they would produce a definite click. 18 Carl Ferdinand Braun studied the properties of such junctions in 1874 and Jagadish Chandra Bose employed crystals as microwave radio detectors in 1894. 19 The crystal detector, too, exhibited “one-way” conduction, though this was not recognized at first. Even after that behavior was established, the physical principles underlying it were not understood until the 1930s and 1940s; the researches of that period would provide the basis for modern solid-state electronics. �Fleming proposed a circuit like this one for the detection of electromagnetic wave 13 signals. I have reduced his circuit 20 to its essentials for clarity. When an electromagnetic wave is intercepted by the antenna, it induces an oscillating current. If the red portion of the circuit were absent, that current would simply flow between the antenna A and the earth, E. But in Fleming’s circuit, it must first pass through the vacuum tube O, then through the telephone receiver T, and finally to earth. And so long as the incoming wave makes the antenna positive with respect to the earth, the plate of the tube will likewise be positive, and current will flow. But in the next phase of oscillation, the antenna will be negative; the plate too will therefore be negative; and we saw that current cannot flow through the tube under those conditions. Only positive current pulses will pass through to the telephone; and together these will produce the “click” that announces the commencement of a “dot” or “dash,” followed by a slightly different “click” when it releases at the end of the train of pulses. You can see, then, why Fleming called his device a “valve”—a term which British usage retains to this day. By permitting flow in one direction only, it functions like the “check valve” used in plumbing and other applications. In the mechanism depicted here, a hinged flap opens to permit liquid flow from left to right, but any flow from right to left bears on the flap to close it and halt the flow. Fleming’s explicit adoption of the terminology of plumbing suggests that he had in mind the “fluid flow” image of electric current. Indeed, to the extent that we seem to have demonstrated the existence of a stream of charged material particles in the vacuum tube, flowing from filament to plate, it may appear that we have confirmed the fluid image quite literally. Yet I hesitate to say that, for there are some vital differences. What, exactly, constitutes a “fluid”? For Archimedes, a fluid is a continuous whole, capable of sustaining pressure; and when it moves, it is driven by a difference of pressure between one region and another. Recall his definition at the opening of On Floating Bodies: Fleming, J. A., Principles of Electric Wave Telegraphy and Telephony, 3rd ed. (1916), p. 531. The diagram in my slide is highly idealized; an actual receiver circuit would incorporate additional coils and capacitors to utilize the incoming current more efficiently. 20 �14 Let it be assumed that a fluid is of such a nature that if its parts lie evenly and are continuous, that part which is pressed the less is thrust out by that which is pressed the more... 21 But in the vacuum tube we have a multiplicity of individual charged particles; they do not seem to form a “whole” in any obvious sense; and their motion is by no means the result of anything like Archimedean pressure, for we envisioned each of the particles as being accelerated independently by the electric field. In an Archimedean fluid, every portion moves in virtue of being part of a moving whole; but in the vacuum tube, each particle enjoys its own private treaty 22 with the field. So it is not at all clear that we ought to call this stream of particles a “fluid.” Furthermore, what kind of relation do these particles have to the electricity which distinguishes them? Is electricity itself an immaterial substance, which just happens to be embellishing each of these particles like a coat of paint? Or is electricity these very particles—not a fluid at all, even though they seem to resemble a fluid when they are confined to a “conductor”? No doubt Fleming meant his “check valve” allusion to convey only what the vacuum tube does, not to explain how it does it. Despite any resemblance between a stream of charged particles and the flow of a fluid, we will not expect to find a mechanical “flap” anywhere in the vacuum tube. How, then, shall we understand the halting of current when it is applied in the forbidden direction? We may find some guidance in another example of “one-way” flow, one that does not rely on mechanical contrivance; that is the waterfall. Water (or any material) can fall down, but it does not fall up. If you tried to swim up the falls you would be “blocked,” so to speak—not by a hinged flap, but by the gravitational field. Nevertheless, a field can be overcome with a sufficiently great store of kinetic energy—as salmon do when they ascend waterfalls in order to reach their ancestral breeding grounds: When we try to direct a reverse current through the vacuum tube, we have the equivalent of a salmon run; for instead of the kinetic energy of the fish being converted to gravitational potential energy, we have the kinetic energy of the emitted particles—Richardson’s thermions—being expended through conversion to electrical potential energy due to the reversed electric field. But whereas most salmon have enough kinetic energy to overcome the potential-barrier, in a well-functioning vacuum tube the ions do not. We have seen how Fleming’s vacuum tube blocked half of each cycle of the alternating current induced in the antenna; thus the remaining half-cycles, having a common direction, could exert a cumulative effect upon a telephone receiver or a galvanometer. A disadvantage, of course, is that since the incoming wave current is very weak, only a weak current was available to energize the telephone or galvanometer. 21 Archimedes, On Floating Bodies, Book I. “Private treaty” was the bon mot used by Clinton J. Davisson in his paper, “Are Electrons Waves?” Franklin Institute Journal, 205 (1928). 22 �15 But not long after Fleming introduced his vacuum valve, the American inventor Lee De Forest dramatically altered the technical possibilities by adding a third electrode. Fleming, offended by what he saw as a trespass upon his specialty, regarded the new electrode as a merely gratuitous embellishment of his own invention; and there ensued a 37-year succession of court battles over patent rights—a real-life Jarndyce v. Jarndyce, that celebrated legal saga immortalized in Dickens’ Bleak House. 23 Do we know why De Forest added the third element? That is not clear from his initial efforts, since he originally positioned the additional electrode outside the tube, first in the form of parallel foil sheets, shown on the left, and later as a wire coil wound around the glass tube, in the middle sketch; in both drawings the third element is highlighted in red. 24 Eventually, however, he located the added structure within the tube, between the filament and the plate, as drawn on the right. De Forest named these devices Audions; 25 but since his definitive version included three electrodes within the glass, it quickly acquired the informal name “triode,” distinguishing it from Fleming’s two-element tube, which (despite Fleming’s objections) won the name “diode.” 26 Let us see what De Forest’s third element accomplished; for I think we will find that the triode, even more than the diode, implicitly draws upon the imagery of a stream of particles rather than a flowing fluid. In this diagram of one of De Forest’s triodes, taken from an early patent application, F represents the filament and b the plate, just as in the Fleming diode—although De Forest preferred the term “wing” rather than “plate.” Between them is the third electrode—not an impenetrable plate, but a wire bent into the serpentine shape drawn here. It was thought to resemble a cooking grate or griddle; and on that basis the third electrode acquired the name “grid.” As I mentioned before, it isn’t clear whether De Forest adopted this configuration with any specific expectation in mind. About its operation, he is reported to have said, “I don’t “The Radio Tube Goes to Court,” The New York Times, 23 November 1930. The ongoing cases described in the article would finally be resolved in 1943. 23 24 Adapted from Gerald F. J. Tyne, Saga of the Vacuum Tube (1977), p. 59. Considering the wide range of designs upon which De Forest bestowed the name “Audion,” it would seem that he regarded that title as a the name of a brand more than as the name of a device. 25 26 Fleming insisted that the name of a device ought to convey its essential function, as his epithet “valve” did. “Diode,” which expressed only the number of elements in the tube, would fail to meet that criterion. �16 know how it works, it just does!” That is rather hard to believe, since I think that even we can anticipate some of the effects of mounting any element whatsoever within the tube itself. Nevertheless, the grid does exhibit some behavior that is bound to seem odd at first sight. Let us study that initial puzzle in a triode of our own, before going on to the “working” that De Forest would have had chiefly in mind if he really did say, “it just works.” This lovely objet d’art is an Eimac 250TH triode, first introduced in 1938. It is practically the twin of the diode we have been experimenting with, except for the added grid; you can see the grid connection at the side of the glass envelope. I am applying 3.9 volts to the filament, which heats it to glowing; and I am making no connection to the plate. Now when I connect a meter between the grid and the positive end of the filament, we measure a negative current; it is 1.29 milliamperes. No doubt this is our old friend the Edison current; and if so, we should expect zero current between the grid and the negative end of the filament. Let us try . . . Yes, we are reading zero milliamperes. But I am not using the most sensitive range of my meter; let’s see what happens when we switch to the microampere range . . . �17 Well, it is not zero! Instead, we have a negative current of 1.7 microamperes. True, this is a very small value—only about one thousandth of what we measured for the Edison current—but the account we gave earlier implied that there should be no current whatsoever to the negative end of the filament; because if the Edison current really consists of a stream of negative ions, they should be repelled by the negative end of the filament, and therefore repelled by the grid which is connected to it through the meter; they should not be flowing to the grid! What on earth can be causing this “maverick” current? Our earlier explanation of the Edison Effect appealed solely to the electric field that exists between the positive and negative ends of the filament. We did not consider any source for the ions’ motion other than their acceleration by that field. But remember that the charged particles were emitted from a heated filament. They will therefore have acquired kinetic energy by heat, wholly apart from experiencing any electrical attraction. So it is not at all far-fetched to suppose that some of them might have enough energy to strike the grid and adhere to it, making the grid slightly negative. Thus when we connect the ammeter, negative electricity will migrate from the grid to the rest of the circuit, constituting the current that seemed at first so surprising. 27 We can test this conjecture. If this “maverick” current to the negative end of the filament is indeed powered by thermal energy, it should diminish when I reduce the filament heating. Let us do that . . . Yes; you see I am lowering the filament voltage, and the current has already dropped by about half. With a further reduction it drops to 4/10ths of a 27 This well-known phenomenon is called “grid leak” in radio circles. �18 microampere. So this constitutes additional confirmation, if we still needed any, that the current within the vacuum tube is constituted by a stream of material particles— Richardson’s thermions. The thermionic current, then, makes itself evident at De Forest’s third electrode. But that could scarcely have been any part of De Forest’s thinking when he introduced the grid. For the chief function of the diode was to conduct or block current through the plate. Surely when De Forest added a third electrode, he would have been looking for some result bearing on the plate current. Let us try to envision what such an effect might be. Imagine, then, a current of negative ions departing from the heated filament and streaming towards the positive plate under the influence of the electric field between the filament and the plate. If the grid is electrically neutral, it should have no effect on the stream of ions. But what will happen if the grid is made negative? It will then repel negative ions; and some of the less energetic among them will be brought to rest, or even turned back toward the filament, with a consequent reduction in plate current. This resembles the operation of two devices that I mentioned in February’s talk: the relay and the coherer. In both of those contrivances a weak current controlled a second, much stronger current. That is exactly what happens in the triode: a negative current supplied to the grid controls a completely separate current to the plate, whose strength can be far greater, since it is powered by its own battery. No wonder De Forest was convinced that his alleged trespass against the integrity of the diode was in fact a spectacular advance! Let us see for ourselves this remarkable capability in action. I am applying 120 volts to this light bulb—not directly, but through the triode, so that the current has to pass through the tube in order to light the bulb. The grid supply is off, so the grid is neutral; and as you see, the tube is passing enough current to light the bulb fully. But watch the light bulb when I turn on the grid supply to place a negative voltage on the grid. The current is blocked, and the bulb is extinguished. When I remove that negative voltage, current flows again and the bulb lights once more. You see how a mere two or three volts applied to the grid is able to control a 120-volt lamp. We can carry out a more quantitative study of the triode’s action by measuring both the plate and the grid currents. �19 Here I am applying positive 120 volts to the triode’s plate, but zero volts to the grid. As you see, the plate is carrying a current of 1.658 milliamperes; and the grid current is (negative) 0.3 microamperes. I begin increasing the grid voltage (that is, making it more and more negative), and the plate current decreases, just as we expected. But so does the grid current. That might have seemed surprising earlier, but now we know what is going on: the grid power supply is sending negative current to the grid, thus opposing the thermionic current that was coming from the grid; that’s why the grid current decreases when the grid voltage rises. Finally, when the grid current has been brought to zero, the plate current has diminished to .362 milliamperes. When we graph these measurements, the results show plainly how the plate current depends on the grid current. The graph is a straight line; thus changes in the plate current are proportional to changes in the grid current. Do you see what this means? If I send to the grid of the triode a signal which causes the current to vary between 0 and .3 microamperes according to any pattern whatsoever, the plate current will reproduce that very same pattern—but at a larger scale: a .3 microampere variation of grid current will produce a plate current variation of 1.293 milliamperes, that is, a signal more than 4300 times greater! This has implications of great consequence for electromagnetic wave detection. It means that even if the received signal current is merely a microampere, it will be able to initiate a current greater than 4.3 milliamperes—enough to operate a telephone receiver, a meter, or some other indicator. De Forest boasted that this, in effect, amplifies the tiny incoming signal—since even though the triode does not actually make a small current larger, the result is the same as if it did. I will have much more to say about amplification in the next lecture. For one thing, we can combine the detection function of the diode and the amplification function of the triode to make a far more sensitive wireless telegraphy receiver. But the triode’s capability for �20 amplification will do much more; it will also become the basis for a new method of electromagnetic wave generation, far superior to the spark transmitter. This in turn opens up the first truly practical prospects for transmission and reception of sound—the heart of what we now call “radio.” That will be the topic of my next talk, the final lecture in this series. I hope many of you will be able to join me then. �21 “Rectification” Handout to accompany the lecture “The Vacuum Tube” As the distinctive capabilities of the diode became more perfectly understood, one technical term came into particularly prominent use. “Rectify,” to put right or correct, but literally to make straight, became the standard term for the conversion of alternating current to direct current. If the diode really had been capable of converting a current from the profile (A) to profile (B)—making “the crooked straight, the rough places plain,” 28 the term would indeed have been well-chosen; but as I noted in the lecture, the vacuum diode does not produce current having the rectilinear profile (B) but the pulsing profile (C). It is “direct current” only in the sense of having a single direction, not in the original sense of being constant or straight; it ought really to be called “pulsating” current. Nevertheless the term “rectify” quickly acquired the meaning “to make unidirectional”— with no regard whatsoever to straightness or constancy. Alternating current had been the standard power source in the United States since 1892; and alternating current was very well suited to applications such as incandescent electric lighting and many kinds of electric machinery. But tasks like arc lighting and electric welding still for the most part required direct current; and this either had to be generated by dynamos or obtained from the standard power system by rectification. Long after incandescent bulbs became the dominant light source, electric arcs remained standard in high-intensity instruments such as searchlights, theatrical follow-spots, and cinema film projectors. As late as 1959, my high school’s movie projectors employed 28 Handel’s adaptation, in The Messiah, of Isaiah 40:4. �22 electric arc units like the AmproArc model shown here. The arc housing is the large grey cylindrical structure. The rectifier is on the floor at the right. Storage batteries were another source of demand for direct current. Unlike the batteries of Faraday’s time, which were recharged literally 29 by discarding the used acid solution and replacing it with new, storage batteries are recharged by passing direct current through them in reverse. Such batteries became standard automotive equipment with the advent of electric starting; and rectifying battery chargers like the Tungar model pictured here were found in nearly every service garage and even some households. The large glass bulb on the right is the rectifying diode. (The bulb on the left is an ordinary light bulb, used to regulate the current.) A third area demanding direct current was that of vacuum-tube electronics, especially for radio receivers and transmitters. I will be discussing this technology further in the next lecture, and we shall see why these devices required direct-current power sources. The diode I experimented with for this lecture, the Eimac 250-R, was designed to supply direct current for high-powered radio transmitters. H. Fisher 5 December 2025 29 The root meaning of “charge” is to load or fill. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 22 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Vacuum Tube Description An account of the resource Transcript of a lecture given by tutor emeritus Howard Fisher on April 15, 2026 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "A puzzling phenomenon observed by Edison in one of his early light bulbs seemed to disclose a kind of electric current that did not fit the received ideas, for it was not confined to a conductor, and it would flow in one direction but not in the reverse direction. From this 'Edison effect' arose the modern vacuum tube, which enabled, first, the detection of electromagnetic waves by a process readily understood (rectification), and second, a means for effectively multiplying the strength of an electric current (amplification). Both of these capabilities brought dramatic improvements in wireless telegraphy, and they also introduced the prospect of radio transmission of voice and sound. This is the second of three talks on early radio technology." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2026-04-15 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Vacuum-tube circuits Relation A related resource <a href="https://digitalarchives.sjc.edu/items/show/8163" target="_blank" rel="noreferrer noopener">Telegraphy and Radiotelegraphy</a> Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_The_Vacuum_Tube_2026-04-15 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/da1f508026d92e2df9b586037d088a9f.pdf d5f566071fcda915ec4fa85c23ab5e68 PDF Text Text Telegraphy and Radiotelegraphy1 Howard J. Fisher Welcome, everyone, to this first of three talks on early radio technology. The talks are three in number because the history of radio communication may be said to fall into three periods. The first was the era of signaling, in which radio transmission was limited to the dots and dashes of Morse code; accordingly, this first talk will be on telegraphy and radiotelegraphy. The third period, which dates from about 1920, is that of transmission and reception of voice and sound; so that the final talk in this series will be on broadcast radio. Between these periods there appeared one magnificent invention, which alone made it possible to advance from the first era to the third. This was the vacuum tube, and that will be the topic of my next talk, to be given in April. Thus the three periods of radio development are more or less defined by three specific technologies. These bodies of knowledge and practice are highly interesting, both historically and in their own right; but I wish to pay special attention to what we might call their rhetoric—because each of them embodies, in its own way, an implicit image of electric current. Each of them implies a picture of electric current as some sort of motion; but they appear to paint with different palettes when they try to contend with what, exactly, is doing the moving. Let us begin by considering—or, for some of us, remembering 2—the telegraph. If you have ever made an electromagnet from a nail, a length of wire, and a battery, you have already constructed the heart of a potential telegraphy system. The telegraph “sounder,” on the left, is basically a pair of electromagnets; they pull down the pivoting bar when current flows through the coils, and the bar springs back when the current ceases; both the pull and the release make a distinctive click. The telegraph “key,” on the right, is 1 2 Lecture delivered at St. John’s College, Santa Fe, on 18 February 2026. Western Union sent its last telegram on January 27, 2006. �2 simply a switch that closes when the knob is pressed. (The shorting switch, marked with a red arrow, keeps the key closed when it is engaged; we will see in a moment why this is useful.) I have connected this battery and key to the sounder at the side of the room; you can hear it respond as I press the key: a short press for a dot, a long press for a dash. Does anyone here know Morse code? . . . I just sent “S J C”: dit dit dit for “S,” dit dah dah dah for “J,” dah dit dah dit for “C.” Here is a diagram of our wired connections. The battery and key are here at the table, the electromagnet is part of the sounder, and when the key is closed we have an unbroken pathway, which evidently transports electricity from the battery to the sounder, and back again. The “back again” element is crucial. If a break occurs anywhere in the path—whether in the red or the black wire in our drawing—the action will cease. We usually speak of this fact as showing the need for “a closed circuit.” The terminology we conventionally use to describe this system is revealing. We say that the wire is a “conductor,” and that a “current” of electricity “flows” through it. These are all, clearly, metaphorical terms, drawn from the behavior of water or other liquid. The metaphor portrays electricity as a mobile substance which, just like ordinary liquids, requires a source of motive power to maintain its flow. Can we be content with metaphorical imagery? Don’t we want a true image of electrical action? But I don’t see how we can escape metaphor. We cannot observe “electricity itself” (whatever that means); so it is impossible to speak “literally” about electricity, or to imagine it in any terms other than what we had previously gleaned from other realms of experience. But what we can do, and should, is to remember that our description is metaphorical—and that other metaphors may be possible, perhaps even preferable. Our telegraph system, in its present form, gives perfect expression to the fluid-flow metaphor by treating the wires as paths of circulation for electricity. But as familiar systems expand into new settings, new phenomena sometimes appear, which may not be easily accommodated by the received imagery. When that happens, alternative idioms of expression, images which might previously have seemed merely speculative or idiosyncratic, may perhaps take on new persuasive force. I think we will encounter an example of this, as our telegraphic system becomes more refined. �3 Now, with the apparatus I have assembled, I can send messages to you, but you cannot send back to me. We can fix that by adding a second key at your end and a second sounder at mine, like this: We might also want to add an extra battery, since we will be operating two sounders at once. Now I can send to you and you can send to me. As long as my key is closed, your key is what turns the current on and off; so I can receive the message you send; that was the purpose of the shorting switch I pointed out in our first slide. When it is time for you to receive, you keep your key closed; then my key controls the current, and you can receive the message I send. But a useful telegraph system must be capable of operating over greater distances than the length of a lecture hall! When the first railroad tracks were laid from Lamy Station to Santa Fe, in 1880, telegraph lines were strung along the right-of-way, a distance of some 17 miles. How would you adapt our telegraph system to that distance? Would you suspend the wires on poles, like this? That would work, of course; but it would involve twice as much wire, and nearly twice as much labor, as you really need. For the earth itself can serve as a conductor; in fact, over long distances it is often more effective than a wire would be. So instead of stringing both the red and blue wires on poles, as pictured here, connect the blue wires to the earth by driving a long copper rod into the ground at each station; like this: �4 Now our system seems complete. Is there any reason why it could not in principle be extended to greater and greater distances? True, we find that longer distances require additional batteries; but that presents no great challenge. We understand that the wire in some measure resists the flow of current even while conducting it. Longer distances therefore represent more resistance, and that is overcome by using more batteries—sometimes by simply adding to the ones already in use; but more often by another technique, which I want to take note of here because the same principle will come up later in another connection. To produce a clearly audible signal, a telegraph sounder requires a fairly strong current; mine needs a current of about 375 milliamperes for reliable operation. But the currents which could realistically be maintained in a telegraph wire hundreds of miles long were typically only about one-eighth of this. How then, with a such a small current, could the sounder be activated? That problem was solved by the relay: this is an electromagneti- cally operated switch that requires only a weak current to turn it on. As you see here, the incoming current energizes the relay’s electromagnet and closes the red contacts. Once these contacts are closed, a much stronger current from the batteries can pass and activate the sounder. It is as though the weak current were able to do the job of a much stronger current. Although telegraphers sometimes spoke of this arrangement as “magnifying” the incoming current, you can see that such a description is misleading. The relay cannot “magnify” anything; it is a control mechanism, not a transforming one. It does not alter the incoming current, but only enables it to trigger a more powerful one. With this latest development, it seems that the only barrier to unlimited expansion of our telegraphic system is the ocean: you can’t erect telegraph poles in the middle of the deep blue sea! But poles would be unnecessary if a telegraph wire could be laid directly on the seabed itself. In 1850 a telegraph cable was indeed laid along the floor of the English Channel. The �5 first successful transatlantic cable was achieved in 1865. 3 But not only were undersea cables enormously expensive, their transmission speeds proved to be extremely slow compared to overland telegraph lines; let’s try to see why. The 1865 transatlantic cable ran 2900 miles from the west coast of Ireland to the shores of Nova Scotia. As we just saw, distance in itself is not a novel problem. 2900 miles happens also to be the distance between New York and San Francisco; and that mileage had already been covered with the completion of the transcontinental telegraph line in 1861. But overland wires are surrounded by air. What happens to such a wire when it is surrounded by seawater? The 1865 cable is reported 4 to have contained a central conductor of about 64 thousandths of an inch in diameter, surrounded by gutta-percha insulation to a diameter of 3/8 of an inch. Then the wire proper will have had a circumference of .204 inches. The cable was 2900 miles long, or 183 million, 744 thousand inches; so the outer surface of the conductor would have had area of nearly 37 million, 484 thousand square inches. This means that more than 37 million square inches of copper was separated by a little over 1/8 inch from an equal surface of excellently-conducting saltwater. What do we have when these two conductive surfaces are separated by a thin layer of insulating material? We have the equivalent of a gigantic Leyden jar. 5 3 4 An earlier cable, the first, carried its inaugural message in 1858 but failed within three weeks. Bright, Charles, The Story of the Atlantic Cable (1903), pp. 46, 52. 5 Faraday observed behavior similar to that of the Leyden jar when very long wires were buried in the earth. See Faraday, Michael, “On Subterraneous Electro-Telegraph Wires,” Philosophical Magazine, June 1854. �6 Imagine what would happen if we added such an enormous Leyden jar to our telegraphic line between Lamy and Santa Fe: For simplicity, we’ll consider only a one-way system, with a single battery. When the key is closed, current flows to the Leyden jar as well as to the sounder; the jar becomes charged, and the sounder clicks as its electromagnet is energized. But when the key is released, the Leyden jar will still be charged. It will then discharge through the sounder, keeping the electromagnet energized until discharge is complete. This means that after the key is released, there will be a delay before the sounder is released. If the key operator sent a “dot,” the receiving operator might hear a “dash!” We can mimic this effect. Here I have connected the sounder normally; you can see it respond faithfully when I tap “dots” and “dashes” on the key. Next I will connect a capacitor—the modern equivalent of a Leyden jar— across the sounder’s terminals to represent the result of submerging the cable. The capacitor is only about an inch in diameter; nevertheless it is electrically equivalent to many millions of Leyden jars! Let’s see how the �7 sounder responds . . . Did you notice how sluggishly the sounder’s armature released? When I tried to send “S J C,” the sounder quickly became paralyzed, and the message was completely garbled. The only way to avoid confusion is for the key operator to allow time for the jar to discharge between each “key up” and the next “key down.” But this requires sending very slowly. The 1865 undersea cable could not carry more than about 8 words per minute. 6 By comparison, skilled landline telegraphers were expected to send and receive between 20 and 40 words per minute; while the lines themselves could support speeds three times as fast using automated sending and receiving apparatus. The problem, then, with an undersea cable is not the cable but the sea! But doesn’t that cast serious doubt on the imagery of electric current as a species of flow, with the telegraph wire as its channel or conduit? For the performance of a pipe or channel is ordinarily determined by the conditions inside it, not the conditions outside it. To take only the most naive example, water flowing through a pipe is majestically indifferent as to whether that pipe is suspended in air, lying on the ground, or buried in the earth—in stark contrast to the submarine cable’s susceptibility to its environment. Moreover, if a telegraph cable immersed in saltwater behaves conspicuously like a Leyden jar, then surely the same cable suspended in air must likewise act like a Leyden jar—but merely to a lesser extent. How, then, can any realistic image of electric current privilege a supposed process of “flow” within the wire, while excluding any representation of the conditions surrounding the wire? Yet that is just what the conventional “flow” image does. Faraday, in his Eleventh and Twelfth Series, especially, found that for purposes of understanding the Leyden jar, an image of electrification that emphasized tension in the surrounding dielectric was far more fruitful than was the image of electricity as residing in or on a conductor. A more Faradayan account of our telegraph system might proceed like this: 6 Even this was a vast improvement over the first, 1858, cable, which required 16 hours to transmit a 98-word message from Queen Victoria to President James Buchanan—a speed of 1/10 word per minute! �8 The telegraph key has just sent a single “dot”—a short pulse. While the key was closed, the battery began to impose tension between the left end of the wire and ground. That tension spread to the right as each region affected the adjacent region. Similarly, when the key was opened, tension at the wire’s end returned to zero; and this condition too propagated to the right from region to region. The result is a zone of tension that has moved from the telegraph key about halfway to the first telegraph pole. A moment later, that zone of tension will have moved just beyond the first telegraph pole... And from that point nearly to the second pole... Finally it will approach the telegraph sounder; and when it reaches that device, the tension will discharge in the sounder’s coils, developing the momentary magnetic field which attracts the armature, thus signaling that a “dot” has been received. Let us view these images as a sequence, to give a better sense of how this wave of tension—for it is a wave—travels. Notice that while the wave may be said to be guided by the telegraph wire and the �9 earth; it is not travelling through either of them. It is traveling through space. You see how the behavior of the transatlantic cable is far more concordant with Faraday’s “tension” image than with the image of flow through a wire. Moreover, by acknowledging the relevance of action outside the wire Faraday’s image provides a more natural rhetorical context for the phenomena encountered in submarine cables. But I would not wish to argue that either image is the “true” one. Both of them bear a metaphorical stamp, in that both work to render unfamiliar phenomena in familiar terms. Returning, then, to the telegraphic predicament, I had suggested earlier that the problem was not the cable but its submersion in the sea. On the other hand, inasmuch as there is no way to suspend a cable above the sea, from a practical point of view the problem was the cable; and the solution would be to eliminate the cable altogether. 7 But how is that possible? Indeed, it is not possible if electric current is essentially flow through a wire, as the conventional image represents it; for then a wire will be indispensable. But if current is essentially a wave of tension, new possibilities for transmission would seem to open up, in imagination if nothing else. I am not sure to what extent Faraday actually contemplated the idea of “separating the current from the wire,” so to speak, though some of his later writings are highly suggestive. 8 But Maxwell certainly did—and largely on the basis of Faraday’s views on the nature of conduction. As juniors will see for themselves later in the semester, Maxwell’s theory implies that every electric current necessarily generates an electromagnetic wave—and although currents cannot travel without wires, electromagnetic waves travel in space. Maxwell’s calculations provided compelling reasons for him to believe that light was an electromagnetic wave, but he did not conjecture whether there might be other kinds of electromagnetic waves, nor whether such waves could be produced in practice by electrical means. That achievement—the generation of electromagnetic waves—was accomplished by Heinrich Hertz in 1888. By no means, of course, have submarine communication cables been abandoned! Dramatic improvements have resulted from the use of better insulation, inline repeaters, and, since about the 1980s, transition from electrical to optical cables. 7 8 See, for example, “Thoughts on Ray-Vibrations,” Experimental Researches in Electricity, vol. III, p. 447; “On Subterraneous Electro-Telegraph Wires,” ibid., p. 521; “On Electric Induction— Associated cases of current and Static Effects,” Phil. Mag, Series 4, vol. 7, no. 44 (March 1854), p. 197. �10 Hertz’s apparatus is shown in this haunting photograph—taken, I believe, by Hertz himself. The pair of long rods with large globes at their ends form the radiating structure—what we now call the “antenna.” 9 The large coil at the far left end of the bench produced pulses of very high voltage which discharged through the spark gap at the center of the rod pair, and so initiated in them a rapidly oscillating current. Although a spark discharge inherently produces waves exhibiting a huge range of frequencies, Hertz’s rods and spheres were sized to favor waves having a length of about 6 meters and a frequency of about 50 million cycles per second. Hertz demonstrated the existence of these waves by showing their effect on a separate wire loop, interrupted by a spark gap of its own; here is Hertz’s diagram. 10 Whenever the main apparatus sparked, a discernable spark developed across the gap at M, even when the loop was held twenty or thirty feet away, thus demonstrating that the waves were indeed traveling through the air. Antenna—Latin term for sailyard, yardarm; later (15th cent.) the long protruding “feeler” of an insect—became the accepted name for such a radiator by 1902. 9 10 Redrawn from Hertz, Electric Waves (1893). �11 Hertz’s apparatus could not transmit waves over distances useful for practical communication. Nor had Hertz harbored any such intention; his interest was in confirming Maxwell’s electromagnetic theory. But a young Guglielmo Marconi saw these “Hertzian waves,” as they were then called, as a potential means of wireless communication. Instead of turning an electric current on and off in short and long pulses, it was only necessary to turn the spark generator on and off. Pulses of electromagnetic waves would then carry messages in the same form as landline telegraphy did; but it would be wireless telegraphy. Marconi began experimenting with his own wave apparatus in 1894; he succeeded in transmitting over distances that gradually increased from one-half mile, to 10 miles in 1897, across the English Channel in 1898, and in 1899 to a vessel at sea 66 miles away. 11 A few years ago I made a spark transmitter from old automotive ignition parts; I even tried to imitate Hertz’s antenna: The two copper sheets correspond to Hertz’s globes, while the brass spheres at the center form the spark gap. The two cylindrical objects are a pair of automotive ignition coils, wired back-to-back. They take the place of Hertz’s enormous induction coil. Here is the transmitter in operation . . . The buzzing sound you heard is a set of contacts in the battery circuit; each time they open, a spark is generated. With the aid of this Hertzian memorial, Mr. Franks and I were able to send a signal from the portico of Weigle Hall to his office in ESL—a magnificent range of one hundred twenty feet! The wave produced by this series of sparks is, accordingly, a series of wave pulses. A new pulse is produced each time the buzzer contacts open— in my transmitter that is about 500 times per second. But each pulse itself oscillates at a frequency of roughly 50 million times per second, and it also decays slightly in amplitude. So if I hold down the transmitter key for the 11 Largely because of Marconi’s single-minded pursuit of wireless telegraphy, it is his name that is most prominently associated with its development; but Marconi was by no means the sole originator. In the year when Marconi began his experiments Oliver Lodge, already a scientific luminary, would introduce the “syntonic” (tuned) transmitter in a memorial lecture on Hertz (Royal Institution, June 1894) and transmit wireless Morse messages between two Oxford University buildings (British Association meeting, August 1894). �12 length of one “dash” (about a third of a second) it produces a wave shape like the one drawn at the top of this slide. A Morse “dot,” which should be about a third of the length of a dash, is drawn at the bottom. With the steady growth of wireless range it seemed inevitable that wireless transmission would eventually extend from continent to continent; it was only a matter of commandeering more electric power for the spark and building more efficient radiating and receiving structures. But the intermittent character of the spark wave was a serious impediment to transmission power. As the slide shows, only about 50% of the transmission time was actually occupied by a wave! This is deplorably inefficient, and various methods were developed to make the radiated wave more nearly continuous. One of them utilized a Leyden jar and a coil, like the ones highlighted on the right, to prolong oscillations in the antenna. Instead of the intermittent pulses generated by the earliest transmitters, this and other improvements resulted in a wave that was nearly continuous, so that Morse “dashes” and “dots” had waveshapes like these: But even with the greater efficiency obtained with improved wave continuity, the power needed for long-distance wireless transmission was �13 far greater than any battery could supply. Marconi’s most powerful transmitters employed steam-powered electric generators like these: 12 As you might imagine, the sparks produced by such enormous machines were distressingly noisy. The residents near one installation 13 complained that the sound of the spark could be heard four miles downwind of the station! The radiating and receiving antenna structures likewise grew in scale and complexity, as more and more came to be known about the behavior of electromagnetic waves. Here are two of Marconi’s transatlantic antenna arrays, the one on the left was located in Nova Scotia, Canada; the one on the right was constructed in Cornwall, England around 1902. But successful reception of the transmitted electromagnetic wave was not, in itself, sufficient to establish communication; the recipient had to know that it was received! The wave had to be detected. How was this to be achieved? When an electromagnetic wave is intercepted by a conductor, an electric current tends to develop in it. Roughly speaking, a conductor will develop a current insofar as it is parallel to the incoming electric field, or perpendicular to the incoming magnetic field. (Recall that Faraday had shown, long before, that a conductor tends to develop a current when it 12 13 Bucher, Elmer. Practical Wireless Telegraphy (1917), p. 295. South Wellfleet, Mass. See https://www.nps.gov/caco/learn/historyculture/marconi.htm �14 “cuts” lines of magnetic force.) Any phenomenon associated with this current will, therefore, signal the reception of an electromagnetic wave. So long as Hertz’s wire loop was within a few dozen feet of the transmitter, the currents that developed in it were strong enough to produce a spark, though a very feeble one (Hertz had to darken the room and use a magnifying lens to see it). But the currents that would develop in a far distant conductor—even in a sophisticated and elaborate array like the ones shown here—would be incomparably weaker. A far more sensitive indicator would be required to discern them. Numerous such indicators were contrived, several of them truly ingenious; but most were either too delicate or too slow for practical use. The first device to gain widespread adoption was one whose operation no one really understood. This was the “coherer,” widely credited in principle to the French physicist Édouard Branly, although many other investigators, including Marconi, contributed to its practical utility. On the left is one of Marconi’s coherers; and on the right you see how it would have been incorporated in an actual wireless receiver, mounted in a mechanism I will explain in a moment. The coherer itself is a length of glass tubing fitted with two electrodes, the space between them filled loosely with metal filings. Although metals are generally good conductors, tiny shards of metal do not contact one another very well, so the coherer does not readily conduct an electric current. But in the presence of a high-frequency electromagnetic wave the metal filings mysteriously “cohere”—hence the name—and the device becomes suddenly conductive. We can see the “cohering” behavior using little more than a heap of metallic spheres. 14 I have squeezed about a dozen four-inch squares of This was one of Branly’s early versions; see Fleming, J. A., Principles of Electric Wave Telegraphy and Telephony, 3rd ed. (1916), p. 478. My version is adapted from a YouTube video whose presenter identifies himself only as “Mr. Whibley”: look for https://youtu.be/ESgoNEH__E8?si=Xw5a7Zzmhb8Q5G6Y—truly a little gem. 14 �15 aluminum foil into spheres about one-half inch in diameter and placed them loosely in the glass beaker. Strips of foil taped to opposite sides of the beaker connect the pile of spheres in the circuit of a battery and flashlight bulb. As you see, the bulb is not lit, even though aluminum is a good conductor. Evidently the loose contact between the spheres cannot pass enough current to light the bulb. But we know from Hertz that an electric spark generates electromagnetic waves. This battery-operated stove lighter generates a series of electric sparks when I press the trigger; it’s what I use to light my propane stove at home. In a wireless telegraphy receiver, the incoming electromagnetic wave would have been routed to the coherer by means of a wire from the antenna; but since my sparker is only a few feet away, we need no special conductor. See? When the discharge occurs, the pile of spheres suddenly becomes conductive, and the lamp lights. But you can also see the great deficiency of this device: it remains conductive, and the lamp stays lit, even after the spark �16 and its radiated wave have ceased. It cannot distinguish a dot from a dash; indeed, so long as it remains conductive it will not be able to detect the arrival of a subsequent pulse, whether dot or dash. What seems to be happening is that the sudden spike of electromagnetic disturbance generates tiny sparks between the foil surfaces, which momentarily raise the temperature at the discharge points sufficiently high to weld the spheres together. But notice that when I give the cup a sharp tap, the current ceases. Evidently a slight disturbance is enough to break the tiny bonds—today they are called “microwelds”—between the foil spheres. Such was the purpose of the elaborate mechanism in which the coherer was typically mounted. The moment Marconi’s coherer became conductive, it passed a current through an electromagnet; this in turn caused a steel tapper to strike the glass tube and “de-cohere” the particles, opening the circuit. The tapper would immediately spring back; and if the received pulse had been a short “dot,” the coherer would now be ready to receive the next pulse. If, however, the pulse had been a long “dash,” the wave would still be present; the particles would cohere again, the tapper would strike the glass once more; and this process would repeat, and the tapping would continue, until the cessation of the “dash.” This principle—the “switch that turns itself off”—is the principle of the electric buzzer, and when the decoherer was in operation it created a similar buzzing sound. 15 The coherer was, however, far from satisfactory. One engineer complained, “It was publicized as wonderful, and it was—wonderfully erratic and bad. It would not work when it should, and it worked overtime when it should not have.” 16 Another described it as “the bête noir of radiotelegraphy,” since it could not reliably distinguish between “legitimate signals, static disturbances, a slipping trolley several blocks away, and even This “buzzer” style of decoherer was only one of many methods that were developed to enable the coherer to control an indicator, or even a recorder. 15 16 Robert Marriott, quoted in Lewis Coe, Wireless Radio, A History, 2nd ed. (2006), p. 6. �17 the turning on and off of electric lights in the building.” 17 But perhaps the most serious deficiency was its sluggishness; the coherer’s cumbersome tapping mechanism limited reception to about 12 or 15 words per minute— barely a third of the standard for landline telegraphers. Still, even with these limitations, the coherer was a tremendous advance over Hertz’s spark detector, because it acted indirectly. Let me explain what I mean by this. The spark Hertz observed in his wire loop detector was the direct manifestation of the current induced in it by the intercepted electromagnetic wave. Nothing intervened between the current and the spark; indeed, from Faraday’s point of view, the current was the spark. Necessarily, then, the perceptible indication of the electromagnetic wave could be no more energetic than was the received wave itself. The coherer, by contrast, allows a weak incoming current to trigger a separate and much stronger current— the same principle we saw earlier in the telegraphic relay, only effected by different means. The relay used electromagnetic attraction to close a switch mechanically, thus completing a battery circuit. In the coherer, heat from tiny sparks creates microscopic welds between the metal shards; it completes the battery circuit by fusing conductive fragments together. The coherer, like the relay, makes possible a much more pronounced effect than the wave itself would have been able to produce; and since the wave-current only triggers the current which produces the sensible outcome, I call the coherer an indirect detector. Notice that the crucial element in the operation of the coherer is production of heat; everything else depends on that. But until about twenty years ago, when research finally confirmed it, 18 the role played by heat in the operation of the coherer could at best be only surmised. Nevertheless, to G. W. Pickard, “How I Invented the Crystal Detector,” Electrical Experimenter (August 1919), p. 325. 17 18 Falcon, Eric; Castaing, Bernard. "Electrical conductivity in granular media and Branly's coherer: A simple experiment," American Journal of Physics 73 (4): 302–307 (2005). �18 early twentieth-century investigators, explanations based on heat would have seemed the most reasonable and natural among the available candidates. We find a striking example of this in the work of G. W. Pickard, an American engineer. In 1902 Pickard was experimenting with the detector shown here on the left, a sewing needle laid across two carbon supports; and perhaps because it involved loose contact between conducting bodies, Pickard thought of his detector as a type of coherer. In contrast to Marconi’s coherer, though, this one was self-restoring; it did not need to be tapped after each received pulse. Nevertheless, Pickard was using it just as he would have employed an ordinary Marconi coherer: the detector was connected to a set of batteries, so that when it became conductive in the presence of an electromagnetic wave, the resulting rush of current from the batteries would produce a distinctive clicking sound in a Bell-style telephone earpiece, pictured on the right. Pickard’s great discovery occurred when he accidentally disconnected the batteries, yet continued to hear clicks in the earpiece whenever a wave pulse was received at the antenna. But with the batteries disconnected, the only source of electrical energy was the current from the antenna. Pickard was astonished to realize that the telephone diaphragm was being operated solely by the energy of the receive[d] signals. 19 What was astonishing about this? I’m guessing here, because Pickard did not say very much more about the course of his thinking. But everyone knew, and he would have known, that the current coming directly from the antenna should have been incapable of activating the telephone. G. W. Pickard, “How I Invented the Crystal Detector,” Electrical Experimenter (August 1919). Emphasis in the original. 19 �19 Let us understand why this is so. Remember that every wave has an oscillatory or, we might say, alternating character. For example, when we send a wave along one of our Bell wave machines, the wave itself travels horizontally; but every point on it moves vertically—alternating between motion up and motion down. Here is a clip from a 1959 instructional film; the presenter is John Shive, who invented the Bell wave device. Keep your eye on any of the crossbars, and notice their alternating up-and-down displacement. A vertical red line will assist you in this. Analogously, each of the components of an electromagnetic wave—the electric field and the magnetic field—undergoes a similar alternation between positive and negative. Maxwell depicted that alternation in this drawing; it is Figure 66 from his Treatise on Electricity and Magnetism. When the incoming electromagnetic wave is intercepted by the receiving antenna of a radiotelegraphy station, the alternating fields induce an alternating current. What happens, then, if this current is sent directly to a telephone receiver? The Bell telephone earpiece of 1900 (and they really haven’t changed all that much, even today) consists of a permanent magnet, shown here in blue, and a coil and a flexible steel diaphragm, shown in red. The diaphragm, �20 being attracted to the magnet, is permanently deflected in the magnet’s direction. But when an alternating current is sent through the coil, the coil’s magnetic field will alternately reinforce and oppose the magnet’s field. As a result, the diaphragm will be deflected now more, and now less; it will attempt to vibrate with the frequency at which the current alternates. But this frequency would have been in excess of hundreds of thousands of vibrations per second, for that was typical of the electromagnetic waves used in radiotelegraphy. No telephone diaphragm could sustain such rapid vibrations; and even had it done so, Pickard could never have heard them! For the limit of human hearing is at best about twenty thousand vibrations per second. What Pickard did hear in the earpiece was a sequence of clicks. What kind of current would cause that? The same kind of current that produced the clicks in our telegraph sounder earlier: a steady current—what we now call a direct current—turning on and off. Just as with the sounder’s armature, the telephone’s diaphragm is pulled towards the magnet when the current is on, then released when the current is off. Here I have an actual Bell receiver mechanism from about 1915; listen in the following video as I repeatedly connect, then disconnect, a battery. In the video I have amplified the sound considerably, and you can also see the diaphragm moving in and out. Here, then, is the problem. Pickard’s antenna was sending an alternating current to the detector—but the telephone was receiving a direct current from the detector! The detector, then, must either be transforming an alternating current into a direct current, or it must be replacing one current with the other. Pickard assumed the second option: replacement rather than transformation. We know this because in his patent certificate of 1906 20 he states: The energy of the received oscillations is converted into heat at the .. . junction.. .. This heat energy is then, according to this invention, regenerated or converted into a direct electric current. 20 U.S. Patent no. 836,531 issued November 20, 1906. �21 Pickard is asserting that the process has two stages: first, the alternating current produces heat; next, the heated junction between two dissimilar materials produces a direct current. Both of these phenomena were well known. Electrical production of heat had been described by Joule in 1840, and the thermal generation of electricity—Faraday called it “thermoelectricity” in his Third Series—was discovered by Seebeck in 1821. On what basis did Pickard offer this interpretation?—or, rather, this conjecture, since as far as I know, there existed at that time no research confirming the idea that heat was really involved. I think his word “regenerated” is telling. In the absence of a clear picture of electric current, he cannot think of “current” as a thing capable of being wrought from a serpentine profile, like this: into a rectilinear profile, like this: Current, for him, does not seem to be the sort of thing that can be fashioned “as clay in the hands of the potter.” 21 Without a clear picture of its nature, electric current can only be conceived as a manifestation of the power that generates it; the only way it can change its character, therefore, is to perish in one form, then be re-generated in another. Pickard’s attempt to avoid the idea of “transformation” would prove to be mistaken. His device actually does transform the current as such—not by converting it to heat and back again, but by a different process altogether. 21 Jeremiah 18:6 �22 That process would finally reveal itself with the advent of the vacuum tube. That invention will be the topic of my talk in April, and I hope many of you will be able to come. The vacuum tube would not only address the problem of wave detection in a much improved way, it would make possible sweeping advances in the production and transmission of electromagnetic waves. It would, above all, enable electromagnetic waves to convey sound—giving rise to the fascinating technology we now call “radio.” Additionally, the vacuum tube will raise anew, and perhaps in a more visualizable form, our former question, that of the nature of electric current. I hope you will join me for that discussion. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 22 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Telegraphy and Radiotelegraphy Description An account of the resource Transcript of a lecture given by tutor emeritus Howard Fisher on February 18, 2026 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "Overland telegraph technology implicitly embodied a familiar image of electricity as being a mobile substance confined to conductors. But phenomena encountered in undersea telegraph cables presented a competing image, one far more suggestive of Faraday’s image of tension in a dielectric. Subsequent production of electromagnetic waves could be seen as the ultimate practical expression of Faraday’s 'tension' image. But if they were to serve as a communications medium, electromagnetic waves had not only to be produced but be detected; and although various detecting mechanisms were developed, their operation was mysterious: the received images of electric current appeared to provide little clarity." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2026-02-18 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Telegraphy_and_Radiotelegraphy_2026-02-18 Subject The topic of the resource Telegraph, Wireless https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/2689666d1da8a00f354c975ab498fc44.pdf acfa92beb17bbdb4ca373b8160a5b135 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Speedy Achilles Description An account of the resource Transcript of a lecture given by tutor Howard Fisher on April 9, 2025 as part of the Dean's Lecture &amp; Concert Series. The Dean's Office has provided this description of the event: "Galileo’s mathematical treatment of 'speed' perennially draws questions about both the validity and the presuppositions of his argument. But perhaps we can make more sense of Galilean speed when we find in it traces of Homeric speed: the speed of 'speedy Achilles.' We may also find some Homeric grounding for an algebraic form of discourse exemplified by the Taylor Series—a form that moves uneasily between favoring <em>being</em> and favoring <em>fact</em>." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2025-04-09 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Galilei, Galileo, 1564-1642. Discorsi e dimostrazioni matematiche. English Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Speedy_Achilles_2025-04-09 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Page numeration Number of pages in the original item. 16 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Speedy Achilles Description An account of the resource Transcript of a lecture given by tutor emeritus Howard Fisher on April 9, 2025 as part of the Dean's Lecture &amp; Concert Series. The Dean's Office has provided this description of the event: "Galileo’s mathematical treatment of 'speed' perennially draws questions about both the validity and the presuppositions of his argument. But perhaps we can make more sense of Galilean speed when we find in it traces of Homeric speed: the speed of 'speedy Achilles.' We may also find some Homeric grounding for an algebraic form of discourse exemplified by the Taylor Series—a form that moves uneasily between favoring <em>being</em> and favoring <em>fact</em>." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2025-04-09 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Galilei, Galileo, 1564-1642. Discorsi e dimostrazioni matematiche. English Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Speedy_Achilles_2025-04-09 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a542b12f4a78f7a46d42b828ecbcaa34.pdf 2bc65410cf3101c56eb6f35f26376361 PDF Text Text ��������� Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Page numeration Number of pages in the original item. 9 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Concept of Truth in the Book of Genesis Description An account of the resource Transcript of a lecture given by tutor Phil LeCuyer on April 14, 2023 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "Join tutor Phil LeCuyer for a lecture on the Book of Genesis with a focus on Chapter 24." Creator An entity primarily responsible for making the resource LeCuyer, Phillip Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2023-04-14 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Bible. Genesis Truth Relation A related resource <a href="https://digitalarchives.sjc.edu/items/show/7745">https://digitalarchives.sjc.edu/items/show/7745</a> Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_LeCuyerP_The_Concept_of_Truth_in_the_Book_of_Genesis_2023-04-14_transcript Friday night lecture Worrell lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/d400f01b6f70d2cf52fe3aaa349b4938.pdf b30c4673b1f9be95f122bb769e87dc7e PDF Text Text Clima(c)tic Change: Twenty Million Hands and the Living, Wavering Globe [Mia slides] Last August, I heard an NPR discussion1 of climate change in relation to the Inflation Reduction Act, a bill designating almost $800 billion to energy security and climate change. The show’s host emphasized that, per the UN’s Intergovernmental Panel on Climate Change (the IPCC), fossil fuel usage must fall dramatically within ten years to keep Earth’s warming by 2100 below 1.5°C.2 She asked the Director of the Bureau of Land Management, Tracy Stone-Manning, “Is this process that we’re currently following going to get us there?” To my astonishment, Ms. Stone-Manning replied, “Yes. Yes is the answer.” The unequivocal answer was surprising given the BLM’s equivocal charge: as manager of federal lands, one of its responsibilities is to award oil and gas leases.3 So, somehow, partly as a result of the Inflation Reduction Act, we will as a nation, albeit a politically divided one, meet requirements to limit warming to 1.5°C while still doling out oil and gas leases and without asking our citizenry to make substantative behavioral changes. Ms. Stone-Manning’s response reminded me of the 2015 Paris Climate Accords, which the United States initially joined, before withdrawing under President Trump and re-joining under President Biden. These Accords set as their upper target 2°C of warming by 2100 and their ideal target as 1.5°C. The latter requires a 50% cut in global greenhouse gas emissions by 2030. What distressed me about the 1 The show was “1A,” hosted by Jenn White. It aired on August 12, 2023. What Ms. White actually said is this: “The United Nations Intergovernmental Panel on Climate Change has repeatedly said that the world has to phase out fossil fuels quickly to keep earth’s warming below 1.5 °C increase, which we are currently not on track to do… Time is of the essence. We have a limited amount of time – again by 2035, the UN Intergovernmental Panel says we have to phase out fossil fuels to keep the earth’s warming below that 1.5 °C increase.” Per the IPCC 6th assessment report, fossil fuel emissions need to be reduced 50% by 2030 and 100% by 2050 to limit warming to 1.5 °C by 2100. Ms. White’s imprecision is symptomatic of a broader problem: the implication that fossil fuels have to be entirely phased out by 2035 can only signify how little she has imagined the difficulty of merely reducing them by half in that time period. And, by the way, what happens after 2100? 3 An even more egregious example of such ambivalence (the word is mild) is how the United Arab Emirates, host of the 2023 United Nations Climate Change Conference (COP28), planned, per a BBC review of leaked documents, to use that conference to pursue oil and gas deals with foreign governments. But is this really surprising when the President of the COP28 happens also to be the CEO of the UAE’s giant state oil company? 2 1 �Accords, however, was that the agreed-upon means in no conceivable way could meet that goal. The modeling studies grounding the Accords clearly showed that. In fact, annual global greenhouse gas emissions are higher now than they were in 2015.4 The Accords set 2030 as their deadline for a 50% cut; over half of that time has elapsed; and we are further from that goal now than when we started (slide).5 Fundamental reasons explain why we’ve made so little progress cutting greenhouse gases, reasons which we haven’t seriously confronted politically or socially. We may think, for example, that electric cars will save us; we even call them, falsely, “emissions-free vehicles.” This falsehood epitomizes the problem. We are too comfortable imagining vehicles have no emissions if we see no tail pipes. We forget that the missing tail pipe is actually a power plant’s smokestack. Such forgetfulness is encouraged by those who understand that part of what they’re selling is a clean conscience, moreover one which allows us to get on with our lives without substantively changing them. Even if we correct this forgetfulness, we too comfortably believe the gases causing climate change are mostly the consequence of vehicles and residences; hence the solution is electric cars and solar panels. Globally, however, ground transportation and residential sources account for only a third of annual anthropogenic CO2 emissions. Our fossil-fuel dependence goes well beyond vehicles and homes.6 4 There’s discrepancy among reported global greenhouse gas emissions, in part depending on whether only CO2 is considered or CO2 plus other gasses and actions (which collectively are called “CO2 equivalents [CO2e]”). In terms of CO2, 2022 estimates range from 36.1 (Liu, Deng, Davis and Ciais [2023] Nature Reviews Earth and Environment 4: 205 – 6) to 36.8 Gt CO2 (the International Energy Agency’s 2022 report, “CO2 emissions in 2022”). Preliminary 2023 data suggest CO2 increased another 1.1% from 2022 (https://globalcarbonbudget.org/fossil-co2-emissions-atrecord-high-in-2023/). As for CO2e, the UN’s emissions gap report (https://doi.org/10.59117/20.500.11822/43922) gives a 2022 value of 57.4 Gt. Whether measuring CO2 or CO2e, these reports agree that the corresponding 2015 value was lower (e.g., 35.6 Gt CO2 or 47 Gt CO2e). Differences in estimates may also reflect treatment of natural but extraordinary emissions, as from forest fires. For example, the 2023 Canadian forest fires collectively burned an area the size of Washington state and produced ~3 times the CO2 emitted annually by all of Canada, but some approaches would omit these sources as extraordinary and natural. Yet it is foreseeable that such sources will soon become ordinary, and whether a source is “natural” or not will have no effect on how much the planet warms. 5 Put otherwise, it’s estimated that we can afford to add a cumulative total of about 250 Gt of carbon dioxide to the atmosphere to give ourselves a 50% chance of keeping warming below 1.5 °C; we’re currently on pace to emit that much carbon dioxide within ~ 7 years. Lamboll et al. (2023) Assessing the size and uncertainty of remaining carbon budgets. Nature Climate Change. https://doi.org/10.1038/s41558-023-01848-5. 6 The children’s book Petro Pete’s Big Bad Dream describes some of the dramatic changes in our everyday lives should fossil fuels abruptly disappear or not be used. It has been widely criticized as Big Oil’s attempt to brainwash 2 �In particular let me call attention to four material mainstays of modern civilization which are fossil-fuel intensive: plastic, steel, concrete and ammonia fertilizers. Collectively they require ~20% of the world’s primary energy supply and generate at least 25% of anthropogenic CO2 emissions.7 Consider all the plastic you handle every day. Consider the steel and plastic required for even an electric car. Consider the steel, plastic and concrete needed for solar-panel installation and building construction, including LEEDcertified green buildings. Consider projections that plastic production will double by 2045 (slide) and the world’s building stock by 2060. That latter is equivalent to building a new New York City every month of every year between now and then.8 Consider the immense quantities of fertilizer needed to sustain 8 billion people and their animals.9 Consider how energy-intensive food production in the western world is, where every calorie of food requires about a dozen calories of fossil fuels, and where in the United States, 30-40% of that food is thrown away.10 Consider, in short, how fossil-fuel intensive a transition to a green economy will necessarily be. Now ask yourself: what dream are we dreaming, which says we can go green without really changing? That message is loudly proclaimed by many on the left: “Let’s have a green revolution,” they say, “so we don’t have to change!” Can there be revolution without profound change? Many on the left say, make fairly minor changes so you don’t really have to change. Many on the right say, it’s all an exaggeration or a hoax, change isn’t needed; if anything, we should change back. Instead of focusing on the disagreement between right and left, notice their substantive agreement. The difference is between not changing and changing very little.11 kids. This criticism is undoubtedly true – but that doesn’t make the cascading effects of our fossil-fuel dependence, to which the story calls attention, untrue. To conclude the latter from the former is its own kind of brainwashing. 7 Smil, How the World Really Works, p. 78. 8 The NYC analogue is from Reilly O’Hara around the 10-minute mark in Peter O’Dowd’s “A ‘concrete’ solution to climate change”: https://www.wbur.org/hereandnow/2023/12/07/concrete-climate-change-solution. 9 ~15% of all human-caused greenhouse gas emissions are due to livestock farming, to satisfy our desire for meat. 10 For the dozen calories claim, see Ghosh (2016), The Great Derangement, p. 147. 30-40% is the USDA’s estimate. 11 Right and left often arrive at opposite conclusions from the same premise, or, as in this case, (basically) the same conclusion from opposite premises. Such phenomena are marks that something other than thinking is happening. 3 �Should we take seriously that climate change is an exaggeration, a hoax, mere, self-interested alarmism? On the one hand, of course we should. The computer models which divide the earth into enormous numbers of grid squares and try to calculate the consequences of every significant biological, chemical and physical process in each square are beyond the critical ken of a human mind. A projection 100 years into the future requires a million billion calculations (slide). Many are simply parameterizations of phenomena we don’t understand. The interdisciplinarity upon which climate research rests is also often authority-driven – collaborative in the mode of deference more than of criticism. Major incentives to jump on the climate-change bandwagon exist, including career and funding incentives. We’d be naïve to imagine all that away, as, even more importantly, we’d be naïve not to wonder about the competence which synthesizes so much specialized knowledge, assuming we can tell what is and isn’t knowledge. On the other hand, warranted skepticism does not deliver us to the knowledge that it’s all a hoax. Far from it. We’ve had warnings, growing in clarity and urgency, since a pioneering paper by Arrhenius in 1896.12 In 1938, Callendar calculated that the planet had warmed ~0.25°C over the previous half-century due to CO2 production by fuel combustion. He then developed a basic model which predicted that around 420 ppm atmospheric CO2 we should expect ~1°C of warming relative to 1938 (slide).13 We are now around 420 ppm CO2 and we’ve had ~1°C of warming relative to 1938. Since then models have become more complicated and refined, offering high-resolution predictions of what global temperature increases mean at regional scales; they’ve added ocean and atmospheric circulation and positive and negative feedbacks; they’ve incorporated responses of the living world; they’ve even charted possible social and political responses. But the problem’s basic contours, the relation of greenhouse gases from fossil fuel combustion to planetary warming and the consequent dangers, have 12 Arrhenius’ Swedish colleague, the mostly forgotten Arvid Högbom, actually called attention to the problem in 1894, which in part stimulated Arrhenius to examine it more carefully in his theoretical 1896 paper. By 1904, Arrhenius concluded that the planet would warm due to human activities, though probably in a beneficial way. 13 See Callendar (1938) Quarterly Journal of the Royal Meteorological Society 64: 223 – 40, particularly Fig. 2. 4 �been reasonably well understood for decades. To some degree we’ve allowed ourselves to be distracted by seeking ever-greater refinements of the simulacra by which we make this problem visible, as if simulacra could stand in for the world where that problem must be addressed and as if the indefinite pursuit of knowledge weren’t itself a deferral of behavioral change. Accepting then that climate change is real, how alarmed should we be? After all, the world has had higher CO2 concentrations than now. The last time the planet had comparable CO2 concentrations was 4.3 million years ago in the mid-Pliocene; the planet was about 3°C warmer than now. That may not sound like a lot, given diurnal temperature changes in New Mexico of ~15°C and seasonal ones of ~20. But those 3°C corresponded to a planet many would find unfamiliar, with sea levels 75 feet higher than today, Florida a mere nub and southeastern Mexico under water (slide). Our nearest relative 4 million years ago was Australopithecus; it would be another million years before the genus Homo appeared. Obviously, there weren’t 8 billion people on earth. Human civilization 4.3 million years ago was still roughly 4.3 million years in the future – a reminder that civilization is young. Besides different atmospheric CO2 concentrations, earth has also had dramatically different past temperature regimes. It has been much colder – I’m not talking Pleistocene ice ages, but something more extreme – Snowball Earth episodes 600 to 700 million years ago when the planet was close to globally glaciated (slide). The planet has also been much warmer, including during the Cretaceous hothouse 110 million years ago and, 56 million years ago, the Paleocene-Eocene Thermal Maximum, when the poles were ice-free and alligators swam in an Arctic lined by palm trees. If it comforts you to know the planet can handle much colder and warmer temperatures, if you think that means there’s nothing to worry about, consider rates of change.14 During the Cretaceous hothouse, the planet warmed about 0.000025°C per century, for a total warming of 5°C over millions of years. During the Paleocene-Eocene Thermal Maximum, the rate of warming was a thousand times 14 The rates which follow are from Kump (2011) Scientific American July: 56 – 61. 5 �faster: 0.025°C per hundred years, again for a total warming of 5°C over tens of thousands of years – a warming coincident with an extinction event.15 In contrast, we’ve already warmed about 1°C in the last century, with future warming projected at as much as 5.7°C by 2100, depending on what we do (slide). That 5.7°C, equivalent to the warming which occurred over millions of years during the Cretaceous hothouse and tens of thousands of years during the Paleocene-Eocene Thermal Maximum, is according to the IPCC’s least optimistic emissions scenario, but that scenario is the one which most accurately matches actual cumulative emissions since 2005, predicting them within 1%.16 The, to-my-eye unrealistic, goal of the Paris Climate Accords is to limit warming to a mere 1.5-to-2°C this century. We’re banking on that, with most of us not realizing how dramatic a rate of planetary warming that actually is, to say nothing of what may happen after 2100.17 These numbers make even more sense from another angle: it took hundreds of millions of years to accumulate fossil fuel reserves which we’ve injected into the atmosphere over a few centuries (and most in one century). In terms of biogeochemical cycles, that’s a discrepancy in rates of six orders of magnitude. In short, the rate of planetary perturbation is extraordinary, especially in the context of the relative stability of climate which has pertained for most of the period civilization has flourished.18 That 15 It’s not surprisingly unclear whether or how the extinction event (which was marine) and the warming are linked. Another dramatic change in planetary temperature occurred during the Eocene-Oligocene Transition between 33.9 and 33.4 million years ago, when the planet cooled as much as 0.01 °C per century (Paleoceanography 11(3): 25166, 1996). This also coincided with a significant terrestrial and marine extinction event. 16 Schwalm, Glendon and Duffy (2020) RCP8.5 tracks cumulative CO2 emissions. PNAS 117 (33): 19656 – 7. 17 The Dansgaard-Oeschger (D-O) events during the last glacial period are the closest analogues to our current situation of which I’m aware, in terms of scale and rate of warning. Ice-core evidence suggests that, around 11,500 years ago, the Greenland ice sheet warmed ~ 8°C in about half a century. How this extreme warming event in Greenland corresponded to the planet as a whole is uncertain; contemporaneous ice cores from Antarctica show less and slower warming. If we assume nonetheless that the warming was global in extent, we can do a back-ofthe-envelope comparison as follows: since the Arctic is currently warming ~ 4 times faster than the global average, an 8-degree warming over Greenland in 50 years is roughly comparable to ~ 4 degrees of present-day global warming in a century – well within what’s possible per current projections. The warming 11,500 years ago was short-lived, however (half a century, followed by centuries of slower cooling) in contrast to present expectations, which anticipate up to 5.7°C by 2100 and quite possibly further warming thereafter. 18 Indeed, abrupt civilization collapses, as occurred nearly synchronously with the Akkadian (Sargonic) empire, the Egyptian Old Kingdom and possibly the Indus Valley civilization, are strongly correlated with abrupt climatic shifts (e.g., Weiss et al. [1993] Science 261: 995 – 1004). 6 �rate of change is a more threatening aspect of climate change than the fact of warming. We are not just facing a world of enhanced drought and intensified storms, where the Great Salt Lake becomes an empty basin and metropolises drain dry the aquifers on which they rely and rest,19 while sea levels gradually rise and forest fires rage. We are facing a world which is fundamentally and rapidly reorganizing; where the last calendar year, 2023, was possibly the hottest in the last 100,000 years but likely won’t hold that record for long;20 where, in most of our lifetimes, the Arctic’s radiation-reflecting, heat-reducing toupee of sea ice, whose maximum extent twenty years ago was 14 million square kilometers, will disappear seasonally;21 and where, partly as a consequence of the melting sea ice, a major conveyer of heat on the planet, the Atlantic Meriodonal Overturning Circulation, will dramatically weaken or even shut down,22 fundamentally changing the distribution of heat on earth (slide). This will matter not just to us but to innumerable organisms who, like us, depend on fairly predictable patterns which already are becoming unreliable. It’s not clear that most organisms and ecosystems can respond fast enough. In the Pleistocene ice ages a primary response was migration – moving north and south and up and down mountains as ice advanced and retreated. But those migrations typically occurred over longer timescales than we face, and, perhaps even more importantly, 19 Per a recent study of ~ 75% of global groundwater withdrawals, aquifer declines have accelerated dramatically in the last 40 years for a third of all regional aquifers, while groundwater declines > 0.5 m/year are widespread (Jasechko et al [2024] Nature 625: 715 -21). Major metropolises, such as Tehran (population 13 million) and Mexico City (22 million), are seeing subsidence rates of up to 50 cm/year due to groundwater depletion, with projections that large sections of Mexico City will be without reliable water within months (which pertains already to some parts of the city; Paddison et al., 2/25/24, “One of the world’s biggest cities may be months away from running out of water,” CNN). Subsidence due largely to water withdrawal is also being measured not in inches but in feet in parts of California’s Central Valley (Dan Charles, All things considered [NPR], 7/23/21), source of over half of all nuts, fruits and vegetables grown in the US. Subsidence poses obvious risks to infrastructure, but even these risks are dwarfed by the impending, local exhaustion of freshwater in large parts of the world. 20 Schmidt (2024) Nature 627: 467. 21 When I started graduate school in 1999, a seasonally ice-free Arctic was predicted for around 2100. When I finished graduate school in 2006, a small number of studies suggested it could happen by 2050. Now it’s looking increasingly likely that it could happen in the 2030’s: e.g., Kim et al. (2023) Nature Communications 14: 3139. 22 Zhu et al. (2023) Nature Communications 14: 1245; van Westen et al. (2024) Sci. Adv. 10, eadk1189. 7 �on a planet much less fragmented by human-created barriers.23 We are currently in the midst of the 7th mass extinction event24 in earth’s metazoan history, with extinction rates a thousand times greater than background. We should worry, how much worse will it get?25 In lieu of imagining all species as rugged individuals with independent fates, we should also worry about how interconnected the ecological world is and how linked many of our fates may be. So here we are, faced with an ongoing mass extinction event and dramatic rates of planetary change. What are we doing? We are persisting in the dream that we can carry on more or less as we imagine we always have, that necessary changes are only on the fringes. Indeed, the Inflation Reduction Act sets aside $161 million to restore landscapes to historic conditions, because, according to the BLM Director, “a restored landscape is more naturally in balance… [T]hat natural cycle that has been with us for the millennia, that’s what we need to strive to restore...”26 Setting aside the question of whether nature has ever really been in balance, setting aside as well the question of what we mean by nature (slide), we still should confront the smaller question – why are we trying to restore natural cycles of past millennia in a dramatically changed and changing world? I see groups dedicated to eradicating alien species. I hear in the ecological register a disturbing echo of a terrifying political project. It really worries me. But I also don’t understand what our thinking is, to eradicate immigrant species when the scale of planetary change likely can be met for many organisms, probably including humans, only by massive 23 One political response to challenges partly due to climate change has been to reify borders – to build walls – and not just in the US (e.g., Poland’s 185-km wall on its border with Belarus; Hungary’s 320-km wall separating it from Serbia and Croatia; Lithuania’s 480-km wall, also on its border with Belarus; Finland’s proposed 1300-km wall on its Russian border; and India’s ongoing efforts to wall 75% of its 4000 km border with Bangladesh.) The Akkadian empire did the same thing, building a 180-km wall to repel migrants, in its failed effort to stave off its own climatechange-associated collapse in the early bronze age. Walls will be catastrophic not only for humans but other animals. Moreover, walls will wall in as well as wall out; in a changing world a wall will inevitably be two-edged. 24 Accepting the Capitanian as a mass extinction event in the mid-Permian. 25 Even seemingly small changes in planetary heat can have surprising consequences. A marine heat-wave of between +0.5 and +3°C in the bottom waters of the eastern Bering Sea in 2018-19 is thought to have led to the deaths of >10 billion snow crab – not because they couldn’t endure the heat, but because the higher temperatures enhanced their metabolic requirements, leading to famine and cannibalism. A water temperature change from 0°C to 3°C can double a crab’s caloric requirements in the laboratory. See Szuwalski et al. (2023) Science 308: 306-310. 26 As quoted in the NPR interview cited earlier. 8 �migration – and I wonder, what is it exactly that we’re trying to preserve, and why? Is this another manifestation of how we want to have a revolution without changing, of how we want, as it were, to spin our wheels? I observed earlier that civilization is young. In roughly 10,000 years, we’ve gone from Neolithic clusters to New York City, while the global human population has exploded (slide). When Socrates lived, the world probably hosted about 100 million people, roughly the same number as now live in just three cities: Tokyo, Jakarta and Delhi. It took all of human history to reach a billion people around 1800; and another century-plus to reach 2 billion when my grandmother was born. When my mother was born, 2.5 billion people lived on earth. When my sister was born 25 years later, there were 4 billion. In 2022 we hit 8 billion, with 11 billion expected by 2100. Let me state the obvious: this enormous growth has occurred in nearly exact conjunction with our intensive utilization of fossil fuels.27 This should impress upon us our extreme dependence on fossil fuels, even while we imagine we can painlessly switch to new energy sources without substantively changing our lives. In fact, per the Inflation Reduction Act, we plan to restore natural cycles to patterns from the last millennia, as if the dramatic transformation of earth by human activities over those same millennia doesn’t really count. If we wish to be serious about climate change, we must do much more. We must let go of many familiar things. We must understand that restoration is mostly off-the-table, or else is a nostalgic aesthetic illusion. We must stop pretending we can achieve a transformation without parallel in human history on the back of only minor inconveniences. Two paths stretch before us (slide). One relies almost exclusively on technology, whether existent technology yet to be built to scale, or technology that doesn’t even exist, in the hopes that it will deliver us from having to make substantive changes (as if technology itself weren’t a substantive change) – and, further, in the hopes that it doesn’t have unforeseen consequences, though history is rife with 27 Likewise, India’s and China’s rise as global powers is not coincidentally linked to their intensive use of fossil fuels. 9 �those consequences – not least of which is climate change. I call this path the path of techno-theology. Rather than praying to the storm god to have mercy on us, we will pray to AI.28 This path leads, inevitably, to geo-engineering – or rather, to further geo-engineering.29 For, against the backdrop of our increasing awareness of its planetary consequences, what has our massive utilization of fossil fuels become but a kind of geo-engineering? The oceanographers Revelle and Suess observed in 1957 that “human beings are now carrying out a large-scale geophysical experiment … that could not have happened in the past nor be reproduced in the future. Within a few centuries we are returning to the atmosphere and oceans the concentrated organic matter stored in sedimentary rocks over hundreds of millions of years.”30 Doesn’t such a large-scale experiment sound like geo-engineering? That is not to say it’s a geo-engineering with a master engineer in charge. Like every human endeavor it’s been some combination of chosen and fallen into, conscious and unconscious. But so will any future geoengineering be as well. I dread the day when a rogue nation or trillionaire unilaterally commits earth to another uncontrolled experiment in the name of salvation. I dread the mind-set which imagines we’re sufficiently self-present to undertake the governance of the natural world, the dominion over the universe which Bacon prophesied,31 or which imagines that dominion is instead the province of AI. I dread the ideologies, so many already in circulation, which will justify any action in terms, not of the existential threat of a changing planet, but of the absolute supremacy of human beings and of sub-groups among them. It seems to me this is the track we are most likely to follow, because of that Sirens’ song it sings to 28 This possibility is already nascent in Bacon, for example in the 129th aphorism of Book I of the New Organon, with its emphasis on the divine honors owed inventors and its claim that the benefits of these inventors’ discoveries serve all mankind and last all time. 29 My objection to geoengineering is in the metaphysical comportment underlying it, as I will discuss later. Insofar as we foresee that our actions necessarily have planetary consequences and modulate (or don’t) those actions in accordance with the consequences we anticipate, we are in some fashion geoengineers – as I am suggesting here. 30 Tellus (1957) 9(1): 18 – 27. 31 Aphorism 129 of Book 1 of The New Organon. 10 �us that we won’t have to change – though that promise is a lie, and we will eventually discover ourselves as changed as Milton’s fallen angels. Among the forms of this techno-theology is the technological creation of illusions we desire – the technological creation of a cave we like. We don’t want just to geo-engineer the world; we want to retreat from the world that is into one of our own devising. This comes to mind when I consider the Fabian allure of massive climate simulations, but no less when I consider the aptly-named Marvel Cinematic Universe. Or when I watch parents at playgrounds engrossed not by children but by phones.32 Or when I ponder why cars are so important to us, those wheeled would-be biospheres with their own entertainment systems and climate control. These are all, at least in part, escapes, caves we retreat to in order not to stare at the sun or to feel its heat. Fear no more the heat o’ the sun, thinks Virginia Woolf’s character Clarissa Dalloway as she prepares her party and contemplates her mortality; fear no more the heat o’ the sun, thinks another of Woolf’s characters, Septimus Warren Smith, even as he is about to plunge holding his treasure (slide). They’re both quoting a line from a funeral dirge in Shakespeare’s Cymbeline; and this retreat I’m describing, this techno-theological embrace of the cave, this path which I think we’re mostly following, is in equal measure that song of sorrow which declares we are afraid to live. Well we should be! It takes gigantic courage to live even a single day. That’s the techno-theological path stretching before us – the path of technological illusion and of god-like faith in technology, but also of retreat into made-up worlds, the bubbles of social media, CGI and cars, while in our hearts suspecting this also deforms us and tears us apart. The other path, the more difficult and the more hopeful one, places the burden of response and responsibility not only on technology, though we will need it, but on ourselves. It rejects that we are helpless to change or that our 32 We can no longer imagine even taking a neighborhood stroll without a computer in our pockets whose memory is millions of times, and processing speed hundreds of thousands of times, that of the computers which first put a man on the Moon. 10% of worldwide electricity generation goes into information and computing systems – according to Roy Scranton (Learning to Die in the Anthropocene), roughly the same amount of electricity as was needed to illuminate the globe in 1985. 11 �only hope of change is to convince politicians and corporations to make changes for us. We have become shockingly accustomed to imagining ourselves as mere followers and our politicians, for whom we frequently express disdain, as nonetheless leaders. They’re not necessarily leaders; in this country, they are first and foremost our representatives. The difficult, more hopeful path begins by taking seriously that they do represent us; that corporations do too; that what we behold in them is a representation of what we are. While it’s comforting and not wholly false to imagine tycoons making gzillions of dollars off of us helpless dupes, we must take seriously that what they sold us we also wanted. Let me underline this point with a recent example. Darren Woods, CEO of ExxonMobil, excited outrage last month when he blamed the public for climate inaction: “The dirty secret nobody talks about is how much all this is going to cost and who’s willing to pay for it. The people who are generating those emissions [i.e., all of us – LW] need to be aware of and pay the price for generating [them]. That is… how you solve the problem.” Setting aside the likely hypocrisy, self-interest and poor faith of Woods’ claim, we nonetheless need to hear its truth, rather than quickly condemning it as, according to The Guardian newspaper, “[an] attempt to skirt climate accountability.”33 Such condemnations are their own variety of skirting accountability. Indeed, the climate activist group 350.org specifically exonerated all of us when it used Woods’ claim to motivate a petition which read, “Fossil fuel giants like ExxonMobil are to blame for the climate crisis – not us. If you agree, add your name” (slide).34 Likewise The Guardian article cited a climate economist who compared Woods to a drug lord “blaming everyone but himself for drug problems.” But if a drug lord is culpable, does that mean drug users are not? I am alarmed by so limited a notion of freedom, according to which we accept ourselves as powerless rather than endure even a modicum of responsibility – as if freedom didn’t imply responsibility but instead was freedom from 33 I am referring to the article by Dharna Noor and Oliver Milman published on 3/4/24 in The Guardian I’m quoting their 3/15/24 email with the subject line “Exxon CEO just blamed the climate crisis on the public.” A subsequent email on 3/23/24 expanded the list of the guilty: “Last month was the ninth consecutive hottest month on record, ever. The climate crisis is here, and the culprit is the fossil fuel industry. But they are not alone – utility companies are also responsible.” Once again, however, we consumers are off the hook. 34 12 �responsibility. This desired freedom from responsibility fuels the dream that we can meet the extraordinary challenges facing us without having to change substantively. We too easily accept that our civic responsibility is to consume rather than sacrifice, to vote on occasion, to write a check to a political candidate, PAC, party or charity or, less often, to go to a protest, even while none of these demands what is more fundamentally needed: that each of us change. Our response can’t be an exception from our normal lives; it has to be what’s normal about them. The assumption that the heavy-lifting can be accomplished only by governments or corporations exculpates us. It’s not entirely untrue, which is why voting, donating and protesting matter, but it also basically assures the status quo while delivering us from having to consider how those governments and corporations truly represent us. Part of what can be recognized in those representations is despair. It seems that we don’t in our hearts believe we can change; it seems that in our hearts we accept that this representation, whether corporate or political, and despite regularly communicated contempt for it, still expresses the dark truth of who we are and of all we can become. We might as well resort to CGI creations of god-like superheroes because, despairing, we can’t imagine ourselves as powerful agents otherwise. We have more faith in technology, including technological illusions, than in ourselves. What if we were to put faith elsewhere than the technological divine? What might that look like? I have three suggestions, each both simple and on the far side of human possibility. If we take them seriously, we must also take seriously why they’re on the far side of possibility. The three suggestions are: Live small, not large. Live slow, not fast. Think differently. Rather than accepting the nihilistic premise that nothing you do matters, consider the exhilarating, terrifying premise that anything you do may matter. You may not know the ways it matters, to whom or to what. Mattering may not be reducible to meaning. But anything you do may matter, at least insofar as anything you do is matter. This is a consequence and expression of finitude. If we and our world were infinite, we could 13 �subtract or add to that infinity without changing it. But we and our world are finite; we and it must change.35 We are like that dream of Pierre described near the end of Tolstoy’s War and Peace,36 where he sees a living, wavering globe consisting of drops tightly packed together. Every drop which moves and shifts causes other drops to move and shift in turn. So it is in our lives: not one of us can move without moving something else. That seeming accident is a condition of possibility of every motion we make. Think about yourself finitely, pushing the earth to push off from it. An oil rig is not just ExxonMobil’s; it also expresses you who are but several drops away. When you drive, your gasoline has travelled vast distances, and that’s now part of your motion. That gasoline was once a fossil-fuel deposit; before that, it was a living being. Its carbon cycled many times through the food web, before getting buried and, over millions of years, being compressed and transformed into something else, which one day we sucked from the earth and transported across the globe so that the controlled oxidation of it would propel your vehicle while once again accomplishing a new transformation, spiritualizing what was once alive into the gas trees breathe and you and I exhale. Being finite means being part of this enormous web which you can’t control or fully foresee, but within which you are not without freedom or responsibility. This web, this unavoidable mediation which finitude implies, is what geo-engineering forgets. It’s arguably what Descartes forgot.37 Both secretly infinitize us; both imagine us as subjects fully present to ourselves for whom absolute agency is possible. For them, the drop each of us is doesn’t shift or get shifted by other drops but is isolatable; no web inextricably entangles us; and hence, surveying all that is, a subject can act with full consciousness. A subject can command itself to doubt; it can doubt away the 35 As recognized by Pascal: “We are floating in a medium of vast extent…; whenever we think we have a fixed point to which we can cling and make fast, it shifts and leaves us behind… Nothing stands still for us. This is our natural state and yet the state most contrary to our inclinations” (Pensées 199, ‘Disproportion of man,’ translated by Krailsheimer). Physical conservation laws can also be interpreted as consequences of the finitude of the world. 36 Vol. IV, Part 3, Chapter XV, pp. 1064-5, translated by Pevear and Volokhonsky. 37 It’s also what our ideology of perpetual growth forgets – an ideology central to capitalism. As for Descartes, his forgetfulness was not shared by his contemporary and critic, Pascal, who regularly linked finitude and mediation (or middleness), as in Pensées 199. 14 �entire world; and having found the seat of certainty in its own thinking, a disembodied and hence definitized thinking, it can then re-create and author the world. This is Descartes’ trajectory in the Discourses and Meditations. It’s also the trajectory of technological illusion, which would replace the world that is with the illusion of a world we author.38 It’s finally the trajectory, if not task, of geoengineering. A famous climatologist39 argued that, given climate’s susceptibility to small changes of initial conditions, the so-called butterfly effect – where the flapping of a butterfly’s wings in one place contributes to a hurricane elsewhere – given this susceptibility, he said, all we had to do was “tame this butterfly.”40 Were this climatologist more of a poet, we might hear irony in his claim; but what we might hear poetically as articulating a hubristic and hopeless project can also be heard as the assignment. It imagines every butterfly can be tamed; but even more dangerously, it posits a tamer outside the web of mediations, a disembodied, decontextualized knower whose command to the legion of tamed butterflies is itself beyond the jostle of any neighboring drop (slide). Think differently. Picture yourself as a drop jostled by and jostling its neighbors. We must relentlessly re-imagine ourselves thus. We must accustom ourselves regularly to undertake the thought experiment Rousseau challenges Emile with, in the book of that title:41 “We [i.e., Emile and his teacher] go to dine in an opulent home. We find the preparations for a feast… All this apparatus of pleasure and festivity has something intoxicating about it… While the meal continues, while the courses follow one another, while much boisterous conversation reigns… I lean toward [Emile’s] ear and say, “Through how many hands would you estimate that all you see on this table has passed before getting here?” What a crowd of ideas I awaken in his brain…! Instantly all the vapors of the delirium are dispelled. He dreams, 38 We tend to confuse our mediated and necessarily partial access to the world that is with our authorship of it. Finding ourselves in a synecdoche, we assume we’re the poet. 39 The climatologist was Kerry Emanuel; he made these comments during an interview on NPR’s Fresh Air, August 1st, 2007. An audio file is available at http://www.npr.org/templates/story/story.php?storyId=12421331. 40 “Is it by your wisdom that the hawk soars and spreads its wings out to the south?” Job 39:26. 41 Emile, Book III, p. 190 in Bloom’s translation. 15 �he reflects, he calculates, he worries. While the philosophers, cheered by the wine, perhaps by the ladies next to them, prate and act like children, he is all alone philosophizing… [W]hat will he think of this luxury when he finds that every region of the world has been made to contribute; that perhaps twenty million hands have worked for a long time; that it has cost the lives of perhaps thousands of men…?” We must think in terms of twenty million hands and thousands of lost lives – not all of them human – and indeed those numbers are surely underestimates. Even when simply walking the placita, we should imagine that what energizes our steps were once living beings who stretched themselves toward the sun.42 We must as best we can make choices worthy of the sun-stretching creatures who enabled them, of twenty million hands and thousands of lost lives. This is not an exhortation not to live, to deny ourself life, to do nothing because there is nothing untouched by twenty million hands and thousands of lost lives. Instead it’s an exhortation to live up to those hands and lives. Derrida described such a comportment in these terms: “Il faut bien manger” he wrote, but also “il faut manger le bien”43 – which as a play on words means several things: it means, of course we must eat, but we must also eat well – and not only that, we must eat the good. We must eat the good well. We must do it because we can’t not do it. To be finite implies, inexorably, that we must eat the good, that no action is without reaction, no gain without loss. We can’t not eat the good, we can’t act in the world without the possibility of harm, we can’t self-consciously leave no trace, nor is the trace which we may leave simply up to us. So what we can’t not do let us do as well as we can (slide).44 If this is not an exhortation not to live, neither is it an exhortation to embrace extravagance, to 42 Mary Oliver’s “Wild geese” comes to mind, especially the last sentence: “Whoever you are, no matter how lonely, / the world offers itself to your imagination, / calls to you like the wild geese, harsh and exciting – / over and over announcing your place / in the family of things.” 43 See Derrida’s conversation with Jean-Luc Nancy entitled “Eating well, or the calculation of the subject” in Points, edited by Elisabeth Weber, especially pp. 277-87. 44 The French uses the impersonal expression “il faut” where I use the pronoun “we.” The “we” is problematic; there is a question of what or who “we” is, as there should also be a question of what eating is. Eating is not simply what subject does to object; eating changes both eater and eaten. The referent of “we” thus necessarily changes. 16 �pronounce, like Louis the XVth, “After me the flood.” Look to the future: look not only at the hands and lives of those who have touched what is already before you, but also to the hands and lives of those yet to come, who not you but your choices will touch. What entangles us stretches from past to future. Our actions may memorialize and prophesize. It is in this context of an imagined future, moreover one which no longer includes you or me, as well as in the context of imagining our global contemporaries who even now live lives much less replete than ours, that it makes sense to live small and slow. Put simply, if you do, you will use less and leave others more.45 What I’m suggesting may sound naïve, as if I’m unaware of the notorious problem of collective action. If each of our contributions is a mere drop in the ocean, why worry about my drop if no one else worries about theirs? This argument of apathy, however, is just the flip side of the infinitization I’m critiquing. It posits that the only options are to be infinite or nothing. Such a coarse dichotomy has long characterized our thought. For example, we have long believed we could dump waste into the ocean or atmosphere and that, in effect, it would be infinitely diluted – that the vastness of ocean or atmosphere would reduce it to nothing. But it was, in fact, finitely diluted. Slowly those drops accumulated, so that now, for example, plastic, invented in 1907, is found in every region of earth from the Himalayas to the deepest ocean trenches, as well as in our food, our bodies and our mothers’ milk, to say nothing of the atmosphere’s rising greenhouse gases (slide). This whole lecture is about taking that disproportion between nothing and the infinite, a drop,46 seriously, about admitting that yes, you are a drop, but every drop matters, you matter. The pretense that drops don’t matter expresses, in one mode, a deluded if aggrandizing infinitization, as if we count but only as infinite; but in another mode, the complement of our aggrandizing delusion, it confines each of us to a cell of impotent isolation, where our drops don’t collectively constitute a wavering globe, much less our exalted, lonely individualities a world. But there is 45 You’ll eat less good more well. “For, after all, what is man in nature? A nothing compared to the infinite, a whole compared to the nothing, a middle point between all and nothing…” Pascal, Pensées 199. 46 17 �a world, a living, wavering globe of drops, in which each of us is a single drop who matters.47 Live small and slow and think differently for that world. This advice is more dramatic than it sounds. We easily dream of circumstances which will enable us to live as we imagine we like without unwanted consequences. This is like imagining ourselves disembodied, mattering without being matter, meaning immaterially. It articulates a desire that we not be finite or mortal. I suspect that we live large and fast partly to distract ourselves from that finitude, from the death which awaits, as a substitute for the gigantic courage it takes to live even a single day. Yet such a response accelerates catastrophe. The illusion that we’re infinite, that change can happen without us changing, is not a salve for our fragile mortality but a destructive expression of it. It secretly confesses that none of us really believes we will die, as perhaps none of us really believes in climate change. There is another option: that we try to believe; that, trying to believe, we make an offering to the future and of the future; 48 that we not insist that future be ours; that we let it be someone else’s.49 The talk of a new green revolution sometimes sways this direction, but so long as it promises immense change which won’t require us to change, it’s a pipedream if not something more malevolent: 47 This may signal a return to the sacred, if we accept Amitav Ghosh’s suggestion (op. cit.): “It is impossible to see any way out of this crisis without an acceptance of limits and limitations, and this in turn is, I think, intimately related to the idea of the sacred…” (p. 161). 48 The French scholar Rémi Brague gave a lecture at St. John’s (Santa Fe) many years ago in which he identified as one of the crises of modernity the decline in the human birth rate. He interpreted this decline as expressing despair about the future, as articulating that we no longer believe in a future worth giving. I agree with such a diagnosis of despair; I agree as well that it behooves us to believe in a future worth giving, and to do the hard things such a belief demands, but, contra Brague, I think it imperative that we appreciate that the recipients of such a gift are not, and cannot be, restricted to human beings. Indeed we can no more know the recipients than the future. 49 One is always at risk of re-infinitizing the subject. Letting the future be someone else’s still secretly assumes the future is mine to yield. Even just letting the future be may fall into this conundrum. It’s reminiscent of the catch-22 in Faulkner’s Go Down, Moses, where Isaac – whose apt Biblical name invokes the possible sacrifice of both past and future – endeavors to relinquish his patrimony but can’t escape thinking that it is his to relinquish. As a result, his may still be a version of thinking infinitely, albeit in a negative mode, rather than thinking finitely. Perhaps Molly comes closer to the latter, she who “dont want nothing” (p. 97), mistrusts a divination which has become of gold rather than god, and at the end of the book keens for her Benjamin, sold into slavery and dead. But the spiritual “Go Down, Moses,” for which the book and the last tale are both named, still suggests a future organized by a promise, a promise clearly not met by the book’s end but, like Isaac’s patrimony, not truly relinquished either. 18 �a future we pretend to yield to an other while secretly and stubbornly insisting it remain ours.50 That pipedream pretends that massive solar arrays, nuclear power, windmills, hydrogen, etc., will allow us to build for the future without giving up that future to someone else;51 it pretends we can be responsible to the future without really changing our present, including without decreasing our consumption. Take energy for example. We may dream that someday it will be boundless and free, but it won’t be. Energy will always come with consequences. Its production and use will require a material infrastructure likely including fossil-fuel-intensive steel, concrete and plastic. It will generate waste, including, for nuclear energy, dangerous radioactive products whose half-lives can far exceed the history of human civilization.52 We shouldn’t overlook that hydrogen, if it’s to be our panacea, is a highly explosive panacea. And if a butterfly can cause a hurricane, what can a windmill do (slide)? Put otherwise, don’t look to get something for nothing;53 that only amounts to the illusion of a good conscience.54 Don’t trick 50 Such a situation, including its unfurling ramifications, is well-described again in Go Down, Moses, as in this passage from “The Bear” (pp.257-8): “…old Carothers’ bold cramped hand far less legible than his sons’ even and not much better in spelling, who while capitalizing almost every noun and verb, made no effort to punctuate or construct whatever, just as he made no effort either to explain or obfuscate the thousand-dollar legacy to the son of an unmarried slave-girl, to be paid only at the child’s coming-of-age, bearing the consequence of the act of which there was still no definite incontrovertible proof that he acknowledged, not out of his own substance but penalizing his sons with it, charging them a cash forfeit on the accident of their own paternity; not even a bribe for silence towards his own fame since his fame would suffer only after he was no longer present to defend it, flinging almost contemptuously, as he might a cast-off hat or pair of shoes, the thousand dollars which could have no more reality to him under those conditions than it would have to the negro, the slave who would not even see it until he came of age, twenty-one years too late to begin to learn what money was.” 51 Hence the college’s thoughtless slogan campaign tying “Building for the future” to “St. John’s forever.” 52 Iodine-129 has a half-life of almost 16 million years, for example. 53 To get something for nothing implies an infinity, while suggesting simultaneously how nigh nothing is to anything. 54 Consider carbon offset programs. The science behind these has always been suspect, with concerns about whether they can be done at the necessary scale in the right places, how long-lasting savings will be and whether savings can accurately be determined. Yet worse is the world’s largest carbon offset firm, South Pole, selling carbon credits for reductions which they knew didn’t exist (Blake, “The great cash-for-carbon hustle,” The New Yorker 10/16/23). More broadly, however, the problem lies in our failure to recognize that responsibility is always excessive. To measure and delimit it is to turn any responsibility into the alibi for irresponsibility, as we saw above in the 350.org petition. Appreciating responsibility as necessarily excessive is a corollary of understanding that what we eat, what we internalize and make our own but also at the same time destroy, is in fact the good, for the loss of which there is no compensation. The pretense at the core of capitalism (but also of prevailing notions of justice), that no loss is without compensation (whether in the mode of offsets or criminal penalties) implies irresponsibility. 19 �yourself into believing you can give to the future without giving something up.55 Here’s a more tangible form of this advice: we must pair attempts to choose energy sources more wisely with concerted efforts to use them more wisely – and specifically to reduce how much energy and matter we use. I’m ambivalent but partly glad that the St. John’s Santa Fe campus has gone solar. I’d be more glad and less ambivalent if our solar conversion were accompanied by a campaign to reduce our energy use, if our murals enjoined us to take the next step rather than extolled the step already taken (slide). Per capita energy consumption in the United States is among the highest of any nation. If we want to limit ourselves to “only” 1.5 or 2°C of warming – as dangerous as that may be – if we want to reduce global fossil fuel emissions by 50% in the next 6 years, we can’t do it without dramatically reducing how much energy we use in ways which will present real but surmountable problems. Other first-world countries56 consume per capita half the energy which Americans consume. I grant substantive differences exist between us and them, but it seems to me the most stubborn impediment to making real reductions in our energy consumption is not those differences but our unimaginative and fantastical insistence that we shouldn’t have to change. As a result, so much of the rest of the world strives to live like us, when we desperately need to live more like them. Gandhi worried in 1928, “God forbid that India should ever take to industrialism after the manner of the West… It would strip the world bare like locusts.”57 But India and China and much of the world have taken to westernstyle industrialism, and that stripped world which Gandhi dreaded is increasingly the reality before us, as the rest of the world follows our example while we refuse to follow theirs (slide). My argument for living small and slow is not just that there’s no other way to face the 55 Technology is the claw by which we bring the future closer. It’s a way of materializing time, even commodifying it. We should wonder whether the technological solution isn’t always at the expense of the future. If we admit, however, that technology is an inevitable aspect of what we are, we will also have to admit that we are always expending the future in some form. This again comes down to eating the good but endeavoring to eat it well. 56 For example, Germany and France. 57 As quoted by Amitav Ghosh (op. cit.), p. 111. 20 �extraordinary challenges before us and no better way to accustom ourselves to the lives we will eventually have to lead. It’s also not just an ethical argument about living up to twenty million hands and thousands of lives past and future. The third prong of my argument is what I’ve been saying about how the world reflects us. As much as we may want to blame governments, corporations, even capitalism, and as much as none of those are without blame, we still must take seriously that these reflect us. They express something that we must take responsibility for as our own. Our words may say, “End fossil fuels,” but our actions take for granted their ready abundance. The only way I see to change that is to change how we consume (eat), live and think. We can’t just talk about this; we have to do it. That means recognizing no one is without responsibility for our predicament. Typically, if no one is without responsibility, no one is held responsible. But I’m saying each of us must hold ourself responsible.58 Change is coming, whether we give assent or not. If we try to affirm it, we need to do more than vote, donate, protest or go solar. We need to change. This may not sound revolutionary compared to a new green revolution – but of the two (which aren’t mutually exclusive), I’m advocating the essential one. I’m not saying it will be sufficient if enough of us do our small part. Other things need to happen, including at corporate and political scales. But those things will not happen if enough of us don’t do our small part. As much as we may like to believe we are victims of capitalism run amok, of ExxonMobil, of politicians, these also mirror us. They show us versions of ourselves.59 They give us what we, at least in part, ask for. Ask for something else. Become something else. Corporations, governments and economic systems aren’t simply controlling us; they’re also expressing us. Any revolution without that admission is just a spinning of wheels. Some of the dread darkness of the political era into which so much of the 58 To suppose that I should not act until those act who have behaved even less responsibly than me is to place the burden of our collective response on those who have behaved the absolutely least responsibly. 59 My insistence, or at least invitation, that we recognize our corporate, political and economic reflections runs the risk of becoming a kind of consumption or subsumption, a gobbling up of the other as secretly, within a naïve idealism, the same; yet the other which reflects us in myriad ways need not only be our reflection, nor is it clear that in a reflection what is seen is same and not other. 21 �world is falling stems from our propensity first to alienate, then to vilify, our own reflections – while also believing that it is only those reflections which have the freedom, power and burden of responsibility.60 What I’ve said today is not just about climate change. It’s about how we are using up and running down the earth and, in parallel, exalting and diminishing ourselves. We see the earth as a pantry for us to raid; and as various stocks within it become depleted, we don’t question our behavior or assumptions but merely look for new stocks. For the most part we have not interrogated our prejudice that we are entitled to the earth, that it is ours to exhaust, that no other beings besides human beings – and perhaps even among them only some – truly matter (slide). This exaltation we’ve implicitly bestowed upon ourselves is a kind of self-infinitization, whereby even the governance of the natural world falls under our thinly secularized techno-theological sway; but in lock-step with it comes our diminishment as isolated individuals, impotent and unfree, incapable of response and of responsibility, in effect nullities. Cast between these overtly dueling but secretly linked poles of the infinite and nothing, climate change becomes an avatar of our finitude. Ignore the avatar and the threat will not retreat; but neither will it retreat should the avatar be addressed. That desired retreat follows from the false premise that our finitude is escapable. It is not. We are neither infinite nor nothing but, drop-like, between the two. We need a response which endures that disproportion rather than, through fantasy and denial, embraces one end member or the other, thereby accelerating and compounding catastrophe. Some will hear me as prophesying doom, but I am actually advocating hope – not an easy hope, not a hope without effort or struggle or loss, not an empty hope which consoles by anesthetizing, but the harder and truer hope which admits that we must change – and that we can. Something new is coming, just as it did at the end of the Cretaceous 65 million years ago in the form of an asteroid. That asteroid augured both calamity and immense possibility, the end or 60 If we live in a benighted age, the question still remains: how far advanced is the night? Could we perhaps even now be on the cusp of dawn? 22 �transformation of the dinosaurs,61 and the rise and transformation of the mammals. From it was born the once-new world we recognize and sometimes love. A thin layer of the element iridium is sprinkled across the planet, marking that event and the boundary separating a world familiar to dinosaurs from one more familiar to us. Many now claim that we’ve entered a new geological epoch called the Anthropocene. Since epochs typically last hundreds of thousands to tens of millions of years, I’m struck by the sheer optimism of that designation, which I find both endearing and repellent. How innocent and how hubristic that we think it’s an epoch! It strikes me as far more likely that the Anthropocene will have been a boundary event like the impact at the end of the Cretaceous,62 marked by a layer not of iridium but of plastic, and distinguishing yet another new world from our now-old and familiar one. This new world may not be ours. It doesn’t need to be ours. A new world which includes us can’t only be ours. Like the dinosaurs, if we63 are to persist, we must become something different, something smaller, something which, though almost as light as air, nonetheless greets mo(u)rning with song (slide).64 61 How to read “or”? Transformation may be an alternative to extinction, to an end, but it is also a specific form of the end. The first reading reads continuity where the second reading reads discontinuity. 62 The Anthropocene as a boundary event was initially suggested by Scott Gilbert, per Donna Haraway (2015) Anthropocene, Capitalocene, Plantationocene, Chthulucene: Making Kin. Environmental Humanities 6: 159 – 65. 63 Don’t overlook that the pronoun in this final sentence, despite its seeming stability, refers to things in flux. Its referent is changing (as is characteristic of pronouns). See also the footnote on p. 16 about translating “il faut (bien) manger (le bien).” 64 Every new beginning also marks a loss. 23 � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 23 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Clima(c)tic Change: Twenty Million Hands and the Living, Wavering Globe Description An account of the resource Transcript of a lecture given by tutor Llyd Wells on April 19, 2024 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "Climatic change demands of us climactic change. We have yet fully to admit it. Perhaps we never shall. Can we be serious about climate change? How? That's the core of this lecture, which will try to be serious scientifically, philosophically and literarily." Creator An entity primarily responsible for making the resource Wells, Llyd Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2024-04-19 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Climatic changes Relation A related resource <a href="https://digitalarchives.sjc.edu/items/show/7925" title="Audio recording of lecture">https://digitalarchives.sjc.edu/items/show/7925</a> Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_WellsL_Clima(c)tic_Change_text_2024-04-19 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/433170265cb61a133a1cd8d0c840eab4.pdf 89b0a8fc6dc76aa3170892a7ff0b03e2 PDF Text Text 1 What is the Measure of Electricity? 1 Howard J. Fisher What is the measure of electricity? The question itself raises questions. For not all things are susceptible to measure; and even when they appear to be, it is not always clear whether “measure” applies to them as wholes, or only in certain respects. For purposes of this talk, let me propose that a measure of something must, at minimum, enable us to speak of that thing in terms of more and less. Faraday inherited an electrical vocabulary that appraised electricity as more and less in two respects: first, in quantity; and second, in intensity. At the outset of Faraday’s researches, neither he nor anyone else had been able to state just what these two characteristics were, nor to explain how they related to one another. On the other hand, everybody had some rough and practical idea of them, as we may gather from Faraday’s unassuming characterization in the Third Series: The term quantity in electricity is perhaps sufficiently definite as to sense; the term intensity is more difficult to define strictly. I am using both terms in their ordinary and accepted meaning. [360, note] If Faraday regarded the term “quantity” as relatively straightforward, it is probably because at the time he began his researches, the conventional idiom of electrical thinking was that of electric fluid, a special kind of substance, thought to be endowed with the power to attract or repel other portions of electric fluid. Electric fluid was either vitreous, like that which could be evolved upon glass surfaces, or resinous, like that which could be produced on rubber, gum, amber, and similar materials. Portions of unlike fluids attracted one another; portions of like fluids repelled each other; and the more fluid there was, the stronger that attraction or repulsion would be. It is easy to know what we mean by “quantity” if electricity is a fluid. But is it a fluid? And how can we know? In contrast, as Faraday implies, the fluid language fails to offer a similarly clear image of intensity. What can it mean for a fluid to be more or less “intense”? Faraday will seek, and perhaps he will find, a clearer understanding of both these terms.  As the Third Series opens, we find Faraday in almost the same position as Socrates of the Meno; for how can we hope to know the properties of electricity unless we first know what electricity actually is? We well remember Meno’s reply when Socrates asked after the “what” of virtue: 1 Meno. “There will be no difficulty, Socrates, in answering that. Take first the virtue of a man: it is to know how to administer the state, in which effort he will benefit his friends and injure his enemies, and will take care not to suffer injury himself. A woman’s virtue may also be easily described: it is to order her house, and keep what is indoors, Lecture delivered 23 February, 2024 at St. John’s College, Santa Fe �2 and obey her husband. Every age, every condition of life, young or old, male, or female, bond or free, has a different virtue....” [71] Meno is positively exultant as he contemplates the rich variety of virtues! How disheartening is it, then, to consider that the electrical science of Faraday’s time, though professing to seek a unitary account of electricity, can offer little more than a Meno-like catalog of “electricities.” These include:  Voltaic electricity, which is evolved by devices like Alessandro Volta’s “cups.” Faraday will study voltaic action extensively in the Seventh Series and will show there its relation to chemical combining power.  Magneto-electricity, obtained through the relative motion of magnets and conductors, and which Faraday had already studied in the First Series.  Thermo-electricity, produced when the junction between two different metals is exposed to heat. A. B  Animal electricity, which is produced by several fascinating families of both freshwater and saltwater fishes. Faraday will study the wonderful electric eel in the Fifteenth Series, one of the most engaging of all his researches. And, finally...  Common or ordinary electricity. This is what we now call “static” electricity: the electricity produced primarily by friction—for example, by rubbing a resinous rod with wool, or a glass rod with silk. But how often do we undertake such highly specialized activities as these, except in a classroom or similarly contrived setting? In our day there would seem to be nothing at all “ordinary” about the electricity that arises from friction; but I assure you that when I was a child, rugs, sofas, and especially automobile seats, could easily give you a very unpleasant jolt if you carelessly walked across a carpeted room, or slid out of an upholstered piece of furniture, and then touched a doorknob or a water faucet. Today, many fabrics contain antistatic materials which greatly reduce the frequency of �3 such experiences; so for us, the terms “common electricity” and “ordinary electricity” are no longer apt, and they are consequently no longer in common use. Unfortunately, today’s more familiar term, “static electricity,” is misleading in its own way; for many of the signs that alert us to the presence of static electricity occur precisely when that electricity is not static! Those unpleasant shocks which lurked in my family’s home and automobile, patiently awaiting their opportunity to strike, represented the discharge of electricity which had previously been built up by friction: they were instances of electricity in motion, not electricity at rest. Faraday’s efforts to demonstrate the identicality of this “swarm” of electricities occupies the first and longer part of the Third Series. Only then does he set out upon the second part, where the topic is measure—and particularly the measure of quantity. Readers may notice a distinctive suppleness in the language Faraday adopts for this discussion: while he does not reject the imagery of electric fluids outright, he never crafts his descriptions in a way that depends on that imagery.  Now, one way we can estimate quantity—whether of electricity or anything that is evolved or produced—is to identify a repetitive element in the process that produces it; then, presumably, each repetition of that action will produce an equal amount afresh. Faraday obtained common electricity from a frictional “plate machine,” in which a large plate of glass was rotated against a fixed “rubber”—which was usually made of silkwrapped leather, rather than what we now call rubber. The appliance shown here is a smaller version of Faraday’s enormous machine, which featured a glass plate of fifty inches diameter—nearly four times as large as this one. 2 At several points in the Third Series Faraday treats each turn of his machine as developing the same quantity of electricity. You can see why such a supposition is reasonable; for it is easy to make sure that all revolutions of the crank are accomplished with uniform effort and speed. And to the extent that individual turns are identical to each other, there is no obvious reason why successive turns would not produce identical results. 2 Photo courtesy London Science Museum. The glass disk is 35 cm in diameter. �4 This “same-again” principle of reasoning is familiar to us in other contexts, such as grinding pepper in a mill. Indeed, in the case of grinding we are rewarded with a clear image of “quantity” in the form of a heap of the ground substance, as shown here. But when Faraday cranks his plate machine, no “heap” of electricity is produced. Is electricity even the sort of thing that possesses “quantity” in the sense of a heap, a pile, or a mound? Once again we are reminded of Socrates’ lament to Meno: “If I do not know the ‘what’ of something, how can I know the ‘such’ of it?” 3 In our present case, if we do not know the “what” of electricity, is it really meaningful to ask the “how much” of it? When Faraday remarked that the term quantity was “perhaps sufficiently definite as to sense,” he meant to acknowledge that we habitually think of “quantity” through images of accumulation or gathering up. But do not overlook the note of reservation suggested by his word “perhaps.” Faraday is far from confident that electricity is really amenable to such imagery. We regularly use such language for electricity without a second thought; but can we point to any body of experience that gives real content to that language?  If electricity does not manifest its quantity directly in experience, might it do so indirectly? Sometimes, for example, we think it natural to express the magnitude of something in terms of the power it exercises. Galileo offers a memorable instance in the Two New Sciences; Sagredo is speaking: “Thus a vast number of ants might carry ashore a ship laden with grain. And since experience shows us daily that one ant can easily carry one grain, and it is clear that the number of grains in the ship is not infinite, but falls below a certain limit, then if you take another number four or six times as great, and if you set to work a corresponding number of ants they will carry the grain ashore and the boat also. It is true that this will call for a prodigious number of ants...” [67] That delightful phrase, “a prodigious number of ants,” seems to employ the imagery of number; but its rhetorical burden is rather the sheer magnitude implied by the ability to move “the grain and the boat also.” The phrase expresses huge undifferentiated totality, whose greatness is known primarily by what it can accomplish. It is an indirect representation of quantity. 3 71A �5 Frictional electricity, too, seems to express quantity only indirectly. When a rubber rod is stroked with woolen cloth, it acquires the power to attract a small ball of cork or I \ pith. We say that the rod has been electrified, or charged with electricity; and in the left-hand sketch, the electrified rod has succeeded in drawing the ball aside through a moderate angle of perhaps 9 or 10 degrees. But after receiving additional strokes with the wool, the rod is able to urge the ball to a greater angle—perhaps as much as 18 or 20 degrees, as shown on the right. Is it not reasonable to believe that the rod on the right exerts more attractive force precisely because it has acquired more electricity? But this is conjecture, not direct experience. Any notion of quantity we can gain from this experiment is limited to what we can surmise from the angle of the suspended pith ball. But angle is no image of “muchness,” and it shares none of the straightforwardness of such eminently legible figures as heap, mound, or—in the fluid case—puddle. If not the pith ball, then, might some other electrical instrument offer a more immediate experience of electrical “quantity”? The distinctive power of electrified bodies to attract or repel other electrified bodies is the principle of several electric indicators that are considerably more refined than the pith ball. Two early instruments operate on the principle of mutual repulsion. The leaves of the gold-leaf electroscope, pictured here on the left, diverge from one another more or less, . depending, partly, on how many times the rubber rod has been stroked. On the right, Henley’s electrometer calls even sharper attention to angle by incorporating an obvious pointer and protractor in its design; when the instrument is mounted on the electrified conductor of a plate machine like Faraday’s, the pointer is repelled from the body, just like the leaves of the electroscope. With its angular scale, the Henley instrument emphatically announces its rhetoric of numerical measurement—and hence its name “electrometer” rather than “electroscope.” But what, exactly, does it measure? The �6 angle of the pointer, even when expressed numerically, still seems far removed from a direct image of quantity. In fact, one of Faraday’s experiments in the Third Series suggests that the electrometer is better understood as indicating some other electrical attribute—an attribute rather different from quantity, though it may be related to quantity. Faraday describes that experiment in paragraph 363 of the Third Series. It involves an array, or “battery,” of fifteen identical Leyden jars, like this one. You see that the central conductors, which are connected to the jars’ inner coatings, are all joined together. Within the wooden container, the outer coatings rest upon a conductive plate that is connected to the flexible chain B, which in turn is connected to the earth. Faraday will charge these jars using the plate electric machine. Notice the Henley electrometer mounted on the prime conductor; this was one of the chief applications of the Henley device. At first Faraday connects only eight of the jars, charging them by thirty turns of the plate machine. This causes the electrometer to rise to some position A. Does that position represent the quantity of electricity supplied to the jars? Certainly that quantity must be considerable, since Faraday noted that merely one revolution of the plate will, in his words, “give ten or twelve sparks from the conductors, each an inch in length.” 4 At a later stage of his experiment, Faraday charges all fifteen jars, again by thirty turns of the machine. This time, he reports, The Henley’s electrometer stood not quite half so high as before... 4 Paragraph 290. �7 Obviously the electrometer is not measuring quantity! For the quantity of electricity was the same in both cases—the result of thirty turns of the machine. Yet with a greater number of jars, the electrometer reading was lower by more than half. What electrical characteristic was it, then, that the electrometer measured when it registered that striking reduction? In hopes of answering this question, let us conduct an experiment of our own. Recall that Faraday noted the generous number of sparks produced with each turn of the plate machine. This should give us pause: why does the machine produce a series of sparks rather than one continuous spark? To study the conditions under which spark develops, I will use an electrometer of still greater refinement—one which, although invented long after the Henley device, does not differ greatly from that instrument in the essentials of its operation. The electrostatic voltmeter operates on the principle of attraction rather than repulsion. On C the left is a photograph of our meter. It dates from the 1950s, and is therefore calibrated in units whose defining assumptions would have had little meaning to Faraday. But we can regard the scale divisions as arbitrary units of attractive force; let me explain this. On the right is a much-simplified diagram of the meter’s internal mechanism. A movable plate B is mounted on a pointer which pivots at C and is held in an equilibrium position by a very light spring. Plate A is fixed in place. When the plates are oppositely electrified, they attract one another; and plate B will move upward until its force of attraction is balanced by the spring. The pointer’s angle of displacement then reflects the amount by which the spring has been stretched, and therefore, also, the force of attraction between the plates. The scale divisions are so marked as to represent, broadly, equal increments of that force. 5 We will connect the electrometer’s plates to a Wimshurst machine. I have separated the machine’s terminals by about a millimeter or so (VIDEO BEGINS). This is not really accurate, since true volt-meters must take into account both the plate separation and effective plate area, both of which vary as the reading increases. But in the meter we are using, the correction can be ignored for our purposes. 5 �8 Next, I will slowly crank the machine—and notice that the meter rises until a spark develops, at which point the needle suddenly falls. As I continue to crank, the meter repeatedly exhibits this pattern of rise to a maximum, followed by abrupt descent when the spark passes. The maximum is not always the same; but there always is a maximum, and the subsequent descent always coincides with the spark. The regular association between the meter’s descent and the spark suggests a more pointed question: “What is the condition between the terminals just before the spark passes?” Whatever that condition is, it evidently results in spark each time it occurs. And since the electrometer consistently develops a maximum reading just prior to each spark, it seems very likely that the electrometer is indicating precisely that condition which, when it reaches a certain degree, results in spark. What, then, is the nature of that condition?  Faraday thought of the spark—and, for that matter, all instances of electric discharge—as the breakdown of an antecedent state of stress in the region where the discharge takes place. Faraday calls that region, or the material which may occupy it, the “dielectric.” Here is his description in the Twelfth Series: All the effects prior to the discharge are inductive; and the degree of tension which it is necessary to attain before the spark passes is therefore ... a very important point. It is the limit of the influence which the dielectric exerts in resisting discharge; it is a measure, consequently, ... of the intensity of the electric forces in activity. This golden passage finally lends imaginative content to the term “intensity,” which seemed so questionable to Faraday at the outset of the Third Series. The chief manifestation of electrical action is a condition of tension in the region between two surfaces, and that action is said to possess intensity commensurate with the degree of that tension. “Intensity,” then, characterizes the action; “tension” the region or material that experiences that action. The distinction between intensity and tension is a subtle, but a natural one. We find a comparable distinction in two descriptions of Odysseus’ great bow in Book 21 of the Odyssey. The suitor Antinous knows the bow in terms of its own strength, which makes stringing it so difficult. He warns the crowd: 6 6 Homeric passages translated by Gilbert Murray. �9 “For not easily, I think, is this polished bow to be strung.” (line 90) (The image in this slide is that of a fifth-century Theban coin.) But once the bow is strung and in action, it is known by the thrum of its string, the sign of surpassing tension: 7 And Odysseus held it in his right hand, and tried the string, which sang sweetly beneath his touch... (line 408) Just as Odysseus’ stout bow reveals its strength through the superlative degree of tension it creates in the string, so electric action reveals its strength, or intensity, in the form of tension in the material between oppositely-charged electrodes. Intensity and tension are two different rhetorical aspects of electrical action: “intensity” characterizes the action itself (corresponding to the bow); “tension” characterizes the material or region which experiences that action (analogous to the bowstring). Do not underestimate the scientific importance of such metaphorical images as those of string and bow. Without them, or something like them, our understanding of natural powers would degenerate into a merely formal correlation of numbers with numbers. But any reader of Faraday quickly discovers that Faraday has little interest in symbols, numerical or otherwise. Faraday is constantly alert for legible images that convey the essential character of nature’s beings and powers. What is so remarkable about Faraday’s experimental practice is how much of it consists in allowing the phenomena to reveal their own images. 8  7 8 Illustration: detail from an etching by Theodoor van Thulden, part of a series produced in 1632–33. Fisher, Howard, “The Great Electrical Philosopher,” The College, XXXI,1 (July 1979). �10 Faraday’s interpretation of electrical discharge as being essentially a release of antecedent tension departed sharply from the then-accepted account, represented here on the left. Conventional thinking posited a buildup of opposite electric fluids on the 0 5 ed buildup of . Sup_ 1• e (+) electric fluid pos11v sed buildup of . Supo. (-) electric fluid negative j Tension surfaces between which spark took place. As those fluids accumulated—or so the account maintained—the inherent repulsion of like portions of fluid, combined with the mutual attraction of unlike portions, would eventually propel the electrical substances across the gap to combine with and nullify one another. Notice that the conventional view recognizes no role for the space or material between the charged surfaces; all action is ascribed to the electrical fluids. Faraday’s view—represented on the right—reverses the order of priority by focusing on the gap rather than the bodies which it separates, ascribing tension to the gap, but assigning no causative role to the adjoining bodies, nor to any supposed buildup of electricity upon them. If the dielectric material occupying the gap is capable of sustaining high degrees of tension, it constitutes what we call an “insulator”; but all known insulators, including air, have a limit to the tension they can sustain, and when this limit is exceeded, they break down, electrically speaking. The release of tension associated with that breakdown is disruptive discharge, or spark. In contrast to insulators, the materials classed as “conductors” are incapable of withstanding any tension at all; they break down under the slightest degree of electrical tension, and the condition of continuous breakdown under tension is how Faraday understands “current” in a conductor. Thus the electrometer’s pattern of rise and sudden fall in our spark experiment gives us reason to believe that the electrometer measures that very tension—or its rhetorical counterpart, intensity. 9 How does it do so? If you recall our earlier diagram of the electrometer’s inner workings, you will remember that the needle’s Throughout the Eleventh and Twelfth Series we find Faraday using the terms “tension” and “intensity” almost synonymously. 9 �11 displacement indicated the degree of extension of the internal spring, and hence the force on the moving plate—or, rather, the tension in the region between the plates. But of course the condition of the electrometer’s own plates is not what we are interested in! If the electrometer is to function as a measuring instrument, the pointer’s displacement must tell us about some other object—the object whose condition we wish to measure. How is that possible? Consider, from the standpoint of tension, what must be the case when the electrometer plates are connected to the terminals of the Wimshurst machine. When C the machine is operated, electrical tension is established in the air between its terminals D and E. I say that equal tension must therefore develop in the region between the electrometer plates A and B; for if the tensions were not equal, the conductors DA and EB would together have to bear the difference between those tensions. But recall that, for Faraday, a conductor is incapable of sustaining electrical tension. Thus the tension between A and B must be equal to the tension between D and E; and the needle’s displacement will therefore reflect not only the tension between the electrometer plates but the tension between the Wimshurst terminals as well.  Have we gained any fuller understanding of those troubling electrical terms, quantity and intensity? Faraday’s study of the forms of electric discharge, especially spark, led to the idea of electric tension; and that image of tension, in turn, does indeed seem to offer a firmer notion of intensity, namely, the action producing a certain level of tension in a dielectric. But what about quantity? Initially, we looked to the electroscope as an indicator of quantity; but successive refinements of that instrument brought us, not closer to, but farther and farther away from the expected imagery. All our attempts to find, in experience, the imagery that a material substance would ordinarily demand—a localized heap, mound, or puddle—have led us instead back to tension. Why do the phenomena of static electricity seem to lead us so persistently away from “heap” imagery and toward the vocabulary of tension? Might that be a sign that tension is actually more fundamental than quantity? In fact, Faraday already has ample grounds for this view; for if electrifying a body really represents the accumulation of electric substance upon it, we ought to be able to �12 electrify a body “absolutely," that is, without relation to any other body—just as we can fill a glass with water regardless of whether or not we fill any other container with water. But Faraday’s famous Cage Experiment, along with other investigations, showed definitively that no body can be in a “charged” condition at all except through a mediating relation with some other, oppositely charged, body. This means that there is no such thing as a quantity of electricity in itself. Every instance of electric charge is but one element of a mutual relation to which Faraday gives the name “induction”; and in a striking passage in the Eleventh Series he explicitly elevates the relation over the things related: All charge is sustained by induction. All phenomena of intensity include the principle of induction ... All currents involve previous intensity and therefore previous induction. INDUCTION appears to be the essential function both in the first development and the consequent phenomena of electricity. [1178] Furthermore, since all of what Faraday calls the “phenomena of intensity” involve tension in a dielectric, then it is the dielectric, not the so-called “charged” body, which is to be counted as the principal entity in static electricity. In Faraday’s words, In the theory of induction founded upon ... action of the dielectric, we have to look to the state of that body principally for the cause and determination of the ... effects. [1368] 10 If the dielectric is indeed the principal entity in static electric induction, it is easy to see why Faraday devoted so much of the Eleventh Series to studying the dielectric specifically. To that end, he designed the special “inductive apparatus” illustrated here. The appliance on the left is an historical reproduction; 11 Faraday’s own diagram appears on the right. Today we would call this contrivance a spherical capacitor; but it In an omitted term Faraday characterizes the action in question as “molecular.” By this he merely means action at the level of small portions of the dielectric. He does not refer to chemical molecules of the sort propounded by atomic theory—as readers of his 1844 paper, “A Speculation touching Electric Conduction and the Nature of Matter,” will appreciate. See Experimental Researches in Electricity, Vol. II (1844), p. 284. 10 Photograph generously supplied by Dietmar Höttecke; see Höttecke, Dietmar, “How and What Can We Learn From Replicating Historical Experiments? A Case Study.” Science & Education 9, 343–362 (2000). 11 �13 is essentially a Leyden jar consisting of an outer and an inner conductor, with electrical connection to the inner conductor established by a conductive wire terminating at the little sphere on top. Faraday’s experiments established for all time the pre-eminent role of the dielectric in induction. We can emulate Faraday’s induction experiments. 12 In place of his spherical capacitors, we shall use a pair of our adjustable plate capacitors, set to equal plate separations and thus electrically identical. Faraday placed his two identical inductive devices on a grounded metal work surface, so that their outer conductors were permanently connected to the earth while their inner conductors remained free. We will use a heavy copper wire for the same purpose by connecting it to the earth. The righthand plates of our capacitors are joined to it, and are thus in permanent electrical contact. The lefthand plates will be isolated from one another, except when I briefly connect them later. To measure the electrical tension that developed when his devices were charged, Faraday employed a sensitive torsion balance, pictured here on the left. That fine instrument balanced the tension between two electrified spheres against the elastic twist of a slender thread—just as our modern electrometer, as in the diagram we saw earlier, balances the tension between two electrified plates against the elastic stretch of a spring. Both instruments, therefore, serve to measure electric tension. 12 Faraday describes this series of experiments in paragraphs 1208–1214. �14 Faraday possessed only a single balance with which to measure both his inductive devices; but we have the luxury of using two electrometers, one for each capacitor, A B To earth designated A and B, respectively. Let me first outline the procedure we shall be following; then I’ll show some videos of the actual experiment. Faraday began by charging only one of his devices. Similarly, I will connect the Wimshurst machine to capacitor A alone, and crank it until the electrometer B To earth approaches its full scale reading. Capacitor A will thus sustain a definite tension, indicated by the electrometer. Capacitor B, of course, will remain uncharged and will sustain no electric tension. Next I will momentarily join the ungrounded capacitor plates. Now, think about To oatlh what must happen when I do that. The joining wire is a good conductor, so it cannot �15 sustain tension; therefore when contact is made, the electrical condition of both capacitors should instantly change to make their respective tensions equal, and we should expect both electrometers to read the same. That will constitute the first part of our experiment; so now, let us carry out the steps I just described (VIDEO BEGINS). Here is the setup. The copper wire that is appearing on the left will connect capacitor A to the Wimshurst machine... Now I am cranking the machine, and you can see the electrometer rise almost to its full scale. And here is a closeup view of the electrometer; it shows that Capacitor A is sustaining a tension of 2.80 units. I could not fit the second electrometer into this view, but it reads zero—as of course it must, since Capacitor B was not charged. Now I join the capacitors momentarily ... and the tension in Capacitor A falls; we’ll take a closeup look at the electrometer to see the new value... The tension in Capacitor A has fallen to 1.37 units, while the tension in Capacitor B has risen to the same amount, as it must—though, again, I could not include both meters in the same view. Now, this change in tension took place when I allowed Capacitor A to share its electricity with Capacitor B. But since the capacitors are identical, they ought to divide that electricity equally—so that each capacitor should now embrace half the quantity of electricity that resided originally in Capacitor A alone. And the tension in both capacitors is 1.37 units, that is, almost exactly half the initial tension of 2.80 units. Thus as the quantity of electricity in Capacitor A diminished to half, so too its tension diminished to half. Evidently tension is here proportional to �16 quantity! But doesn’t this contradict what we saw in the Third Series? For there, when Faraday charged first eight Leyden jars, and then fifteen, with the same quantity of electricity, his Henley electrometer gave two different readings; and obviously if one magnitude can take on two different values while the other remains unchanged, those magnitudes cannot be proportional. This reasoning, though, overlooks a critical difference between the two experiments. In the Third Series, Faraday was comparing the tension of a fixed quantity of electricity distributed first over eight jars and then over fifteen jars, as illustrated here. The electrometer readings are indeed very different, just as Faraday reported. But our experiment, like Faraday’s in the Eleventh Series, compares the tensions of different quantities of electricity in one and the same capacitor. The two experiments are not comparable, because in the earlier exercise the physical environment underwent significant change—from a smaller number to a greater number of jars— while in the later experiment the environment did not change: the electrometer measured the variation of tension in one and the same capacitor. Clearly, the physical environment affects how much tension a given quantity of electricity will develop. This should not surprise us, since that environment includes the dielectric; and we have already seen how central is the role of the dielectric, according to Faraday’s thinking. The next step in Faraday’s experiment, and in ours, will confirm that central role by showing that different dielectric materials develop specifically different tensions. Faraday filled the air space in one of his devices with various substances; and we shall do the same to our capacitor B by inserting a sheet of glass between its plates. Then we will run through the same experimental sequence as before; but remember that this time, our capacitors will no longer be identical. (VIDEO BEGINS.) You see I have mounted a glass sheet between the plates of Capacitor B. �17 And again we connect Capacitor A to the Wimshurst machine, and charge it to an initial tension.... Its electrometer reads 2.83 units, nearly the same as before, while of course the other electrometer continues to read zero. Again I briefly join the two capacitors together; and the electrometers once more display equal deflections—as they must, since the tensions have to be equal. But notice that this time the tension is not equal to half the original tension... Instead the tension is only 1.02 units, roughly one-third of the initial tension. How shall we understand this? When Faraday obtained a similar result with his spherical capacitors, he concluded that the apparatus containing a solid dielectric had, in his words, “a greater aptness or capacity for induction” than the apparatus whose dielectric was air. To see what he means by this phrase, let us analyze our results in the same way that Faraday interpreted his. When I joined the two devices, the charged capacitor gave some of its electricity to the uncharged capacitor. Specifically,:  The capacitor with air dielectric lost a certain quantity of electricity, and its tension decreased by 1.81 units.  The capacitor with glass dielectric gained that same quantity of electricity, but its tension increased by only 1.02 units—a much smaller amount. Air dielectric-greater change Glass dielectric-smaller change I~ \ I~ \ by 1.81 units by 1.02 units �18  Thus one and the same quantity of electricity is associated with lower tension when the dielectric is glass, and higher tension when the dielectric is air. Evidently, then, “greater capacity for induction” means the ability to sustain the same quantity of electricity at a lower tension. Or, equivalently, it denotes the ability to sustain a greater quantity of electricity at the same tension. We could go on, as Faraday does, to show that a dielectric’s “capacity for induction” depends on its dimensions as well as its specific material. But the main point is clear: where static electricity is concerned, our only access to electrical “quantity” is indirect—through the measurement of tension, 13 taking account of the medium’s capacity for induction. And thus we must regard electrical quantity as only an alternative rhetorical expression for tension—a special figure of speech. Recall Faraday’s earlier remark, that we have to look principally to the state of the dielectric for the determination of the electric effects. In contrast, he described the supposedly “charged” conductors in this almost dismissive way: The conductors ... may be considered as the termini of the inductive action.... [1361] Charged bodies, then, are merely the boundaries of electrical action, not its cause! To say that a body is “charged” no longer labels it as the source of electric effects, but merely the place where a medium that does sustain tension switches to a medium that does not. With this characterization, Faraday has effectively turned the conventional order of causal priority on its head. Charge is no longer prior to tension; rather, tension is prior to charge. Whatever else this may mean, it fatally undercuts the notion that “charge” is the name of an electrical substance, for—to use an Aristotelian formulation that would have been quite foreign to Faraday: “How can a non-substance be prior to a substance?” 14  I hope I have conveyed how thoroughly Faraday’s account of electricity inverted the conventional understanding. At the same time, I hope it is clear that Faraday did not arrive at his unorthodox view through polemic or disputation. He did not marshal evidence so as to refute the established conceptual scheme. In fact, at least in the Experimental Researches, Faraday hardly ever engages in “collecting evidence,” any more than he engages in symbolic mathematics. Instead, he looks directly to nature showing itself. Classic doctrines of scientific “method” emphasize putting hypotheses and conjectures to the test, establishing a preponderance of evidence for or against them. For electricity undergoing discharge, as Faraday shows, the ballistic galvanometer offers an alternative measure of quantity. But while it might seem obvious that when electricity discharges, its quantity in discharge must be the same as its quantity prior to discharge—when it was still static—the problem of correlating the measures of static and dynamic electricity would prove to be a knotty one. It would eventually become the problem of relating the electrostatic unit to the electromagnetic unit, the problem that would lead Maxwell to his electromagnetic theory of light. 13 14 Aristotle, Physics, Book I (189a34) tr. Cornford. In the present case, how can tension (not a substance) be prior to electric fluid (a substance)?—implying that electric “fluid” is not actually a substance after all. �19 Such an approach is suited to an alien world, a world indifferent to human understanding, a world in which, as has been said, “nature loves to hide.” 15 Faraday’s world, on the contrary, shows itself in forms that may challenge our understanding; but they are not incommensurable with it. Faraday’s science flourishes in a world that is fit for us, a world that is preeminently knowable. How did Faraday manage to nourish a scientific outlook so little influenced by conventional scientific doctrine? A customary answer to this question singles out Faraday’s lack of a conventional education. To be sure, Faraday had little formal education and was largely self-taught; but the materials of his self-education were steeped in established knowledge. As a bookbinder’s apprentice, he read volumes of the Encyclopædia Britannica while engaged in binding them. By his own account he benefited greatly from Jane Marcet’s Conversations in Chemistry, a lovely book which, however, reliably held to established and accepted teachings. 16 Through the generosity of a friend of his employer, Faraday was able to attend lectures by Humphrey Davy, an establishment figure in science if there ever was one. I do not think it was ignorance of established science that explains Faraday’s relative indifference to it. Much of his practice in “reading the book of nature” 17 points instead to his religious tradition.  Faraday belonged to a very small Christian denomination, the Sandemanians, a dissenting offshoot of the Church of Scotland. Sandemanians eschewed theology and had no established clergy; instead, the Bible was the central source of guidance in every aspect of their lives. Reading the Bible demanded no special credentials, for it was written in human language for the sake of human understanding. 18 Similarly, they saw the natural world as having been created as a gift and a fitting home for mankind. Like the biblical text itself, the created world was seen as a channel of God’s communication with the human race. You can see how such views concerning nature could inform Faraday’s methods of natural investigation. If natural phenomena show themselves in terms we can grasp, they will not need to be expressed mathematically—or, for that matter, through any other external symbology. We see from Faraday’s own example that the study of nature requires patient and prolonged labor—but much of that labor stems not from nature’s recalcitrance but from our own sluggishness to put familiar thought patterns aside—what Faraday once called “mental inertia” 19—and allow the phenomena to speak to us directly. For Faraday, at least, the means for cultivating an ear for nature’s 15 Heraclitus, B123 Jane Marcet never sought to break new scientific ground; but by composing instructional texts that were explicitly directed to young women, she conspicuously broke new social and educational ground. 17 Geoffrey N. Cantor, “Reading the Book of Nature: The relation between Faraday’s Religion and his Science” in Faraday Rediscovered: Essays on the Life and Work of Michael Faraday, 1791–1867. The Macmillan Press, Ltd. (1985). 16 See David Gooding, Michael Faraday, 1791–1867: Artisan of Ideas. http://www.bath.ac.uk/~hssdcg/ Michael_Faraday.html, 15 June 2002; accessed 4 September 2023 through the Wayback Machine. 18 See Faraday’s “Observations on Mental Education” (1854) in Experimental Researches in Chemistry and Physics (1859), p. 463 19 �20 dialect and an eye for its forms are practical rather than analytical. Before he asks questions in speech, he asks them in practice; such are Faraday’s experiments. Nevertheless, while Faraday’s mode of experimenting clearly reflects central elements of the Sandemanian outlook, it would be a mistake see him only as dutifully putting the Sandemanian creed into action. Faraday just doesn’t write as though he were feeling the weight of doctrinal obligation. His prose, both in his laboratory Diary and in the published Researches, is simply too fresh, too lively, too responsive to what just happened. There is a palpable difference between being open to nature and observing a code of being open to nature. I invite you to think about that difference— the difference between responsiveness and responsibility 20—and how it plays out both in consciousness and in speech. But for now let us return to the terms “quantity” and “intensity,” the two candidates for electrical measure; for as regards their lucidity, I think we will have to acknowledge that the terms have effectively exchanged places. The term intensity, which Faraday initially found “more difficult to define,” has gained considerable clarity, since Faraday has been able to assimilate to it the figures of speech associated with tension; and we may now understand electrical intensity as commensurate with the degree of tension developed in a specified region. But the term quantity, which Faraday previously thought “sufficiently definite as to sense” has instead become highly questionable. For the “definite sense” of that term rested on the image of heaping up or accumulation of electrical substance; and we have seen how that image has repeatedly failed to find any grounding in experience. Moreover, now that Faraday has identified the primary electrical entity as being the dielectric under tension, not the so-called charged body, any idea of “quantity of electric substance” can only be regarded as a merely verbal one—a figure of speech. Under such circumstances, would it not behoove any responsible thinker to avoid the term “quantity of electricity” altogether? And yet Faraday continues to speak of “quantity of electricity” throughout the remainder of the Eleventh Series, and in the Twelfth, Thirteenth, and Fifteenth Series. Why would he do this? Faraday nowhere speaks directly to that question as regards electrical terminology; but he does address a similar one in connection with the language of atoms. Some of you have read, and some of you will read, his 1844 paper, “A Speculation touching Electric Conduction and the Nature of Matter.” 21 In that essay, after having reviewed his many reservations about the theory of atoms, and hence also the atomic language that takes their existence for granted, he nevertheless admits, I feel myself constrained, for the present hypothetically, to admit them [that is, atoms], and cannot do without them. Here, then, is another instance where Faraday feels obliged to make at least provisional use of a terminology that has not been grounded in phenomena. A doctrinaire purist would have avoided such a compromise; but Faraday’s openness Contrast, for example the Knight of Faith in Kierkegaard’s Fear and Trembling with the rule-inferring “insomniac” who, reflecting on Abraham’s willingness to sacrifice Isaac, confidently deduces, “Oh, I see how it works: you raise the knife, and then suddenly there’s a ram!” 20 21 Experimental Researches in Electricity, Vol. II (1844), p. 284, esp. page 289. �21 extends to language as well as to experience, for each of these must evolve along with the other. Natural phenomena show themselves in forms and images that human beings can apprehend; and those images continually try to shape a language that is anchored in the phenomena. But such a language requires discovery, interpretation, and adeptness; and these in turn require time, patience, and love. As we do not expect to take in a dialogue, or a drama, on first reading, we must not expect to “perform” experiments once only and then set them aside. We must live with them, enter into them, and try them again and again. The idea is less to get the right answer, than to capture the right idiom. The book of nature deserves multiple readings; and no two of those readings are likely to be quite the same. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Page numeration Number of pages in the original item. 21 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource What is the Measure of Electricity? Description An account of the resource Transcript of a lecture given by tutor Howard Fisher on February 23, 2024 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "Faraday made use of numerous electrical measuring instruments; but what, exactly, did they measure? What properties of electricity are "measurable" at all? Faraday's efforts to identify these properties raised a question which Meno would have recognized: how can we know the properties of electricity unless we first know what electricity actually is?" Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2024-02-23 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Electricity Faraday, Michael, 1791-1867 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_What_is_the_Measure_of_Electricity_2024-02-23 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/b227db32913978575ac398583d8bd9cb.pdf da0c883f6eacef07a660e0b643760a2f Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Page numeration Number of pages in the original item. 24 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Alternating Current Description An account of the resource Transcript of a lecture given by tutor Howard Fisher on April 26, 2023 as part of the Dean's Lecture & Concert Series. The Dean's Office has provided this description of the event: "The technology of alternating current was developed almost entirely in response to the emerging demands of large-scale power generation and transmission. Perhaps for that reason, educational institutions tend to treat it as a subject of practical rather than theoretical interest: alternating current is studied extensively in engineering schools, more perfunctorily in physics departments. But alternating current reveals a number of perplexing electrical phenomena that carry deep theoretical significance. Maxwell understood these phenomena through several remarkable analogies, couched in a highly metaphorical terminology that later generations viewed with suspicion and disdain. Maxwell, though, understood that analogy and metaphor are the most intellectually responsible forms of expression when we are reaching towards theories that lie, as yet, beyond our grasp." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2023-05 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Electric currents, Alternating Maxwell, James Clerk, 1831-1879 Relation A related resource https://digitalarchives.sjc.edu/items/show/7751 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Alternating_Current_2023-04-26_text https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a6b0e707f0e7b6e1e9935ba88756f3b4.pdf 3db26f32f3d7a5bb28d89ba2438c6768 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data paper Page numeration Number of pages in the original item. 26 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Faraday's Galvanometer Description An account of the resource Transcript of a lecture given on February 22, 2023 by Howard Fisher as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "Today we think of the galvanometer as a device that measures electric current. But Faraday’s galvanometer principally measured charge rather than current. That distinctive capability (which modern instruments do not possess) played a vital role in shaping Faraday’s understanding of magnetic lines of force—and especially their physical character. Faraday’s galvanometer shares essential mechanical characteristics with our old friend the pendulum. I will discuss the design of Faraday’s instrument and demonstrate its distinctive action with the aid of a simple homemade model." Creator An entity primarily responsible for making the resource Fisher, Howard J., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2023-02-22 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Subject The topic of the resource Faraday, Michael, 1791-1867 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Faraday's_Galvanometer_2023-02-22 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/687a2f5ea81153a3c2ddfaf656313fca.pdf 475f45ae3bea6ffa4783000b6cc26a39 PDF Text Text William Donahue, November 5, 2021 What would Kepler say to Einstein? The more I think of this topic, the more appalled I am at my hubris in proposing it. “Really, Mr. Donahue (you might be thinking), “aren’t you just using Kepler’s name to throw out your own rash thoughts into an arena in which you have no business contending?” As for the arena, I have no defense, other than that here at St. John’s we routinely contend in contests where, by standards accepted elsewhere, we have no formal qualifications. We do it anyway, unapologetically. We do not expect to establish new truths, nor to overthrow established theories. But we do hope that in this rather mad undertaking we may gain for ourselves a little more understanding, both of the amazing universe we live in and the powerful thinkers and their remarkable insights into that universe. In presuming to offer advice to Einstein (ostensibly in the persona of Kepler), I am on much shakier ground. The theme of this lecture sprang from a lecture I gave some years ago at Johns Hopkins, which had to do with Kepler’s introduction of physical principles into astronomy. In the question period, I was asked what could be learned (if anything) from Kepler’s views on what constitutes a good hypothesis. I replied that, more than merely accounting for phenomena, it would have to be based in physical reality. This occasioned much rolling of eyes, presumably at my extreme naïveté, which led me to ponder, in the ensuing years, what Kepler might have understood by “physical reality,” and whether a similar mode of understanding might be applicable to more recent physical theory. At length I thought about Einstein’s replacement of Maxwell’s “luminiferous ether” with a postulate asserting the constancy of the speed of light in all possible inertial frames of reference. This seemed to me a theoretical move about which Kepler would have been extremely skeptical. The idea of the ether, after all, was �introduced into electromagnetic theory in order to provide a medium for electromagnetic waves, whose existence could be demonstrated. These waves were conceived as being similar to sound waves in air and other fluid media, and the idea of a wave in nothing, without any medium or substrate, was (I believe) calculated to raise objections from the community of physicists. And with them, I believe, would be Kepler, were he still around to express his views. Now of course Kepler did not say anything about waves, so I must say first of all what criteria he used in evaluating a scientific hypothesis, and why I think he would have raised an objection to Einstein’s rejection of the ether. One of the best examples of Kepler’s thinking about the physical reality behind the apparent motion of the planets is found right near the beginning of Astronomia Nova, in Chapter 2. This chapter is a deliberate reprise of Book I Chapter 3 of Ptolemy’s Almagest, but Kepler takes the argument a giant step farther, asking how each of the two kinds of hypotheses (epicyclic and eccentric) could be understood to function in physical reality. And for Kepler, the limits of physical reality are very broad, but are also clear and definite. So here is where we have to begin. Kepler takes it as demonstrated that there are no real, rigid epicycles and deferent circles or spheres in the heavens. The question, then, is how the moving power (supposed for the moment to reside in the planet) could make the planet move in a circle. Like Ptolemy, Kepler considered both the eccentric hypothesis and the concentric-with-epicycle hypothesis; in the interest of time, we will consider only the former. This power, whatever it is, would have two jobs: it would have to be strong enough to move the body of the planet, and it would have to know where to move 2 �it. This would require knowledge or perception of the circular path in the unmarked aethereal air—a function of intelligence and not physics. Now a circle, Kepler says, even for God, is nothing but equality of distance from some point. Citing Avicenna, he says that the planet will either have to “imagine for itself the center of its orb and its distance from it,” or use some other property of a circle to establish its distance and direction. Here’s the example Kepler gives. The planet’s assignment will be to move on the path γεδ, with center β (not occupied by any body or perceptible mark). This center is fixed in position with respect to point α, which is occupied by a body. To make this motion possible, the planetary mover might be endowed with some means of perceiving the angular size of the body α, as it should appear, in succession, at γ, ε, η, δ, continually adjusting its distance from α while moving with uniform speed on the resulting circle. Somehow, it will have to match continually the distances γα, εα, ηα, δα, and so on (calculated by observing the apparent size of the body at α), with the angles γαε, γαη, γαδ, respectively. It will also have to know the direction of the line of apsides, αγ, in the sphere of the stars. As Kepler says, “The planet’s mover will thus be occupied with many things at once,” his implication being that this arrangement makes no sense. “To escape this conclusion,” he continues, “one must assert that the planet pays attention to the point β, entirely empty of any body or real quality, and maintains equal distances from that point.” “Body or real quality”: that is the kind of thing that Kepler was looking for. It would have to be perceptible, somehow, even if it were not the same as objects that we interact with every day. Objects of pure mathematics, such as points and 3 �lines, would not do. Beyond that, Kepler was extraordinarily open to analogies, both mechanical and living. Earlier in the chapter, he considers at length the operations of muscles in the human body, and concludes that animal locomotion would not be a good model for planetary motion; nonetheless, he is very much open to a role for minds in the heavens. In passing, it is fascinating to note that Newton, too, found that the supposedly “natural” circular motion could not be produced by central forces alone. The moving body would have to take into account, in addition to its velocity and its distance from the center of force, the chord from its present position through the center to the opposite side of the circle. Thus it would have to know where it would be on the other side of its orbit! We are now prepared to imagine what Kepler might think while reading Einstein’s world-changing paper “On the Electrodynamics of Moving Bodies.” He would certainly want to know what this “electrodynamics” is. Maybe the best explanation would be to display the two experimental demonstrations that Einstein describes, which (I hope) are now familiar to all seniors. [show two videos, one with a hand moving a magnet through a coil of wire, the other with a hand moving the coil with the magnet fixed.] Kepler might be surprised to see the magnetic needle wiggle, whether the magnet moves inside the coil of wire or the coil moves along the magnet. He might be more surprised to learn that, despite the identical effects of the motion upon the magnetic needle, the theory requires that there be two very different accounts of what is happening. Here is Einstein’s description: 4 �“If the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.” [emphasis supplied] (This is clearly shown by the deflection of the needle in the two cases.) Kepler would, I think, be reminded of the physical difference between planetary motion as conceived by the geocentrists and the motion as seen by the heliocentrists. The phenomena as observed by the astronomers would be the same in both cases, but the physical reality would be radically different. I imagine he would say, “Well, two contradictory accounts cannot both be true, so one of the descriptions must be the correct one.” Einstein, however, went off in a completely different direction. Ignoring conventional ideas of space and time in Newtonian physics, he adopted two principles, which he initially raised as conjectures, but immediately (in his own words) “raised to the status of postulates.” 1. The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. 2. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. A “postulate,” we should recall, is how we usually translate the Greek Ἀιτήμα, meaning “demand, request.” It is a statement that the author requests us to accept 5 �as true, without proof, as a basis for the demonstrations that will follow. Einstein states that he will use these postulates to attain “a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies.” In other words, he proposes to fix the inconsistency between the theoretical accounts of electromagnetic induction, noted above, not by adopting one or the other account as “true,” but by a radical reformation of the foundations of all of physics, by adding these two postulates to Newton’s three “Axioms, or Laws of Motion.” Now I do not think that Kepler would be troubled, in principle, with the idea of a radical reform of physics. But I do think he would be troubled by Einstein’s next sentence. Einstein wrote, “The introduction of a ‘luminiferous ether’ will prove to be superfluous inasmuch as the view here to be developed will not require an ‘absolutely stationary space’ provided with special properties.” Since Kepler believed that God had created a finite, spherical universe with the sun at its center, he clearly was an advocate of a stationary space with special properties, such as privileged places. But aside from this, I will argue that, theology aside, Kepler would have philosophical or methodological objections to abandoning the Maxwellian ether. To do this, I will have to make an excursion into the considerations that led Maxwell and other physicists of the nineteenth century to espouse the ancient idea of an ethereal medium filling all space. The excursion I propose will lead us into some rather elementary physical considerations. These may be the sort of thing that Einstein would think of as “superfluous,” but this is exactly the kind of inquiry that Kepler enjoyed. So let us invite him to join us in considering the “simple” pendulum. [video of pendulum] [video of two loosely linked pendulums] 6 �[video of a number of loosely linked pendulums] [video of Bell wave machine, first with all but one bar clamped, then with the clamp removed so as to create a wave] Kepler was very good at constructing mathematical models of physical actions, but (as we saw in Astronomia Nova) he wanted more from a sound physical explanation. In several places in Astronomia Nova he set out a three-leveled structure: the observational evidence, geometrical modeling of the observations, and a physical account that could underlie the geometry. As for what could serve as a “physical account,” Kepler was open to a very wide range of examples: animal joints and muscles, magnetism, whirlpools and other examples of water flow (such as Heron’s fountain), amusement park rides, oars and paddles, and so on. He seems to have sought examples that would be generally acknowledged as physically real and that could be understood as constituting an analogue to a phenomenon that is felt to be in need of explanation. In proposing such examples, he was often not claiming that the analogy provided a full and adequate account of the phenomenon, only that the physical reality might be somewhat like the example. Sometimes he combined two different analogies in a single diagram, as here (from Astronomia Nova Chapter 59), where a magnetic planet (the big black circle) with a vertical axis (note the arrowhead at the top) is alternately attracted and repelled by the sun, but is also being propelled by a boatman with a pole (or perhaps an oar). He also candidly admits the provisional or conjectural nature of some of his analogies: for example, in Ch. 57 he writes, “I am satisfied if this magnetic example demonstrates the general possibility of the proposed mechanism. 7 �Concerning the details, however, I have doubts...There may be absolutely no material, magnetic faculty that can accomplish the tasks entrusted to the planets individually…” His point is, that wherever possible, it is preferable to invoke a physical force or power such as magnetism or weight, but if all else fails, it is permissible to invoke mental or animate powers. So let’s think about what Kepler might hope for as a generalized physical metaphor or underpinning for wave phenomena. To help us, I’d like to bring back the Bell wave machine. [Bell video] We may think of the machine as an assembly of linked pendulums. Each crossbar is attached to a longitudinal torsion bar that runs the length of the machine. If the crossbar at the end is displaced and released while the bar next to it is clamped in place, it oscillates while all the other crossbars remain at rest. [video with just one bar moving] It is acting as a torsional pendulum. When its displacement is at its maximum, the twisting force is also maximum, and the bar, when released, moves in the direction of the force, towards its rest position. But when it gets to its rest position, it has acquired some speed, which carries it past the rest position to a new maximum displacement. If we remove the clamp from the next crossbar, the motion of the first bar twists the torsion bar, which then imparts that twist to the next crossbar, which in turn adds a twist to the torsion bar, and thus the twisting motion is passed on. It’s much like those pendulums we saw earlier, that were connected by springs. So it appears that in order to have a wave in some medium, two things are needed. 8 �1. Individual places in the medium have to be able to move like pendulums: when given an initial push, they will depart from their position, but they then experience a restorative force in the medium that pushes back towards whence they came; 2. The pendulums have to be linked so that the swinging of each of them is communicated to neighboring pendulums. When they are tightly linked, the wave move quickly through them; when loosely linked (as the pendulums were), the motion is communicated more slowly. This can be applied to electromagnetic waves in a general way quite directly. Consider this simple assembly of two collinear pieces of metal (called a “dipole”), colored red, connected to the output of a device (the transmitter) that makes electricity slosh back and forth between the two sides of the dipole. The transmitter/dipole assembly is our initiating pendulum: the natural tendency of the electricity is to create an equality of tension or “potential” between the two sides of the dipole; the transmitter provides the pushes that keep the electricity oscillating. What happens, as we know from Faraday, is that the electric and magnetic forces generated by this assembly ripple out through the surrounding space. Electric tension builds up in space, and as the tension is released by the restoring force of the medium, this release constitutes an electric current, which generates a magnetic force, which grows and decreases in a similar way, generating an electric displacement current, and so on. This action continues, and constitutes what we call a “radio wave.” And there is strong evidence that light, too, is just such a wave. 9 �So the wave metaphor, built up out of linked motions that act like pendulums, evidently applies to electricity, magnetism, and light too, in a direct and comprehensible way. In the face of such evidence, Kepler might say, is it not a retrograde step to dismiss as “superfluous” the medium in which the actions foundational to the observed phenomena take place? Isn’t adopting the “postulate” that the speed of light in a vacuum is constant too much like postulating (as astronomers had done for thousands of years) that all celestial bodies move with uniform circular motion? This, then, is what I think Kepler would say to Einstein. And I could stop here, but in all fairness, we need to let Einstein respond. I think Einstein would point out two problems, one cosmic in scale and one inherent in electromagnetic theory as it was then formulated. The first is related to the ether itself: are we moving through it, or is it moving along with us? If we are moving through it, then waves would seem to us to be moving faster in some directions and slower in others. But when we measure the speed of light, it seems to be pretty much the same in all directions. It gets worse: the speed of electromagnetic waves, as deduced from Maxwell’s equations, is determined by the ratio of the two fundamental electrical and magnetic constants. These are the physical constants that seem, both conceptually and experimentally, to be independent of coordinate systems, and that determine the “springiness” of the medium. In junior lab, every spring, we do a lovely experiment that gives us a number for this ratio, and our number (perhaps surprisingly) is pretty close to what the textbooks say it should be. And further, when (in a second beautiful experiment) we measure the speed of light directly (using a tape measure and a tuning fork), the number we get is not too far from the ratio in the previous experiment. So we are left with a dilemma: either the fundamental electromagnetic 10 �constants are somehow coordinate-system dependent so that they match the measured motion of our coordinate system through the ether, or there is a preferred coordinate system for the entire universe in which we happen to be absolutely at rest. It’s hard to imagine what the first horn of the dilemma even means, while the second horn basically throws out all cosmological thinking since Copernicus. We can call this problem the “ether wind” problem. The other thing Einstein would say is that in his view, this whole dilemma associated with the ether is just a side issue: his concern, which he thinks was a much more fundamental problem, was the asymmetry in the way Maxwellian electrodynamics applies to magnets and wires. As Einstein put it, in the passage quoted earlier in this lecture, “If the magnet is in motion and the conductor is at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.” Restating this in a slightly different way, if you have an observer who sees the magnet moving through the coil of wire, she sees an electric field in the space surrounding the magnet, and this field embodies a definite amount of energy. Another observer, moving uniformly along with the magnet, will see no electric field, and the same space will now be devoid of energy. But energy is a conserved entity. So we have a theory that has lost what later physicists have called “local 11 �reality”: for one observer some real thing is there which according to the laws of physics is not there for the other observer. Thus, the whole idea of objective reality has broken down, which, you may imagine, is a big no-no for a physical theory. So Einstein’s response is to adopt the two “postulates,” mentioned earlier, by which he will solve both the ether wind problem and the objective reality problem. His claim is that abandoning the idea of a physically real medium in which electromagnetic waves occur is a price worth paying for saving the claim of physics to represent objective reality. So, Kepler, what might you have had to say in response to Einstein’s powerful reply? It is clear from our brief look at Astronomia Nova Ch. 2, and from many other places in the book, that Kepler believed it was wrong to allow a physically unsupported postulate (uniform circular motion) to overrule principles, even if they are provisional or conjectural, that are supported by physical arguments. As he put it, we need “a body or real quality” as a foundation. Although he was open to a wide range of examples and analogies that would constitute “real qualities,” a simple rule lacking such support, such as the constancy of the speed of light, would not do. Although Kepler would have acknowledged it as a clever and ingenious solution, it would remind him too much of the many astronomers of his day who rejected his “celestial physics” (a term featured prominently on the title page of Astronomia Nova) and reverted to the circular tracks and angelic movers of the old astronomy. What would have to be done instead, Kepler would say, would be to solve the ether wind problem and the local non-reality problem without abandoning the physical 12 �basis of electromagnetic radiation. This would surely be a difficult task. But would it be more difficult than establishing a sound physical basis for planetary motion? I’ll finish this lecture by saying a few things in support of Kepler’s advice. Not that I consider Kepler’s position is in need of support—it seems to me one of the really deep questions—but to show that, despite what seems to be the unanimous acceptance of Einstein’s two postulates, there has been, and continues to be, a quiet but respectable undercurrent among physicists, a willingness to wonder whether despite the remarkable success of Einstein’s relativity theory, its unsupported second postulate might turn out to have been a mistake. The first direct attempt to determine a possible motion of the earth through the ether was carried out in 1881 by A. A. Michelson, in Potsdam, Germany. Michelson nonetheless noted that the problem had already been approached by Stokes (in 1846) and later by Maxwell (1878). The more famous Michelson-Morley experiment, which used a much larger instrument, followed in 1887. These purely experimental results, which showed no measurable motion, set a problem for the theorists to solve. The approach that appeared most promising at first was that some of the ether was being dragged along by the earth; however, no one succeeded in demonstrating such a phenomenon. A competing account was suggested by Oliver Heaviside’s conclusion, on the basis of Maxwell’s electromagnetic theory, that electromagnetic fields contracted along the direction of their motion. In 1895, H. A. Lorentz published an article in which he proposed that, like electromagnetism, the forces that hold the particles of material bodies together also contract in the direction of motion. He writes, 13 �“Thus one would have to imagine that the motion of a solid body (such as a brass rod or the stone disc employed in the later experiments) through the resting ether exerts upon the dimensions of that body an influence which varies according to the orientation of the body with respect to the direction of motion. … “Surprising as this hypothesis may appear at first sight, yet we shall have to admit that it is by no means far fetched, as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we are able at the present time to make this assertion definitely.” In 1904 (the year preceding Einstein’s Special Relativity article), Lorentz published a more thorough treatment of the same basic idea. In the interim, Poincaré had argued that electromagnetic forces alone were insufficient to produce Lorentz’s contraction, and added an additional hypothetical force. However, as Lorentz notes, Poincaré also objected to this piecemeal approach. Lorentz writes: “Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelson’s negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light.” Lorentz believed that by “starting from the fundamental equations of the theory of electrons,” he could “treat the subject with a better result.” The article that followed is a tour-de-force of Maxwellian analysis, packed with equations dealing with such matters as the electromagnetic inertia of electrons. At this point, Lorentz’s fundamental revision brought his theory, which avoided Einstein’s second postulate, into agreement with Einstein, as far as the observations were concerned. But by a strange turn of events, a series of 14 �experiments by Walter Kaufmann (involving the mass of high-speed electrons rather than motion through the ether) appeared to show that the Einstein/Lorentz predictions were wrong, and that rival theories of Max Abraham and Alfred Bucherer produced more accurate results. Lorentz conceded that Bucherer’s theory was “decidedly unfavorable to the idea of a contraction, such as I attempted to work out.” Einstein, on the other hand, acknowledged that the Abraham and Bucherer theories fit the data better than his own, but wrote, “they have a small probability of being correct since they produce complicated expressions for the mass of a moving electron.” In other words, theoretical simplicity trumps agreement with the data! But now Planck entered the fray, with a meta-analysis of Kaufmann’s numbers, which tipped the balance back in favor of Einstein and Lorentz. And in 1914, refined experiments by Günther Neumann (using Kaufmann’s own equipment with some modifications) appeared to favor Einstein decisively. The curious result of this was that, even though Einstein’s and Lorentz’s theories were essentially in agreement in most of their predictions, Einstein’s ether-free approach came to be viewed as the victor. Nevertheless, attempts to find an “ether wind” continued. The most extensive work was by Dayton Miller, a prominent American physicist who was president of the American Physical Society in 1926. His measurements extended over nine years and, by one account, comprised over five million individual measurements. His primary aim was to show a difference between the ether drift at low elevations (essentially none) and at the summit of Mt. Wilson. He claimed to have found that the solar system is moving towards the constellation Dorado through the ether at a speed of 227 km/s, a result that was similar to independent measurements by the French astronomer Ernest Esclangon and the Swiss 15 �astronomer Leopold Courvoisier. These results have been questioned on various grounds, but were well-received at the time and have never been adequately repeated, according to one scholar whom I know personally and whose work I respect. It appears that the remarkable success of both the special and general theories of relativity have made a search for ether-drift an unattractive career move. However, questions have more recently crept in from an unexpected source: quantum mechanics. Physicist John Bell, in 1964, came up with a purely mathematical theorem that established certain numerical limits to the relatedness of states of certain particles (in this case, polarizations of so-called “entangled photons”). The assumptions upon which the theorem was based were, first, that the particles involved really and actually possess the properties involved (the criterion of “reality”), and second, that communication among the particles cannot occur at speeds faster than light. Naturally, this set a challenge for experimenters: violate Bell’s theorem! The definitive experiment, by Alain Aspect, came along in 1982. It violated the conditions of Bell’s theorem, while remaining entirely consistent with quantum mechanics. In practical terms, this meant that one or both of the assumptions that Bell made would have to be abandoned or modified. Physics would have to give up the idea of local reality, or of what Einstein called “spooky action at a distance,” or perhaps both. One is reminded of what Kepler wrote in Astronomia Nova when he showed that the classically formulated hypothesis of Chapter 16 is inconsistent with the Tychonic observations. He wrote, Therefore, something among those things we had assumed must be false. But what was assumed was: that the orbit upon which the planet moves is a perfect circle; and that there exists some unique point on the line of apsides 16 �at a fixed and constant distance from the center of the eccentric about which point Mars describes equal angles in equal times. Therefore, of these, one or the other or perhaps both are false, for the observations used are not false. So, in light of the Aspect experiment, what gets thrown out? A variety of solutions have been proposed, but I will conclude this lecture with what John Bell himself said, in a discussion with BBC producer J. R. Brown and physicist P. C. W. Davies. Question: Bell’s inequality is, as I understand it, rooted in two assumptions: the first is what we might call objective reality—the reality of the external world, independent of our observations; the second is locality, or non-separability, or no faster-than-light signaling. Now, Aspect’s experiment appears to indicate that one of these two has to go. Which of the two would you like to hang on to? Bell: Well, you see, I don’t really know. For me it’s not something where I have a solution to sell! For me it’s a dilemma. I think it’s a deep dilemma, and the resolution of it will not be trivial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolution is something like going back to relativity as it was before Einstein, when people like Lorentz and Poincaré thought that there was an aether—a preferred frame of reference—but that our measuring instruments were distorted by motion in such a way that we could not detect motion through the aether. Now, in that way you can imagine that there is a preferred frame of reference, and in this preferred frame of reference things do go 17 �faster than light. But then in other frames of reference when they seem to go not only faster than light but backwards in time, that is an optical illusion. Question: Well, that seems a very revolutionary approach! Bell: Revolutionary or reactionary, make your choice. Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant. Question: Of course the theory of relativity has a tremendous amount of experimental support, and it’s hard to imagine that we can actually go back to a pre-Einstein position without contradicting some of this experimental support. Do you think it’s actually possible? Bell: Well, what is not sufficiently emphasized in textbooks, in my opinion, is that the pre-Einstein position of Lorentz and Poincaré, Larmor and Fitzgerald was perfectly coherent, and is not inconsistent with relativity theory. The idea that there is an aether, and these Fitzgerald contractions and Larmor dilations occur, and that as a result the instruments do not detect motion through the aether—that is a perfectly coherent point of view. Let me finish by briefly summarizing the main points of this lecture. Kepler strove mightily throughout his life to oppose the prevalent idea that astronomy must be founded on hypotheses, and that the fundamental and indispensable hypothesis is the principle of regular, uniform circular motion of all 18 �heavenly bodies. He proposed instead that all attempts to understand the cosmos must be founded in some way upon physical reality. As to what constitutes physical reality, we must use familiar examples to try to understand what is less accessible to us, and we may be led to consider accounts or examples that may at first seem far-fetched. But it is a mistake to limit the range of possible explanations within the boundaries of arbitrary postulates. This is what I believe is the advice he would most want to give Einstein. And this is advice that, despite the rolling of eyes at Johns Hopkins, may retain a degree of cogency today. 19 � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data Word doc Page numeration Number of pages in the original item. 19 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource What would Kepler say to Einstein? Description An account of the resource Transcript of a lecture given on November 5, 2021 by William Donahue as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "Einstein once wrote that there is little value, other than the satisfaction of intellectual curiosity, in studying scientific works of the past. Kepler would not agree, and were he alive today he would criticize Einstein for repeating errors of early science. This lecture will begin by exploring some of Kepler’s views on past failures, and then will apply Kepler’s criticism to Einstein’s views, especially his rejection of the ether. The inquiry will then consider the alternative account proposed by H. A. Lorentz, showing how the contraction of bodies at high velocities was deduced from Maxwell’s electrodynamics without abandoning the ether. The lecture then concludes by considering remarks of more recent physicists, most notably John Bell, on the possibility of reviving the ether to solve the local reality problem in quantum physics." Creator An entity primarily responsible for making the resource Donahue, William H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2021-11-05 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Kepler, Johannes, 1571-1630 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_DonahueW_What_would_Kepler_say_to_Einstein_2021-11-05 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/679f2fc19ff471e71070118826634435.pdf 94b6f125e62c0acb4bc443108886ac78 PDF Text Text On Incommensurability This is an attempt to think about a question that is considered settled and so I must warn you that I may seem to be wasting our time. Like other such ventures you may encounter at St. John’s it is less concerned with what we might be said to know either these days or at any time and more interested in how we think we know it. It is also, as a Wednesday Afternoon Lecture, a bit more of a work in progress than a Friday Night Lecture might be. You and I are in the enviable position of not having to worry what each other will think if you just virtually slip out of this virtual Hall. You are putting up with a good bit already in these odd times; if what is on your screen late on a Wednesday afternoon is not at least moderately absorbing then do something else. Incommensurability in its most general sense describes the relation of two or more measurable things that have no common measure. We might say that the length of a road and the loudness of a shout have no common measure: the road is not twice as long as the shout is loud, or any other multiple or fraction of the shout’s loudness either, though both length and loudness can be measured. Must the two things, road length and shout volume, have some common measure in order that we be aware of both of them at all? Do we have to be able to measure a thing to know it is there? We commonly suppose that the existence of things is found in the evidence of our senses. If I can see the road or hear the shout, then it is there. Seeing and hearing do not have to be the same thing in order to testify equally well to the presence of the things we see and hear. Nobody is meanwhile scandalized that many things have some kind of relation of incommensurability to one another. The situation grows more interesting if two things that could easily be �imagined to have a common measure can be shown to have none. Suppose I could show you two straight lines that cannot possibly have a common measure. They would not be lines in pencil lead or chalk; but if those are the only kind of lines you will accept you had better virtually slip out now. No matter how small the equal pieces may be that I can divide one line into, I would show that no such piece would ever fit a whole number of times into the other line. It seems unimaginable that this should be so. It may be even harder to imagine than the possibility – shall I call it the “opposite possibility”? -- raised in the Meno that all human excellence could be grasped as a simple unity. Let us begin with what measuring is. It seems to be a laying out of something next to something else in order to tell which one is more in some way and which less. Euclid says that a small line measures a larger line if it fits into it some whole number of times. We can, using a looser notion of measurement, easily see who is taller when two stand side by side. But how do we know who will make a better ruler? If there is a unit by which leadership may be measured, it has not yet been discovered. Creon in Sophocles’s Antigone knows that he is being measured by the hard times he must rule in, but he does not imagine that it might be in yielding to Antigone that he would show his worthiness to rule. “To what must I show myself adequate?” cries Creon. Or more literally “What is it? To what kind of happenstance do I arrive commensurable?” The Greek word translated by “commensurable” here, “summetros”, means “sharing a measure with.” Symmetric things in English might seem at least to have their parts in proportion to each other: things can be in balance and hold together by having a common measure. Creon seems worried that there may be situations he cannot rule. He is not wrong to worry: rulers need not only strength but understanding. The human soul itself could turn out to contain parts that share no common measure. Do our powerful fears and desires have any common language with our cautious reasonings at all? To find oneself �incapable of judging a situation for want of a proper measure is perhaps not an uncommon fear. It might be like an anxiety dream in which you have to take an exam in a language you don’t recognize. Anxious democracies often invoke the importance of the things we all have in common, our commensurability; we say that is our strength, what unites us. But those things must be so often invoked because of the strength of what divides us: each of us is a unique self. Some of the things we want most cannot be shared. We find ourselves sometimes desperately wishing a very particular someone could see something in us that is to be found in nobody else. Sometimes we boast of how well we know that no-one else can die for us; it seems to make not only our deaths but our whole lives look at least for a moment like inalienable property and to give us permission to do whatever we want with every precarious moment we are alive. Have I persuaded you that incommensurability has much more to do with us all than we might usually think? Here is one last example, from the Hebrew tradition. Adam, speaking to God after God has tried every other animate possibility before finally solving the question of where to find Adam a companion and has divided the first human in two: “This, at last, is bone of my bone of my bone and flesh of my flesh!” Hear the deep relief. Are the Tyrant and the ordinary Citizen of one bone and flesh as well? Socrates reminds us early in his conversation with Meno that Meno’s father was a friend of Xerxes, the Great King of all Persia, and he means us to glimpse Meno’s inner Xerxes, and our own. Meno has grown up knowing that he is two degrees separate from the absolute ruler of all the lands from Egypt to Turkey to the edge of India. Those who unashamedly take Xerxes as a measure in their drive for greatness are likely to refuse to be run-of-the-mill examples of the generically Human. Sophocles minces no words in titling his paradigmatic tragedy about a man who introduces �himself as “Oedipus the Great”: the play is called Oedipus the Tyrant. We each have something in us that wants to be a God, utterly unconstrained. Whether that leads to philosophy or tyranny or eternal salvation is not clear. Each looks, and may seek to look, incommensurable with our daily life. Incommensurability and the Golden Section My wonder at the Incommensurable was re-kindled a few years ago in a Freshman Mathematics Tutorial when we came to Euclid’s proposition eleven of Book Two, which shows how to divide a line so that the rectangle formed by the whole line and the smaller piece of the division will equal the square on the larger piece. The more famous example is already hiding in I, 47, the Pythagorean theorem. In neither proposition does Euclid say anything explicit about the topic. But it somehow came to mind for me in Book Two. When we have learned how to make the division of the line in Proposition 11, we may wonder, even if we do not recognize what we can now do as dividing a line in the so-called “golden section”: is there a numerical relation between the two pieces into which we divide the given line? Whether or not it exists among numbers the Golden Ratio has been found in countless examples of the great works of art and architecture that have survived their Ancient Greek makers. The Parthenon is only the most famous example. But is that ratio to be found among numbers? That is, could the two pieces into which Euclid’s line is divided be measured by some common unit and would there thus be a fixed numerical ratio between the two pieces? Are there then two numbers whose relation to each other names a specific kind of beauty? Maybe so. We might on the other hand be disappointed to find beauty of any kind to be, as we say, formulaic. And yet, perhaps the example of maximum commensurability, instantly recognizable in its unsurpassed simplicity, is the ratio of the double. Aristotle is not afraid to say that the sound of the Octave is �the embodiment of a 2 to 1 ratio and that it is not only agreeable but beautiful. Why should we like the relation produced by doubling? This is the sort of question that may conceal more depth than we think: what does it mean that beauty could be a kind of order? The two sets of vibrations that coincide every second beat: ONE two, ONE two, ONE two, hundreds of times per second are a kind of hyperallegro march or two-step of 440 against 880 beats per second that we hear as two pitches, a perfect mingling and repetition of Same and Slightly Other. We may not even perceive more than one pitch, the two blend so well. Two faces, neither especially noteworthy, placed side by side and seen to resemble each other will often make us laugh with pleasure; who knows why? What can it be that makes twoness beautiful? “I am dying, Egypt, dying,” repeats Marc Antony to Cleopatra, in Shakespeare’s play, calling her “Egypt” in a rhetorical figure, as though the nation she rules has produced in her an image of itself, a kind of double. “Dying…” he repeats, “Dying.” To say something twice is already to begin a poem, because in a well-crafted poem nothing is only itself, and everything implies everything else. If it were sufficient to call a thing by its name just once in order to say all that it is or all one may have in one’s heart, then perhaps there could be no poetry. The verse structure of the ancient Hebrew Psalms, attributed to King David, is a kind of doubling or rhyming in which the sense of a line is repeated in different form to make a pair, or to reveal that One is Two and vice versa. One could call it “Octavic structure”. The two beginning lines of Psalm 23 :“The Lord is my shepherd/ I shall not want”, might be rendered in prose: “ Since the Lord is my shepherd, I will have all I need.” The first thought is completed and becomes one of a unified pair as we hear that to have the Lord as one’s shepherd must involve freedom from want. The logical undeniability of the second line, given the �premise of the first, retroactively transforms the first line into a proclamation of complete trust and proud allegiance. “I am dying, Egypt, dying, only/ I here importune death a while, until/Of many thousand kisses the poor last/ I lay upon thy lips …” says Antony; the first five words could be his very last if taken to mean: “I have killed myself because they told me you were dead,” but he speaks again, “… Dying…” that is, ‘it is nothing I can draw back from now, it is fully real that I will soon cease to be real.’ And perhaps he is asking himself and his love how a story that seemed so full of strength and new possibility has now brought inescapable death. The poetry of the drama perhaps plays with incommensurability here: the lines are spoken as Cleopatra and her two women servants are about to hoist Marc Antony up to the window of their tower. He is almost immovable and will very soon be unreachably far away, among the dead, and yet he and they are asserting the greatest nearness between him and Cleopatra that these great lovers have ever felt. Two are becoming one as they near the vanishing point. If the question of whether it can be numbered does occur to us about the golden rectangle I think Euclid supposes that we ourselves may be able to supply the surprising answer in the negative with only a little reasoning. It is unfortunately not the kind of reasoning I am confident will succeed in this lecture. I have tried several times to write out a brief and perfectly clear account that shows the two pieces of the Golden Section have no common measure; but without the assurance that audience members could be free to pause the lecture and ask a question wherever something is not making sense to them I have felt that my attempts have failed. There may be an unexpected commonality between Plato and Euclid in that the real life of both authors’ work only manifests itself fully in conversation, and in deeds. I will include my proof �that the Golden Section makes two lines that have no common measure in the written text of this talk, which will be available through the Dean’s Office, and maybe another short proof that the diagonal of a square has no common measure with its side. Neither is terribly difficult to follow. Even one of them might still take up too much of our time together today. For now I ask you to grant me that both these things are provable. [Shall I belabor your sense of wonder with one more corollary? If line A and line B should be shown to have no common measure, then the line that is their sum could be measured neither by any equal division of A or of B. Add them together to make a new line and call their sum C. So we will have three lines, none of which shares a measure with either of the others. If we add A to C then we have a new line, D, the largest of the four, which likewise has no common measure with C and thus with any of the others, and so we may proceed ad infinitum, making new lines, each of which has no common measure with any of its components: an endless array of mutually incommensurable lines growing larger and larger forever. We could have proceeded by subtraction and produced the same result in the direction of the infinitesimal.] Perhaps the beauty people say is to be found in the Golden Section and made visible in the dimensions of the Parthenon and of numerous Ancient Greek sculptures and vases, and paintings would be a kind of opposite to the beauty of the Octave. The Two to One ratio is commensurability at its clearest and simplest, expressed in the very smallest numbers, while the Golden Section is a ratio we cannot express but only continually approach in even the very largest numbers. Yet it too has a kind of sublime geometric simplicity: a line has been divided in two pieces so that the area made by a square on one piece and that made by a rectangle of the whole line with the other piece are equal. �In the language of ratios, which the Freshmen will learn very soon, the smaller piece has to the larger piece the same ratio as the larger piece does to the whole. In that same language the side of a square equal to a unit will have the same ratio to the diagonal of that square as the diagonal has to two of those units. The diagonal is the mean of One and Two, somewhat as the larger piece of the Golden Section is the mean between the smaller piece and the whole. Measure and Counting It was one of the complaints of the traditional moralists of Athens that Socrates was a Sophist, a teacher of slippery arguments by which to win debates in courtrooms or public assemblies and that he could make the weaker argument look like the stronger; that he could make you think that day was night or even was odd. These were some of the accusations brought against him in the trial that led to his death. When Meno’s slave has seen that doubling the side of a square has not given us the length to produce a double square, but rather has produced a quadruple, Socrates innocently asks him to say exactly how long the line would need to be to give us a double of the original square, or just to point to it if he prefers. Plato is well aware that there will be no possible naming of the size in any units that also measure the side of the square, and yet that it is very easy to draw a line, the diagonal of the original square and then to point to it as the right size for the side of a double square. But the side and the diagonal are incommensurable. This would be a good moment to consult a written-out proof, one or two of which I will append to this talk. [ Appendix: Euclid I, 47, which shows that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares of the other two sides can be applied to an isosceles right triangle that is half of a square. Then the squares on its two equal �sides will add together to make an area equal to the square on its larger side which we may call the hypotenuse or the diagonal of our half square. So the square on the hypotenuse has an even number of square units. This is only possible if the hypotenuse itself is an even number of units long. But if the ratio of side to diagonal is expressible in numbers then it has an expression in the lowest possible terms, as e.g. 3:2 are the lowest terms for 6:4. One or both numbers will be odd in such lowest-term expressions. Let our half-square triangle’s sides in their relation to its hypotenuse be expressed in lowest terms. Since this case has a diagonal with an even numbered length, the length of the smaller side must be odd. But since the diagonal’s square is even then we know its length must be also; now to be even is to be two times some number, for that is the definition of “even”. So when we multiply it by itself to get the area of the square on the hypotenuse we will be making a square whose area is ( 2 x N) x (2 x N) where N is just whatever number it needs to be to give us, when doubled, the even numbered length of the hypotenuse of our original triangle. That means our square on the hypotenuse, or on the diagonal is (4 x N x N) square units. So each of the squares on the sides of the isosceles right triangle will be half of that, or (2 x N x N) square units. Aha! Each smaller side of our original triangle, since it has a length whose square is 2 x N x N, must after all itself be even for no even square can come from any but an even side. But we said it had to be odd because the hypotenuse was even and the sides of our triangle were to be expressed in lowest terms. If the side and the diagonal of a square have a common measure, i.e. can both be expressed as whole numbers of the same ‘unit’, then we have shown that the same line must have a length measured by a number that is both even and odd. Since there is no such number, we conclude that our premise must have been mistaken and there is no common measure.] Something happens to a mind that has begun to follow geometrical demonstrations; something that looks like a choice as �it regards the truths of different realms; maybe it shows itself most clearly in the response to the absurdity that results when the square and the diagonal are assumed to have a common measure. It does not look like peace of mind. The Imagination presses to remind us that we may divide the side or the diagonal as finely as we like, and insinuates that anyone with sense would see that somewhere in the realm of the very small there is bound to be a piece so small as to measure whatever other piece from wherever else we might assign it to. The Reason insists that no possible common measure can be found which does not involve the result that an even number must also be odd. Aristotle points out that as to the role of wonder in Philosophy, although Socrates may say Philosophy begins in Wonder, we can see how in some cases wonder must give way to a kind of familiarity and that what would now really produce wonder would be a demonstration that diagonal and side were after all commensurable. More wonder would be there than in the known outcome, namely that they are not and never can be. But I am afraid that the original wonder has not ceased to work on me, and even some familiarity with the proofs has not driven away a sense of aporia about their result. Is the incommensurability that lurks so near the beginning of Geometry really just something to get over? Would that make it like the contemporaneous horrifying discovery that lies could prevail over the truth among the audience of a fair and open discussion? Day can be made to seem night. One must still make up one’s mind if that discovery means no persuasion by words is ever to be trusted, or if something like truth can somehow still be approached. That is the choice I am thinking of. If, as I am suggesting, Euclid well knows the problem of incommensurability, and even expects his more discerning readers to perceive it very early in his book, then we see it has not discouraged him. Is there a common unit that measures both our �thoughts and the world? Let us return to this matter of measuring in its primal form of counting. When we count we could be said to measure the “how much” of something in units. We want to know how big our herd is today, perhaps to see if any lambs were carried off in the night, and we lay down a kind of measuring stick called “one sheep” next to the herd and see how many we can find in it. The unit, one sheep, is not exactly like a fixed length: it may match a small lamb in one instance, a large ram in the next. They are equally sheep and there are two of them. Cattle ranchers talk about how many “head” of cattle are on their ranch, so that the head has become the unit, since cows, bulls, and calves each agree in having one head and so can be counted quickly by counting heads. Each particular sheep’s head is not exactly like any other, any more than each sheep was. “A unit,” as Euclid will say in Book Seven, “is that by which each thing is called ‘one’.” Actual unity is like perfect doubleness, not a thing one finds in one’s hands, (although hands may seem quite a good example of the double,) but it is a way of seeing with the mind’s eye and of talking. Things appear to come in kinds. A kind is a natural unity. It may not be saying too much to say that different kinds must always be somewhat incommensurable. Money is a fiction that lets us pretend that three days of the labor and materials and skills of shoe-making could be equal to one day of the labor and material and skills of housebuilding. Money and the market price allow us to set things as equal that in fact have no common measure. Euclid says that ratios can exist (only) among things of the same kind and that things are of the same kind which can by multiplication exceed one another. No number of houses can exceed a shoe, or even equal one, whatever Mother Goose may say.. Comparing houses to shoes is, as we say, like comparing apples to oranges. What lets us count heads or noses or whole sheep of different ages and sizes without difficulty is the notion of the pure unit: the oneness that any thinkable, nameable thing must have to be a thing at all. Things �that are generically of the same kind can be counted by reference to ones that are all exactly the same as each other. Do we invent these units? Different assemblages of these ones are the different whole numbers. Each such assemblage has in addition to its component units a single unity peculiar to itself: three is one number and is different from all others. It is not possible even to be as a multiplicity without at the same time being a kind of one. Can numbers image The World of Being? It sounds absurd. Yet there are enough numbers for every individual being there is or ever has been. And perhaps there would turn out to be enough groupings of numbers to mirror the groupings of things by kinds, by larger and smaller categories. If everything that is real can be named and reckoned with by computers then we are already deep into the project of mirroring the entire world in numbers. Does it matter if we suppose that numbers are simply a convenient labelling device we have in some unspecified way dreamed up or if we think that the three of three horses and the three of three frogs are both able to be three by some active power, a unifying force at work on them as we might imagine the activity at work on making a horse be and stay a horse at all? It might matter quite a bit. [The Freshmen will soon read a book in which Socrates offers a line divided in four as an image of all visible and knowable things; the primary twofold division is between knowable and visible, and each of those two parts is again divided in two “in that same ratio”, however we may like to think of it, between knowable and visible. Within the example we have to represent the relation between visible and knowable simply as a matter of the sizes of our two pieces of an original line, which itself we may suppose to represent all that is. Should the visible be larger than the knowable or vice versa? If we draw a line, divide it first in two in any way we choose, and divide each piece in the same ratio as the first division, we will produce four lines, the greatest, the least, and the two in the middle. It can be easily proven that the two in the middle will be of equal size. In Socrates’s example the upper division of the �lower part of the whole line represents among the visible those things we would call “visible originals”: trees, people, animals, stars etc.; while the lower division represents “visible images” of those things: shadows, reflections, paintings, and so forth. The lower piece of the upper part of the original line thus must represent in the realm of the knowable the knowable images, which Socrates suggests are mathematical beings: triangles, lines, points, drawable figures and representable numbers, etc. while the uppermost piece of all will represent whatever could be the “knowable originals” or the actual things that are known, whether always through images or perhaps sometimes directly through themselves. The mathematical or “learnable” things would thus be the shadows of the knowables, and the quantity of images of the knowables would match that of the originals of the visibles. Or would they overlap? The things both knowable as images and immediately visible as things would then be the same.] The appearance of things coming in kinds and our capacity to see and grasp it grounds most of how and what we think. When we recognize something at all we are finding it to be a part of some whole, an example or a fragment of a certain kind of unity. Here too we seem to have a choice: shall we notice this power we seem to have, give it a nod, and just get on with our work of understanding and re-shaping the world, or shall we try to dwell upon it and wonder at it? Maybe returning to it would modify our urge to reshape the world a little. On the other hand it may be so near to our root that to dig it up would leave us nothing to get near it with. Just the same, let us try a little right now. When we encounter something that appears especially unified, that brings together many parts in many ways to make a One, we feel delight. We call it beautiful, whether it be a painting or a melody or a story, or a face, or a sunset. We locate unities within larger unities: person, family, city, nation, Cosmos. When we meet something that strongly resembles something of a different �kind in some respect, we delight to bring the two together in our speech. We call it making a metaphor. “All Flesh is Grass”. Is it a stretch to describe this behavior as a kind of counting or measuring? It seems to have little to do with the measuring about which Nietzsche complains. That, he says, is a devaluing of the only world we have, by the false invention of another better world somewhere else, whose beings we claim to glimpse and by which we judge our world and find it wanting. This measuring that lets us see how flesh is grass might be no slander on real flesh or grass, but a primary encounter with both. People sometimes express disappointment that the earliest ancient examples of writing – the wedge-shaped marks in tablets of clay that were baked or dried in Mesopotamia many thousands of years ago -- seem to be restricted to inventories: mere lists of things or quantities of grain. But counting or measuring in a primal sense is the essence of our grasp on the world, at least as that grasp is found in language itself. We need not shy from calling those lists the first written poetry. The transition from listing names (i.e. units) and numbers of things to writing poetry seems very slight indeed compared with the transition from an unbroken sequence of immediate stimulus-reactions, to names and numbers of things at all. Indeed ancient poets as well as modern ones are notoriously fond of lists as such. Homer gives us the Catalog of Ships. The Hebrew Scriptures list who begat whom. And both are masters of metaphor and measure. Homer speaks unforgettably of the sword or spear blade cutting flesh as “Pitiless Bronze.” If we are not explicitly measuring by unities, or counting -- “giving an account” as we say -- we are measuring still. Saul was by head and shoulders the tallest among the men of Israel. Thomas Hobbes says in Leviathan that all thinking is counting and computation: adding and subtracting. It is a surprising agreement with the Platonic insight that our capacity to see unity is what makes us human. And somehow we can see unity in what refuses to break into natural units. The Continuous must have a unity of its own: it is imaged �in the line in which every point has another as near to it as you please. Could the Unlimited be another ingredient of the World, like wholeness being present in everything that is? It could never be visible without having already undergone some unification but it might always maintain its indeterminacy in a kind of refusal of any permanent allegiance to particular unity or identity. Nothing is immune to change and everything must decay. Today’s Ponderosa Pine tree seems fully formed and vitally involved in being what it is; it seems almost to breathe if you look at it in the sun and the wind. Botanists will tell you that it does breathe. But some few years from now it will be lying on the ground, rotting into dirt, relinquishing the noble form that seemed completely to possess it. In Incommensurability we seem to have found a thing we cannot imagine, if to imagine means to give a unified identity to something, but must nevertheless think as true. This is already remarkable and may encourage us to be more careful in distinguishing the imaginable from the true in other cases. The imagination is not a perfectly reliable guide even to the possible, let alone to the true. Is it the infinite divisibility of the line that leads our imagination and our reason in opposite directions? Perhaps we can know some things to be true regarding what is infinite without being able to imagine them. Moses Maimonides points out that Apollonius proves a curved asymptote approaches a limiting straight line so that the distance between them is forever diminishing without limit and yet without ever entirely disappearing. They can get infinitely nearer forever without meeting. Maimonides says that we cannot imagine this but that we can know it. Even to look down railroad tracks includes imagining we see that they meet at the distance of the vanishing point. We may get so used to knowing something, that we think we are successfully imagining it, or we may decide to discard our imaginations as any help at all to knowing, but both alternatives are likely to be mistaken. �What about the wider implications of incommensurability? There are many. One seems to be a fundamental distinction between the continuous and the discrete: two different kinds of magnitude, represented in our thinking here by lines, which are continuous and by numbers, which are assemblages of discrete units. Of course a line may be divided into as many equal parts, or artificial units, as we please, or as few as two, so it is not in every way unavailable to the language of number, but it has no natural unit and can be thought of apart from the notion of an assemblage of units. It may be this absence of natural units that is at the heart of incommensurability. When we want to measure distances in the physical world we begin by inventing a unit length like the inch: roughly the top joint of a king’s thumb. Everything that has a length must thereafter submit to being so many thumbjoints long, measured to the nearest half-thumb-joint. But that original thumb joint reveals its peculiarity when we ask how we would measure its length. Strictly speaking it has none since there is no agreed-upon unit by which we would measure it. It is one unit long. How big that unit is cannot be determined. Would we say, “One is one”? Calling One a number might be like claiming to know how long an inch is: it only works for us as a measure in multitudes of itself. If we really want to say anything satisfying about how long an inch is we must invent a centimeter and say it is 2.5 of them, but then we cannot say what the ‘absolute’ length of the centimeter is. So we do seem to make, or find, a multiple thing, number, amid things, namely units, each of which contains no multiplicity. Can the One be incommensurable with the numbers of things it counts? Socrates on the day of his death says that he gave up the study of natural science when he realized he still did not understand how one and one made two. Do we understand it? We may simply not know how to come any closer to understanding unity and so we proceed to get farther away, to make progress in some direction �rather than seek we know not what from the origins. We might remind ourselves that although lines are limited or determined by points, they are not made of them. Wherever there can be two points we can think of a line that joins them but since the line can always be divided then there is always another point between any two so that if a line were made of points it would have as many as we like and we could add as many more without expecting it to change in length. That is not the way a wall is made of bricks at the very least. Is it the way a brick is made of clay? We can think so many remarkable things if we do not linger too long at the beginning that perhaps it would be wrong not to get on with our deductions and further explorations merely because we do not really understand unity. Perhaps there is room for both directions of thought? Let me offer another image of a kind of incommensurability. Do we know what allows us to use words? Can there be untranslatable words? What do we mean when we say that a particular word, say, of Greek or French, really has no equivalent in English? We might like to say that such a word in its own language brings together as parts of the same whole several different thoughts or meanings which nowhere exist together in any single word of English. We can still list those meanings and instruct the learner to think them together; and we may have the learner’s experience of beginning to feel as if after all the different meanings do deserve to have their own single word to unite in. We may start to think that “Deinos”, the Ancient Greek root of our word “Dinosaur”, doesn’t have to be heard as: “Terrible but possibly also in other cases “Wondrous” and in yet others “Clever and Effective”. We begin to hear “Terrible AND Wondrous, Clever AND Effective” all at once, in a way not really captured by our own recent and over-used “Awesome”. But perhaps our minds have jumped a gap or discontinuity between Greek and English rather than finding a common unit of measure, and we are briefly thinking in Greek? �Perhaps all foreign words are strictly speaking not incomprehensible but yet never perfectly translatable? I should address a doubt. What shall we say to someone who tells us we are making mountains of molehills and that there is no real problem with translation or even with expressing the diagonal in terms of the side? The Doubter will say that if the side is called “one” then the diagonal may be called “the square root of two”, or “that number which when multiplied by itself will give us an answer of two”. We may ask if the doubter can tell us how many times we will be multiplying it by itself and receive the somewhat mysterious reply “1.4142 …” with the further explanation that the dots represent a continuing fraction that never actually ceases. We may feel as if something is peculiar about a number that can never finish being named. But any number ending with a finite fraction will when multiplied by itself give us an answer that is either bigger or smaller than two. Inventing a symbol that means “find the number which when multiplied by itself gives the number under this sign” and calling it a “square root sign” does not guarantee that there is such a number corresponding to any number I put under the sign. For 49 we find 7 but for 2 we find “1.4142…” and the dots go on forever. If this infinitely continuing fraction is our way of reconciling the continuous with the discrete, or letting the same number be even and odd, we may wonder what might be getting lost. Perhaps there are no two things so close that the mind cannot find a gap nor so far apart that the mind cannot find a bridge? What about Being and Nonbeing, or life and death? Or Right and Wrong? Are they incommensurable once and for all? What is at stake for us when we try to know? Incommensurability and Meno I want to turn now again to the Meno for some help with the question of how we might come to know that something is true or of how we might learn. Why does Socrates use the example of �proving something about the diagonal of a square when he wants to encourage Meno to suppose that it is possible to learn, and even perhaps to learn how to be good? My thoughts are not terribly well-organized on this topic but let us begin with some possible connections. Virtue is proposed to us in the Meno and elsewhere in the dialogues as having four parts: Courage, Moderation, Justice, and Wisdom. Sometimes it is suggested that none of these can be separated from the others, that Courage without Wisdom is mere rashness, or Moderation without Justice mere cold selfishness. They are compared to the parts of a face, unable to exist as themselves except when all together. Suppose we imagined them as a square in which their equality and co-dependence might be imaged. Then when Socrates asks what unites them or plays the role of that by which each deserves to be called a virtue, we could imagine that he is asking if there is another line that touches each and all of them. That line could be a circle around the square, or it could be the diagonal. It is a line which by making two triangles in its division of the square would prevent the collapse of the square under pressure, as carpenters all know. It is inside the square yet bigger than any of the lines whose particular arrangement makes the square. It turns out – and you may as I have said want to consult a written version of this lecture to see why this is so – that its precise size is not nameable or measurable in terms of the sides of the square; and yet it has a perfectly well-defined size. How did Socrates’s and Meno’s attempt to define and unify human excellence lead to a mathematical problem about incommensurability and disharmony? The road leads through Power. From the first they have disagreed on a fundamental level: Socrates wants to know what human excellence is, and Meno insists that the most needful thing is finding out how to acquire it. Socrates seems to promise that really knowing what it is or at least making a real attempt at learning that might be the only way to begin acquiring it and Meno fears that �insisting on insight will lead to paralysis; and so he seems to recommend settling for anything that looks a lot like the path to acquisition: say, Power. If you have power, then you can enact whatever looks excellent to you, but without it, no quantity of insight will help; you will be a victim or a bystander, no real doer. Socrates helps us and Meno to see that Meno is after all more interested in power than in excellence; since everyone wants what is good or excellent but few seem capable of acquiring it, it must be, thinks Meno, that those few are the powerful; and a corollary must be that what is excellent is what can be acquired by power: gold and silver and a place in the councils of the city. Meno and the reader are shown the consequence of the definition of Virtue that says it is “For the one desiring fine things to have the power to get them”. Socrates does not here suggest that a more important difference among people than the division into who has or lacks the power to get good things might lie in the question of what things are really good and what others only look good. Perhaps he is not surprised that the distinction between those who only think they know what is good and those who really do know is not a familiar one for Meno. Like most of us Meno thinks it is easy to know what good things are. He also thought it was easy to imagine that some of the unhappy or the unlucky might actually want for themselves things they knew were bad for them. Socrates must carefully remind him what would really be involved in wishing to harm oneself without any counterbalancing benefit of any kind, namely a kind of absurdity or impossibility; and then Meno admits we all suppose we are choosing what is best, so that everyone can be said to share the desire for fine things, or at least for apparently fine things. This equality in desiring apparent goods leaves the struggle for excellence to be, as we mentioned earlier, a matter decided according to who has the power to get or get at those apparent goods: Gold and Silver, and honors and powers and offices in the �city. Meno uses the verb “porizesthai” or “ to achieve … procure… make progress ” as the third infinitive we translated in the phrase “ To desire fine things and be able to get them.” “ Poros”, the noun in that verb, is cognate with the English word “ford”, as in “ you can cross the river at the ford”. A Poros in Greek may refer to any number of stratagems or devices for accomplishing one’s goals. It is a word whose privative form “aporia” has great resonances with incommensurability. Aporia is the condition of being without resources in the face of something. It can describe simple lack of money or that more general difficulty that we describe as “feeling completely at a loss”. We are at an impasse and can see no way across some barrier. Here we are near to incommensurability. Meno first uses the word in the dialogue to say that when one knows the different virtues appropriate to old and young, men and women, slave and free, one will never be in any aporia about saying what virtue is. The simple connection of Poros, resource, to money lets Socrates remind Meno that for all the importance of Porizesthai or the “ being able to GET for oneself …” those fine things that virtuous people want; and for all the ways that money lets you have access to nearly anything you might want, there could be situations in which not Poros, resource, is crucial to virtue, but precisely Aporia, resourcelessness. If the only money to be had in a certain situation was money unjustly acquired, then Meno agrees an Aporia of money would then be virtuous. One might go another step and say if the only action that could be taken in a particular situation had to be action taken in complete ignorance of what a truly just outcome would look like then inaction might be preferable. Meno turns out to be more familiar with Aporia, at least in thought, than one might suspect of a very ambitious and not very scrupulous young aristocrat. The things he thinks about are strongly marked by the possibility of aporia. What kind of thinking does he like? He says he likes the way Gorgias explains the functioning of the body’s senses by reference to effluences or “outflows” from the objects sensed, which outflows can be �compared to very small shapes constantly crossing the space between say a sweet-smelling flower, and my nose. If the tiny shapes fit the pores of my nose then I will sense aroma. Other shapes, e.g. those conveying sounds, will not fit my smelling pores but instead will find paths through my ears and I will hear things. Socrates’s Gorgian example ends with him noting that this mode of explanation can be adapted to all of the senses and perhaps many other questions as well. Maybe it explains too much? The element of the Incommensurable is very prominent. Socrates says the little shapes are “symmetroi” with the pores of their proper sense organs, that is they have a common measure. But smells are incommensurable with ears. To survey the microscopic world for a moment through this lens, we must be ready to see a constant flow of all kinds of possible sensations, tiny shapes crossing one another’s paths in all directions constantly and bouncing away from doorways not designed to let them in or slipping neatly into passages through which they fit as smoothly as an old key. A simple test is always automatically going on amid the outflows and the bumps of the myriad tiny shapes, “Are you the right shape to pass?” is asked and answered thousands of times per second. Socrates describes Meno’s pleasure in this explanation as a response to the High Tragic Manner, in which, as he says, this account appears. High Tragic Manner? What is that likely to mean? Does Meno, whose father knew Xerxes, the tragic hero of Herodotus’s History, does Meno have reason to wish for a world built on Tragedy? If he wants to be excellent, to win praise for his power and fine possessions, to be honored as a Homeric Warrior is honored, is it more comfortable than facing the impasse of your actual knowledge or ignorance of your own powers, to think that the best warriors are just the right shape from birth and that if you are fated to be great you will find a fitting passage? If you are already among the leading families of Thessaly then your fate is clearly calling you, and if your father narrowly missed becoming a Persian Satrap over some of Greece, �maybe that has been saved for you! Those who treasure power above all things will recognize you because it takes one to know one, and they will recruit you. You and they will be ready to betray each other if that is the path to greater power, but meanwhile you are commensurable and you both need allies. Another piece of thinking Meno likes is the argument that you cannot usefully seek to learn anything because either you know or you don’t know and so you cannot learn what you already know but you cannot even recognize what you do not know. So await your fate in the confidence that you already have the right stuff. It’s all or nothing at all. The tragic flavor is a flattering spice that suggests that the great human beings must be prepared to do and to suffer things perhaps not bearable for ordinary humans, and that this is why we remember them and follow them and tell their stories. Amid the chaos some shapes are arriving where they fit; praise and blame may be beside the point. The importance of welldirected effort or the possible guidance of insight look like idealistic distractions on a stage full of opportunities for immediate deployment of strategies: cast what grappling hooks you have in all directions and pull in the biggest fish you snag. Plato is showing us a young Meno shortly before he seized what must have seemed the perfect opportunity: an expedition led by the younger brother of the Great King of Persia, intending to take the throne from his incompetent elder. Xenophon, a contemporary of Plato and like him a student of Socrates, has written in his book The March Back of Meno’s corrupt and violent attempt to become a Satrap in the Persian style and of the awful fate it led him to. We may suppose that Xenophon’s report was known to Plato’s first readers. That this young Meno even wants to talk about how virtue is to be gotten and kept is a positive sign that he doesn’t yet simply suppose that his job is to grab every gift of nature or fortune that comes in his reach. He somehow is open to the thought that he might need to do something else for himself, maybe improve �himself somehow, and that Socrates could help him. But the Tragic element perhaps overwhelms him. That Tragedy is finally built on the foundation of incommensurability, the incommensurability of Gods and humans, that may be what will keep Meno safe from the danger of being truly changed by his talk with Socrates. Tragedy for Meno may be about the near-miss when a mortal comes along who could almost be mistaken for an immortal. There is no room on earth for humans who do not die. The great human beings who nevertheless do not yield or resign themselves to being less than the Gods will be wonders in their lifetimes and will be long remembered after their deaths. The shape that is our lot may limit us to either having what it takes or not but a certain kind of defiance of Fate is possible. The very thing that seeks to diminish us, our mortality, can be embraced by the tragic protagonist and can transform us. The insistence by the tragic character that she alone will define herself even if it should cost her life nearly makes her into a God, and it simultaneously kills her. Medea murders her children, her husband’s new wife and father-in-law, taunts Jason and mounts a dragon chariot on the roof to fly to Athens, where she will bear what few can: to lead what remains of a wrecked and miserable life all her own. Something like this may be what Meno loves, hidden in the answers of Gorgias. He does not seem to love learning for its own sake. It is striking how Gorgias’s science anticipates our own: two millennia later our biology is still deep in the process of describing the docking of different tiny chemical particles and molecules in an elaborate process of sending and receiving signals and instructions according to what effluent shapes fit what receptors. And it is still more striking what a difference remains between the recognition that an effluence fits an opening and the experience of understanding a thought. To the Greek listener or reader a similarity of sound appears between the “Aporia” or being at a loss that Socrates will praise as an indispensable part of learning when �he helps the slave boy recollect how to double the square, and on the other hand the the “Aporrhoe”, or flowing outward, that constitutes the whole Gorgian account of how we perceive and possibly even of how we know. Maybe the nearness in sound of Aporia and Aporrhoe is intended to point toward the thought that the Gorgian/Scientific answer always leaves us where we started: on the outside of what may be going on. If we can see color, we say, it must be because little bits of something are flying into our eyes, something we might as well call “color particles” or color photons or color waves if you prefer; but what allows the arrival of these little shapes to become our experience of color remains dark. We posit that something about them carries what we end up calling “color” and that when it arrives in our eyes, or perhaps bumps something in our eyes that can send something that arrives in our brains, well then, color has arrived. Leibniz says that no matter how much we may imagine enlarging the physical pieces of the brain and nervous system, we will not thereby have achieved more than a larger picture of particles moving other particles. But where, he asks, will be our own understanding from the inside of thoughts and sensations in all this sequence of actions and reactions? We may get to a place where we can say, in effect: “When this exact sequence of synapses firing takes place you are remembering your first grade teacher”; but all we will be doing is correlating two separate events: your own experience of memory, and a neurophysiologist’s observations of events among your synapses. I do not mean to speak ill of the enterprise of Science, which surely has shown us many real beauties and marvels; and which is by no means simply identical with Technology; but I wonder if there are important differences in kind among ways of doing what we call knowing. We have now “known” for well over a century that fire happens when particles of combustible substances, like carbon and oxygen, combine in such a way as to release light and heat, and often other gases and particles. We have gone on learning many more details about smaller and smaller particles involved in the process. We have also claimed to know �for much longer than a hundred years that it is a crime to set your neighbor’s house on fire; but we have never ceased from arson. I make bold to say that no matter how thorough an account we can learn to give of fire, it will make no difference to the ways we treat our fellow human beings or indeed any of the living beings of the world. If no other mode of knowing is available to us than the techno-scientific, I am afraid we are doomed. How welcome a kind of knowing of Incommensurability might be to us if it should guarantee justice and equality! Beyond any property I might own, I own myself as an insoluble mystery which may decline to be judged by any other standard than its own. If our dignity is our irreducible otherness from all others then most of our moral experiences will be determined by ways we do not fit in. Resistance and bravely saying “No!” will become the unmistakable marks of human goodness. But maybe this, too, like Gorgias’s answers, solves too much. The simple incongruity of having no common measure with others can stand for an inalienable freedom and an infinite value but how do we know we are not still flattering ourselves in the High Tragic manner? Is it not precisely commensurability we seek when we propose with Socrates that the effort to learn makes us better? All Nature is akin, he says. This is the opposite of severing ourselves from a world of indifferent collisions, or of what is worse, of seeking to become similarly indifferent ourselves. If we are bound to act for the sake of what is or seems better, then that may be a clue to how the cosmos acts. These are only speculations about Meno and about ourselves. Without them I would not know how to begin reading Plato. The Aporia which Meno will call “ numbness” in his image of Socrates the Torpedo Fish has its counterpart in the Gorgian Theory of Everything: in the word “Aporrhoe”, or effluence, outflow. Everything is only connected to everything else by this �constant outflow or stream of shapes by which each thing shares its visible, audible, smellable, tasteable, touchable, knowable self with the various human organs of perception which happen to be commensurate with it. One wonders where the inexhaustible source for such constant outflow can be located, and why it never runs low. One may also wonder if all that seeming incommensurability conceals a suppression of genuine differences of kind in things. Have we really seen very deeply, or heard any divine harmonies when we assert more or less a priori that sight and hearing are at bottom both the same, just matter in motion? There may be more numbness involved in this conclusion than in any temporary impasse that Socrates and his conversationspartners suffer in their attempts to learn. The binary simplicity of the life of an effluence is perhaps part of its attraction to Meno and to us; either a shape fits what it hits or it doesn’t. It is like being told either you already have what it takes or you never will. You are spared the wandering about in some gray area while trying to find common measures or small steps of approximation. You don’t have to start trying from where you are to get someplace different. You do not have to examine opinions to see what may be partly true in them and where that may lead. As in the argument that you cannot learn what you already know nor what you have never encountered, there seems to be a kind of absolute separation between the wealthy possession of knowledge and the impoverished condition of ignorance. Neither one is quite understandable beyond the simple model of property ownership: if you know something then you have it, if not then you don’t. And having means chiefly having the power to exchange one thing for another, to trade up, as we say. So Meno hopes by his conversation with Socrates to end up with some answers in his back pocket that will confound anyone he should have to debate; he is challenging Socrates with the answers he has memorized from Gorgias: “ have you got anything stronger than this?” �If on the other hand it is possible to learn by experience that knowable things are not inert possessions but after all have a kind of life, then knowing must be different from simple possession of property, which one can do while sleeping; knowing must be an active practice, and one must be seeking to take on some of the life of what one wants to know. All nature, as Socrates suggests, must be akin for the learning that interests him to be possible. However true it may be that each being must somehow differ from all others, that no two snowflakes or leaves are ever identical, it is still finally commensurability that we seek. That the example chosen to give us hope about our capacity to learn is the diagonal of the square, whose ratio to its side is not expressible in common units, must be especially important. What may it mean? It seems to say that there is more than one kind of intelligibility; the lack of a common length measure for diagonal and side does not preclude knowing things about the diagonal and its relation to that side. The counting of discrete units is not the only path to knowledge. The discovery of the line that will let us double the square depends on seeing more than what is there; we must begin to see areas, twodimensional beings in order to address a question about finding one dimension: the length of a line. The diagonal is not in the end confoundingly hard to know; it is the side of a square twice as big as the one whose diagonal it is. If it is to be measured we must refer to a second dimension, not to a line but to an area. The Greek language uses the word “dunamis” or “power” in geometrical contexts somewhat as we might use “squared” or “to the second power” in English. A line may be said to be equal “dunamei”( the dative form of ‘dunamis’) or “in power, by means of its power” to some area, meaning that the square on that line is equal to the area in question. The diagonal can said in Greek to be “in power” the double of the square it divides. The echo of Meno’s word for his ideal of the raw power of the tyrant who can grab whatever he wants and hold it is not accidental. But what is echoing what? �The suggestion that the diagonal is knowable by its power to become a double square is a fertile one. It makes us wonder if virtue has a kind of life in it since living things are partly known by their power to duplicate themselves, to grow or reproduce. Does the recognition of virtue necessarily involve a beginning of reproducing it or a desire to see it exercise its power? When we encounter someone good, we are in fact moved to imitation. Virtue would then be essentially active; it would be what it can become rather than a simple inert quantity. This could be connected to the difficulty in saying what exactly it is or even in describing it. A contemporary philosopher has written a book called “The Fragility of Goodness”, but perhaps another could be written on the Power of Goodness. What if the more knowable a thing were the more beautiful, and alive it were? Would the most unified and beautiful of all naturally generate an entire world as a kind of octave of itself? We might then expect the nearness of such a thing to possess active power, to contain a kind of life that sustains and increases itself. Knowing would not mean to behold something over there in calm clarity, while deciding whether to take it or leave it. It would be to feel the effect of the nearness of life, to be drawn to imitate and be informed by order and pattern, so as to become more nearly unified oneself and more aware of the unity of the world. Would such an experience make us suppose that beautiful speeches could have the power to bring about beautiful deeds? One Greek word for “Rascal”, gleefully appropriated by Rabelais many centuries after its birth, is “Panourgos”, which comes from two words meaning “all”, and “work”. A Rascal is someone who acknowledges no limits, who will “do everything” or do anything to have their way. We say, typically about a villain, that he or she would “stop at nothing” on the way to fulfilling their wicked plans. Turn this inside out and suppose a being so fully limited, which we will conjecture could mean so good, as to be in a sense fully at rest: it would not need to do anything in order to be as it was, and �it would lack nothing so that no motive would exist for its taking some particular step, nefarious or otherwise. We might suppose it to be already fully active all the time and indeed to be the principle of all activity everywhere, but in a way that while pervading everything would have nothing to prove and so no need to undertake any new act. It would remind us of Achilles telling Phoinix that he has no need of human honor and hence no need to do any mighty deeds, but that he has honor enough from Zeus simply by being who he is. So humans who resemble this conjectural opposite of a rascal look like Gods, and the full version of such a being might be God. Human Excellence or Virtue would then likewise resemble God in being as self-contained as a social being might be able to be while remaining social. It might be better-defined than most things are, and hence more knowable, if definition is chiefly a matter of limits. Socrates suggests a little of this as he asks Meno about the presumed behavior of a virtuous human. What if becoming virtuous really is a matter of making the effort to learn, not how we can master things but how things really are; and what if finding that out involves discovering that things are more orderly and beautiful than we can ever fully imagine? � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data Word doc Page numeration Number of pages in the original item. 30 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Incommensurability : Meno and the diagonal Description An account of the resource Transcript of a lecture given on October 21, 2020 by Cary Stickney as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "Can beautiful speeches give rise to beautiful deeds? Please join the SJC Campus Community for our virtual Dean’s Lecture & Concert Series featuring St. John’s College Santa Fe Tutor, Cary Stickney. Cary will be giving a lecture titled 'Incommensurability: Meno and the Diagonal.'" Creator An entity primarily responsible for making the resource Stickney, Cary Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2020-10-21 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Plato. Meno Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_StickneyC_Incommensurability_Meno_and_the_diagonal_2020-10-21 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c31006301547cf74718332e6c2c7b0bf.pdf 5476000426cfcb7dd3df21a39e549846 PDF Text Text In the year 610 of the Christian Era, a merchant of the prominent Quraysh tribe sat meditating in a cave on Mt. Hira near Mecca. He heard a voice saying, Recite: In the Name of thy Lord who created. created man of a blood clot. Recite: And thy Lord is the Most Generous, who taught by the Pen, taught Man that he knew not. [96.1-5] Thus began the youngest of the major world religions and one of the most successful lives in world history. As a religious, political, and military leader, Muhammad (570-632) is without equal. Only Moses comes close, but Moses was not allowed to enter the Promised Land, while Muhammad returned to Mecca as a victorious conqueror. We are. moreover, fortunate to have better documentation for his life than for that of Moses, Jesus, or the Buddha. On any reckoning, Muhammad’s biography is one well worth studying. If you read the Qur'an, you may want to read along with it the most important early biography, the Life of Muḥammad by Ibn Isḥāq. Today, however, our primary goal is to become acquainted with the Qur’an. While some light may be shed on this great book by a fuller knowledge of its historical context, nothing replaces study of the text itself. Thus, most of my talk will focus on the primary text, though I will first discuss some of the major events and issues that form the background of the Qur’an. Muhammad was an orphan. His father died before he was born and his mother when he was six years old. His grandfather took care of him for two more years before he died #1 �as well. Thereafter his uncle Abu Talib, head of the Banu Hashim clan, assumed guardianship of the boy. Thus Muhammad grew up as something of an outsider within Meccan society. Although he did belong to its most prominent tribe, the Quraysh, he was a weak and vulnerable member of it. He rose to prominence, however, due to his skills as a caravan trader, as well as for his reputation of honesty. When he was 25, the wealthy widow Khadija, rather impressed, asked for his hand in marriage, was accepted, and became his first wife. Mecca was a major hub of the Arabian caravan trade routes that connected the Byzantine Empire in the north with the spice-exporting Yemen in the south. The Quraysh not only dominated Meccan trade but also were custodians of the Kaaba, the central shrine for the still largely pagan Arab tribes. The word Kaaba, related to our word "cube", refers to the cubical structure enclosing the Black Stone, a sacred object traditionally venerated by the pagan Arabs and possibly of meteoric origin. Mecca and the Kaaba were already sites of pilgrimage before Muhammad's time, the time that Muslims refer to as Jahiliyya, or the time of ignorance. During their sojourn there, the Arabs would hold fairs, including competitions in poetry, still a largely oral art. Several of these pre-Islamic poems survive. Some of them are known as the "Hanging" or "Suspended" Odes and were supposedly hung up in the Kaaba as a token of honor. Although Arab polytheism still flourished at its major center of Mecca, monotheistic religions were common not only in the surrounding areas but even with Arabia itself. Orthodox Christianity was the official religion of the Byzantine Empire, while the #2 �Sassanid Persian Empire supported Zoroastrianism, arguably a monotheistic faith, although a highly dualistic one. Many Christians, of various sects, were spread througout Arabia, and there was a sizeable Jewish community in the city of Yathrib. Thus when Muhammad brought forward his monotheistic message, he had many enemies. Although he had hoped to find a receptive audience among the “People of the Book”, i.e, Jews and Christians, in this hope he was largely disappointed. The fiercer and earlier struggle, however, was against the leaders of his own city and tribe, the polytheist Quraysh, for Muslims, like Jews and Christians before them, not only believed in the existence of one God, but held that God to be a jealous god, a god who would “have no other gods before him.” Polytheism was not simply mistaken, but even a direct affront to God and could not be tolerated. Polytheism is more tolerant than monotheism. The chief god of the Arabic pagan pantheon was Allah, or "the God." "Allah" simply comes from a common Semitic root for "god" and is cognate with Hebrew Elohim and Ugaritic El. The pagan Arabs had traditionally associated other gods with Allah and worshipped these other divinities, in particular Allah’s daughters (al-Lat, Manat, and al-Uzza). The polytheists could well accept that Allah was the one supreme god; they could not, however, accept that he was the only god or the only god to be worshipped. Particularly offensive to this traditional tribal society, however, must have been the claim that their ancestors, by worshipping associates alongside of Allah, were now burning in hell. Moreover, Muhammad’s attack upon polytheism was a direct threat to their domination of the Meccan trade and shrine. #3 �The polytheists challenged Muhammad to prove his apostleship by performing a miracle. He replied that it was not in his power to perform miracles, but only in God’s power to do so, and that the Qur’an itself was the miracle. A noble, elevated discourse spoken through an illiterate merchant, the Qur’an impressed both believers and nonbelievers alike. Muhammad challenged his opponents to sit down and produce something like it. If they could not do so, the argument goes, then the Qur’an must be a work of greater than human creation. Besides the Qur’an itself, there is one other miracle involving Muhammad that cannot be passed over in silence, since it is the basis of the Muslim claim on Jerusalem as a holy city. It is reported that one night as he was sleeping in Mecca, Muhammad was transported by the fabulous winged beast Buraq to the Temple Mount in Jerusalem, whence he was allowed to ascend the seven heavens and discourse with Abraham, Moses, and Jesus. Thence he was brought back to Mecca the same night. More than half a century after the Muslims conquered Jerusalem from the Byzantine Christians, the Umayyad Caliph Abd-al-Malik, had the Dome of the Rock constructed on the Temple Mount, known to Muslims as Haram es-Sharif. The hostility of the Quraysh leadership could well have led to the murder of Muhammad, if it had not been for the protection of his still pagan uncle Abu Talib. The killing of somebody under tribal protection would have led to a blood feud. So instead of attacking Muhammad directly, the polytheists persecuted his followers. Despite persecution, Islam grew, attracting in particular many of the alienated members of Meccan society, such as freedmen and slaves. When Abu Talib died, however, (619) and #4 �the new leader of the Banu Hashim, Abu Lahab (another uncle of the prophet) withdrew protection from him, Muhammad looked for another home for the Muslim community. When an opportunity for refuge and alliance presented itself in nearby Yathrib, he and his Muslim followers migrated there. This migration, or hijra, is the beginning of the Muslim epoch. Up to this point, Muhammad had been a religious leader. Now he became a political leader by founding the nascent Islamic state in Yathrib, now known as Madinat an-Nabiy, that is, the City of the Prophet, or Medina. The revelations of the Medina period show a much greater concern for political matters and laws relevant to the foundation of a state. The hostility between the Muslims and the polytheists of Mecca did not end then, however. Muhammad insisted that the Muslims be allowed to worship at the Kaaba, which he claimed had been originally a monotheist shrine founded by Abraham and his son Ishmael. The Meccans had also confiscated Muslim properties in Mecca and the immigrants to Medina turned to the Arab tradition of caravan raiding to make a living. This hostility broke out into open war when Muhammad led the Muslims in a raid on a Meccan caravan at Badr (624). Engaging with reinforcements from Mecca and outnumbered by more than three to one, the Muslims won a decisive victory. After further battles with mixed results, Muhammad entered Mecca as a conqueror in 630, pardoned nearly the whole population, and purified the Kaaba of its idols. Muhammad only lived for two more years. In that time he completed the conquest and conversion of Arabia and unified the Arab tribes for the first time in history, a unification made possible perhaps by religion alone. He thus provided the basis for the #5 �astonishing Arab military expansion that was to explode onto the world scene shorty after his death. He had no surviving sons, however, and his only significant failure as a leader was that he did not appoint a clear successor or establish a clear policy of succession. This failure resulted in a series of civil wars after his death and in the schism of the Islamic community into Sunni and Shi’ite sects that has remained of fateful importance even to the present day. The majority sect, the Sunnis, accepted Abu Bakr as the caliph or successor to Muhammad, whereas the Shi'ites believed that Muhammad's nephew and son-in-law 'Ali should have been recognized as the first caliph. Even if Muhammad had only united the Arab tribes, he would be remembered as an eminent political and military leader. But his importance as not merely an Arab leader, but also as a world leader rests on his prophetic mission. For although the Qur’an is in Arabic and addresses Arabs most directly, its message is of universal import. From the beginning, Islam, like Christianity, has seen itself as having a universal mission. So without further ado, let us turn to the Qur’an. When we first encounter with the Qur’an as Westerners, we are likely to be puzzled. This is not a book like the books we are familiar with. It does not tell a story like the Iliad or War and Peace. Although it has many themes in common with the Bible, it lacks the narrative frame that organizes many, if not all, of the books of the Bible. Although it has chapters, or suras, there is little or no apparent connection between a given chapter and the one that comes before or after it. Even within a given sura, one can encounter a bewildering mixture of prophetic warnings, stories, and legal stipulations. So our first question is, “What kind of book is the Qur’an?”. #6 �Just as the Bible is not one book, but a collection of many books, so too the Qur’an is not a single revelation but a collection of several revelations. If one were to sit down and read the entire Bible, one would be rightly puzzled if one were to find the book of Joshua next to the Gospel of Matthew, the Song of Songs next to Paul’s Letter to the Romans. It is not surprising to find diversity within the Bible, a collection of texts spanning some thousand years, written by different authors, addressing different audiences in widely divergent circumstances. Since the Qur’an, however, was all revealed within a span of some 23 years, and to one man, Muhammad, we might have expected a high degree of uniformity, and while there is more uniformity in the Qur’an than in the Bible, there is still a surpising amount of diversity, as we shall see. When I say that the Qur’an was revealed to Muhammad, I do not wish to take a stance on the question of divine authorship, but I do want to emphasize that Muhammad did not compose or write this book. According to all accounts, both those supportive of and hostile to him, Muhammad spoke forth individual suras while in a kind of trance or ecstatic state. Some believed that he was receiving communication from the angel Gabriel, others that he was possessed by a genie or demon. The former, of course, took him to be the latest prophet and became his first followers; while the latter accused him of being a “poet possessed,” alluding to the traditional Arabic view of poets as being possessed by some divine or demonic spirit. The Arabic word for "crazy," majnun derives from the same root as jinn or genie. While some thought that he spun old wives’ tales, there is no contemporary accusation that he was simply “faking” an ecstatic state for some ulterior motive, e.g., a #7 �political one. This, I have no doubt, is how Machiavelli sees Muhammad, thus joining him with Numa and Moses as political leaders who feigned divine communication in order to bolster a political order. But telling against this view is the fact that when the Quraysh offerred Muhammad political leadership in exchange for ceasing to preach monotheism, he refused. Muhammad spoke forth individual revelations or suras when he fell into an ecstatic trance. He and many of his followers were illiterate, so although some may have been written down by his literate followers, by and large the revelations were passed on by word of mouth, until they were all written down and collected by the third caliph ‘Uthman (c.656). Although traditions had passed down some information about when the various suras were revealed, in particular whether during the Meccan or the Medinan period, ‘Uthman did not attempt to arrange the suras chronologically. Instead, by and large, and with the exception of the first sura, the suras are arranged from longest to shortest. It turns out that the Meccan suras tend to be shorter than the Medinan suras, so the Qur’an roughly moves in a backwards chronological order. Thus the traditional Muslim way of learning the Qur’an in Arabic—beginning with the end of the book—also makes chronological sense. A concern with chronology, however, is a largely Western concern, for Muslims would deny that there is any change or development in the message revealed in their holy book, whereas Westerners are always looking for development, even where there is none to be found. Although I would argue that there are interesting differences between the Meccan and Medinan suras, it is still debatable how significant those #8 �differences are. The Meccan suras tend not only to be shorter, but also often use beautiful natural imagery to discuss the coming Day of Judgment. The Medinan suras, by contrast, are not only longer, but often deal with many of the social and legal issues that needed to be addressed by the nascent Islamic state in Medina. So the Qur’an is not a composition, if by “composition” we mean an arrangement ordered according to a certain principle, so that it would be impossible to move pieces around and still have the same thing. Exodus cannot come before Genesis, the death of Patroclus cannot come before the anger of Achilles, Proposition I.47 of Euclid cannot come before proposition I.1. Nothing is lost, I would argue, by reading the Qur’an backwards. This is another way of saying that the Qur’an is a collection rather than a composition. But perhaps a more important point to emphasize is that each sura is meant to stand on its own. The longer suras, one might argue, are even meant to present the whole truth. Thus to go from one sura to another in sequence is not like adding pieces together to form a whole picture but is like revisiting the same truth again and again, sometimes from a slightly different angle. Thus a key feature of the form of the Qur’an is repetition. While this may be tedious for a Western reader who is used always to encountering something new in the next chapter, this formal feature also reinforces one of the central points of the content of the Qur’an: human beings’ central failing is that they are forgetful. Prophets come to remind us of the truth that we have forgotten or that we would like to forget. And as anybody knows who has tried to learn a foreign language, repetition is the key to remembering. #9 �To fend off the accusation that Muhammad was just another “posessed poet,” the Qur’an itself is claimed not to be poetry, altough it does make use of many poetic techniques. The suras are composed of verses and make extensive use of end rhyme. I will now play for you a recitation of the first sura, “Al-Fatihah”, or “The Opening.” Notice the end rhyme on “-im, -in.” I hope this excerpt, even through the medium of a foreign language, gives you a sense of the beauty, power, and appeal of the original. These features of language, in particular of poetic language, suffer the most in the process of translation. Nor are they thought to be extrinsic to the essence of the Qur’an. For the Qur’an tells us more than once that it is written in clear, noble Arabic. The incomparable beauty of the language is the main argument for the Qur’an being a divine revelation. The verses are called ‘ayāt,’ which literally means “signs.” Just like the beautiful and powerful cosmic signs such as the sun, the moon, and the stars, the verses of the Qur’an are taken to be signs that point to the power, goodness, and wisdom of the Creator who made them. Having touched briefly on the form of the Qur’an, I will now turn to its content. The first and most essential part of this content is the theology. A concise statement of its theology is provided by sura 112: Say: ‘He is God, One God, the Everlasting Refuge, who has not begotten, and has not been begotten, and equal to him is not any one.’ #10 �Thus God is one and without associates. That he neither begets nor is begotten not only rules out the Arab polytheist beliefs that he has daughters but also the Christian trinitarian doctrine. He is eternal and absolute. Elsewhere we are told that he is allknowing and all-powerful. He created everything, not only inanimate things like the sun and moon, stars and earth, but also the different orders of living things—the angels, the jinn, and human beings and plants and animals. God is not only just but also "compassionate and merciful." He commands human beings to do good and resist evil, but is compassionate towards those who turn to him and ask for forgiveness. On the Day of Judgment, human beings will be resurrected and summoned before God. Their good and evil deeds will be recorded and weighed in a balance. Those whose good deeds prevail will be rewarded will eternal life in Paradise. Others will be cast into the pit of Hell to suffer eternal torment. When God created Adam he commanded the angels to bow down before him. All did so except for Iblis (Satan), who thereby became man’s bitter enemy. Adam and Eve were cast from the Garden for eating of the fruit of the tree of life, contrary to divine prohibition. There is no Islamic doctrine of original sin, however. We are not being punished now for the sin that Adam and Eve committed. We have, however, inherited their forgetfulness. In particular, human beings get caught up in pursuing their individual self-interest, such as accumulating wealth, and forget divine warnings. We will all die and cannot take our wealth with us. We will all be judged and our wealth will not help us. We are commanded to provide for the more vulnerable members of society—the #11 �widow, the orphan, the poor. We are commanded to do so by paying the alms tax, the zakat. Failure to do so will result in grievous punishment in the hereafter. Prophets have been sent to all peoples and have by and large been ignored. Even after punishment came upon certain cities that ignored a prophet’s warnings, others did not heed those examples. God has even sent down two books, the Torah and the Gospel, to be constant reminders. The people who preserve those books, the “People of the Book” (i.e., Jews and Christians), continue to bear witness to the one true God, although even they have altered the true message by corrupting the divine text with human interpolations. During to these corruptions, Islam, unlike Christianity, does not regard earlier biblical texts as part of its canon. All the truths of the Torah and Gospel are also to be found in the Qur'an itself. Muhammad has now been sent as the final prophet, as the “seal of the prophets,” so this is humanity’s last opportunity to finally get the message. The message has been essentially the same ever since Abraham, the first monotheist, brought it to human beings. By submitting his willing to Allah, the one God, Abraham became the first Muslim, (“one who submits”). The word muslim comes from the same root as the greeting salām, and is cognate with the Hebrew shalom. According to Islam, Islam did not begin with Muhammad but rather with Abraham. Muhammad’s importance lies not in founding Islam, but in restoring it and in being the final prophet. Together with his son Ishmael, the ancestor of the Arabs, Abraham built and consecrated the central shrine of Islam, the Kaaba in Mecca. To receive the message brought first by Abraham, restated by Moses and Jesus, and finally restored by Muhammad, is to be a believer. To ignore or reject the message is to #12 �be a non-believer, or infidel. Since the essence of the message is monotheism, infidels and polytheists are seen as one and the same. Because prophets have been sent to all peoples, there are no “innocent” polytheists: every people has had an opportunity to accept the monotheist message. Since there are clear signs everywhere pointing to the existence of one God, rejecting the oneness of God is taken to indicate not mere ignorance, but willful ignorance. Polytheists reject God because they want to, not because they are clueless. Some passages suggest a doctrine of predestination: "God guides whom he wills and leads astray whom he wills." The “People of the Book” are not infidels, nor are they believers in the proper sense. While they have accepted the core of the message—i.e., that God is one—they have become confused as to other aspects of it. Christians, for example, have mistakenly taken their prophet Jesus to be not a mere messenger of God, but to be God. Jews have wrongly rejected Muhammad’s prophetic mission. Islam asserts a strong dualism of good versus evil and sees them as in constant struggle with one another. Struggle, or jihād, is a central concept of Islam, although it is not quite one of the pillars of the faith, at least for Sunnis. Just as in the universe, so too amongst human beings and in the human soul there is a constant battle between good and evil, a battle that will last until the Day of Judgment, when all will be resolved by God. Since God is good, and believers are the ones who have taken God’s side, believers are inherently on the side of good. This does not mean that believers cannot fall into evil or err, but it does at least mean that they are on the right side of the cosmic struggle. Contrariwise, to disbelieve is to go against God, to side with evil against good. Thus #13 �whatever meritorious action, such as feeding a beggar, disbelievers may do, that action cannot override the fact that disbelievers have taken the wrong side in the battle of good versus evil. While they continue in their disbelief, they cannot be saved. Believers, on the other hand, are not guaranteed salvation, but they will at least receive God’s open ear and mercy when they ask for forgiveness for their sins. The struggle against disbelief and evil in oneself and in the world has important implications for how the Islamic community defines itself in relation to others. During the Meccan period, when Muslims were a perscuted minority in a largely pagan city, the message preached sounds something like a message of toleration, as we can see from sura 109: Say: ‘O unbelievers, I serve not what you serve and you are not serving what I serve, nor am I serving what you have served, neither are you serving what I serve.’ To you your religion, and to me my religion!’ Now this sura can be taken in more than one way. The weakest reading is that it is a mere observation that Muslims and polytheists have different religions. But since this is said directly to polytheists, it is at the very least an act of defiance, for polytheism seeks to incorporate new gods and cults within itself. It may even, as we can see from Herodotus, deny the existence of different religions. This sura may be a way of saying, “You may say that both you and we worship Allah, but in fact we don’t worship the same thing, for #14 �we worship Allah alone, while you worship him alongside of his supposed daughters and other false gods.” The last line is thus an assertion of an impassable barrier between Islam and polytheism. Another intriguing possibility lies in an ambiguous word in the last line. The word translated as “religion,” din, can also mean “judgment,” as in the expression, yawmu d-din, the “Day of Judgment.” Thus we could translate instead, “To you your judgment, and to me my judgment.” This could be a way of saying, “We fundamentally disagree, and God will decide between us on Judgment Day.” Whichever of these possible readings we adopt, something like tolerance is still being proposed, for in this sura the believer is told to speak the truth to the non-believer, rather than to attack, oppress, or kill the unbeliever. It does not, however, go against the idea of a fundamental struggle between good and evil, or between believers and nonbelievers. The Muslim community in Mecca was not in a position to take the offensive against the Meccan polytheists, so the most that can be expected of them is to maintain the integrity of their belief by bearing witness to it, i.e., being martyrs for it, in the face of persecution and oppression. Once the Muslims migrated to Medina, however, and became powerful enough to assert themselves against the Meccans, they did so. And the suras from that period reveal a more aggressive and militant policy against polytheism. Muslims are commanded to fight the polytheists of Mecca until they cease oppressing Muslims and allow them to worship in the sacred mosque of Mecca: “Fight them, till there is no persecution and the #15 �religion is God’s; then if they give over, there shall be no enmity save for evildoers.” (2.193). Thus Islam is not a religion that says “Turn the other cheek.” On the other hand, Muslims are explicity warned not to be the aggressors, “And fight in the way of God with those who fight with you, but aggress not: God loves not the aggressors.” (2.190) Thus only defensive warfare is justified, and it is not only justified but even commanded. Moreover, while Muslims are commanded to spread the word, forced conversion is explicitly forbidden, “No compulsion is there in religion.” (2.256). The People of the Book have a special status within Islam. While conflict between Muslims and polytheists is seen as nearly unavoidable, the People of the Book should be granted tolerance as fellow, although erring, monotheists. Tolerance in this context means that Jews and Christians living in a Muslim society are allowed to practice their own religion under their own laws so long as they recognize Muslim superiority and pay a tax in exchange for Muslim military protection. While this policy is not explicitly stated in the Qur’an itself, it did become enshrined in the shari’a or Muslim law. The Qur’an itself is equivocal on the relations between Muslims and Jews or Christians. To cite a favorable passage: Dispute not with the People of the Book save in the fairer manner, except for those of them that do wrong; and say, ‘We believe in what has been sent down to us, and what has been sent down to you; our God and your God is One, and to Him we have surrendered.’ (29.46) #16 �We also read: Surely they that believe, and those of Jewry and the Christians, and those Sabaeans, whoso believes in God and the Last Day, and works righteousness—their wage awaits them with their Lord, and no fear shall be on them, neither shall they sorrow. (2.62). If we turn to the structure of the Islamic society, we find it bound together by religious and social duties. Although the Qur’an itself does not assign a particular number to these duties or refer to them as “pillars,” different Islamic sects have enumerated different “pillars of the faith.” The majority sect, the Sunnis, enumerate five such pillars. Besides payment of the alms tax, or zakat, that we have already mentioned, we also find the prescription of five daily prayers, or salat, the pilgrimage to Mecca, or the hajj, as well as the fast of Ramadan. The remaining duty, the shahada, or testimony of faith, is not explicitly prescribed as a duty in the Qur’an but may be seen as a precondition for accepting the Qur’an as a revealed word at all. It goes, “I testify that there is no god but God, and I testify that Muhammad is the messenger of God.” What kind of society do these duties promote? First of all, it is one that struggles against the selfishness of individualism. There is nothing wrong with becoming wealthy in itself, but there is if one does so at the expense of others, or if one refuses to contribute to the welfare of those less fortunate. The Qur’an does not seek to abolish or level existing social hierarchies, whether of rich vs. poor, free vs. slave, or man vs. woman, but #17 �it does accept the spiritual equality of all before God and insists that all have a duty to attend not only to the spiritual, but also to the physical, welfare of all others in the community. The opposition between the spiritual and the physical, between the spirit and the “flesh,” so marked in Christianity, is not so strong in Islam. Islamic paradise includes flowing water, flourishing plants, abundant honey, and beautiful virgins and youths. Christians have long been scandalised, but that only shows that Muslims do not war against the flesh as Christians have for so long. Given that God has made both our bodies and our souls, our flesh and our spirit, to reject the physical is to reject part of God’s creation. While Islam does believe in a strong opposition between good and evil and does contrast this current inferior world with the superior world to come, it does not show a marked contrast between flesh and spirit, nor does it brand the “desires of the flesh” as inherently evil. There is nothing wrong with desiring and enjoying beautiful things. This world is inferior to the world to come not because this world is physical and the next world is spiritual. Even Christians, after all, insist on the resurrection of the body, and what would a body be good for in a purely spiritual realm? This world is inferior to the next rather because it is fleeting and filled with injustice and selfishness. To take one particular example. Islam prohibits the consumption of alcohol not because it excessively titillates our appetite for gustatory relish, but rather because it inhibits our ability to act as responsible members of society. Likewise, its sexual regulations, against adultery and fornication for example, are justified in terms of mainting a well-regulated society. There is nothing wrong with sexual pleasure per se, #18 �much less with sexual desire. Modesty in dress is prescribed for both men and women, although it is more strictly expected of the latter. Let us take another example. Islam, along with Judaism and Christianity, prohibits usury on loans to one’s fellow citizens. While economists will rightly point out that prohibiting usury is both ineffective and inefficient, that criticism misses the point, for the economists are presupposing a core human selfishness that Islam is striving to overcome. It is possible to feed the poor to bolster one’s sense of grandeur, or one’s ranking on some list; it may even work well when all in society simply pursue their enlightened self-interest. But to do the right thing for the wrong reason is still not to act morally: one should support charity just because it is the right thing to do. This is much more that one could say about the Qur’an. I hope the little that I have said gives you some sense of the context in which it was revealed, of its form and content, and also of how it conceives of the nature of Islamic society and the relation of Islam to other religions. #19 � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 19 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Qur'an : an introduction for Johnnies Description An account of the resource Transcript of a lecture given on June 16, 2021 by Ken Wolfe as part of the Graduate Institute Summer Lecture Series. Mr. Wolfe provided this description of the event: "In this introduction to the Qur'an, I will explore the context of its composition within the life of Muhammad and 6th century Arabia, its form and content, its relation to other texts and traditions (the Bible, Judaism, Christianity), and its influence upon certain aspects of the Islamic tradition." Creator An entity primarily responsible for making the resource Wolfe, Kenneth Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2021-06-16 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Qur'an Islam Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_WolfeK_The_Qur'an--An_Introduction_for_Johnnies_2021-06-16 Graduate Institute https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4c95870af1e885dd44897c3daea1725e.pdf db6dd1249074f3fe3d21241b6d65c144 PDF Text Text Learning to Love Lincoln: Frederick Douglass’s Journey from Grievance to Gratitude Diana J. Schaub St. Johns College, Santa Fe 4 December 2020 I am delighted to be here to talk about my two heroes: Abraham Lincoln and Frederick Douglass. I will be focusing on what one of them had to say about the other. Frederick Douglass, in his great oration in memory of Lincoln, delivered in 1876 upon the occasion of the dedication of the Freedmen’s Monument, observed that “Any man can say things that are true of Abraham Lincoln, but no man can say anything that is new of Abraham Lincoln.” That is still the case today. Not even by resorting to lies and untruths can one find anything new to say about Abraham Lincoln. The truths and untruths—and maybe most common, the half-truths—have all been around a long time. The task is thus not to be original in one’s appreciation, but to be just. Proper appreciation of Lincoln’s statesmanship, particularly during his lifetime, was rare. The contrast with George Washington is instructive. Although both experts and ordinary citizens now routinely consider Washington and Lincoln the greatest of American presidents, Washington’s rank as a statesman was clear and uncontested from the first—so uncontested that his election to the presidency was unanimous, while the election of Lincoln was so contentious as to provoke civil war. In addition to the seditious opposition of the South, Lincoln encountered plenty of loyal opposition in the North, not only from Democrats, but from those more radical than he both within the Republican Party and outside it (among the various strands of abolitionism). Radicals, then and now, have been particularly stinting in their praise of Lincoln. Some today suggest that credit for emancipation belongs more to those, like Frederick Douglass, who pressured Lincoln to take that decisive step. At the extreme, this position asserts that Lincoln was anti-black, that the Proclamation was basically a fraud, and that Lincoln does not deserve any credit for emancipation since he was “forced into glory.”1 1 �Before signing on to the contemporary radical critique, we might want to examine what the greatest of the abolitionists himself had to say about Lincoln. From his newspaper editorials before and during the war to his speeches and personal reminiscences after the war, the trajectory of Frederick Douglass’s thinking about Lincoln is one of increasing and deepening appreciation, often revising his own earlier negative assessments. Perhaps because Douglass was selfeducated, he remained a lifelong learner, capable of open-minded and rigorous reconsiderations. The way in which the exercise of his critical faculties could lead him to substantive revaluations was evident early in his career when he dramatically changed his opinion about the status of slavery under the Constitution. Repudiating the Garrisonian view of the pro-slavery character of the Constitution, Douglass embraced an anti-slavery reading of the document, thereby transforming himself from a revolutionary, intent on annulling the Constitution, to a reformer, still fiercely critical of American practice, but ever after a staunch defender of America’s founding principles.2 A parallel, but more subtle, shift occurred as a result of Douglass’s encounter with Lincoln—an encounter that taught him to appreciate the statesman (which is to say the prudent politician) as well as the John Browns of the world. Douglass learned to love Lincoln and in his 1876 “Oration in Memory of Lincoln” he recapitulated that intellectual and emotional journey for the benefit of all Americans.3 First Things Douglass’s oration was the keynote address at the unveiling of the nation’s first statue in honor of Lincoln. The statue is entitled “Emancipation” and it was erected in the name of the former slaves and was paid for by their donations. As befitted the ceremonial nature of the occasion, Douglass’s speech expressed gratitude toward Lincoln, but more intriguingly, it reflected on the political significance of gratitude. It is a speech both of gratitude and about 2 �gratitude. Douglass says that “the sentiment of gratitude” which “perpetuate[s] the memories of great public men” is “one of the noblest that can stir and thrill the human heart.” Further, he points out that with the dedication of the Freedmen’s Monument black Americans now “[f]or the first time in the history of our people, . . . join in this high worship.” Douglass wants the world to notice what “we, the colored people” are doing in honoring Abraham Lincoln. As he explains, “First things are always interesting, and this is one of our first things.” Douglass presents the black commemoration of Lincoln as an act that honors the honorers almost as much as it honors the honoree. The story of how the Freedmen’s Memorial took the shape it did, and Douglass’s role in ensuring that his people’s first “national act” came off well, is fascinating. Douglass was asked in 1865 to lend his name to the Educational Monument Association which proposed to raise money from blacks and whites alike to build a black college in honor of Lincoln’s memory. Douglass refused to participate in the project. Here is what he wrote in his letter to the organizers: For a monument, by itself, and upon its own merits, I say good. For a college by itself . . . and upon its own merits, I say good. But for a college-monument, or for a monumentcollege, I do not say good; . . . . The whole scheme is derogatory to the character of the colored people of the United States. . . . It looks to me like an attempt to wash the black man’s face in the nation’s tears for Abraham Lincoln! . . . I am for washing the black man’s face (that is, educating his mind), for that is a good thing to be done, and I appreciate the nation’s tears for Abraham Lincoln; but I am not so enterprising as to think of turning the nation’s veneration for our martyred President into a means of advantage to the colored people, and, of sending around the hat to a mourning public.4 Douglass doesn’t want gratitude—which he calls “one of the holiest sentiments of the human heart”5—to be contaminated with blatant self-interest, for gratitude isn’t even gratitude then. In the proposed college-monument, the problem of impure motives would have been even worse, since there would not just be a mixture of motives but actually a division of motives along 3 �racial lines. Whites would be doing the creditable giving and blacks the self-interested taking. Douglass did not want blacks to enter upon citizenship in that way. Instead of an ennobling display of black gratitude, which would elevate the givers and, moreover, elevate the givers in the minds of white observers, the college-monument idea would reduce blacks primarily to the role of recipients. Douglass was not, in principle, opposed to white philanthropy on behalf of blacks. Years earlier he had sketched a plan for an industrial college in answer to an inquiry from Harriet Beecher Stowe about what she could do to contribute to black advancement.6 However, Douglass was always sensitive to the dangers of ill-timed and overly intrusive assistance, which could have the perverse effect of sapping black initiative, thereby impeding the long-term prospects of the race. Douglass worried that there was always more of benevolence and pity rather than straightforward justice in white America’s dealings with blacks. His preference was for justice—sternly and blindly equal, with no special pleadings or privileges.7 This leads to what at first might seem a contradiction in Douglass’s reaction to the monument-college project. As is well-known, Douglass’s vision of America was fundamentally integrationist. Nonetheless, he wants the monument to be exclusively a black effort; however humble, it should be, he says, “our own act and deed.”8 On the other hand, when it comes to the idea of a college, Douglass speaks against not only the self-serving hybrid of a monumentcollege, but also against the idea of any college being built for the permanent and exclusive use of blacks. Given the discrimination of the day, Douglass admitted the need for temporary recourse to complexional institutions, but he did not want to see the founding of any institution that accepted the permanence of segregation. As he says, “the American people must stand each for all and all for each, without respect to color or race.”9 4 �So, he is in favor of a separately erected monument but opposed to a separate college. Why a Freedmen’s Memorial but not a Freedmen’s College? What accounts for the different judgments on these two endeavors? The explanation, I think, hinges on the nature of the two undertakings and their potential contribution to either lessening racial prejudice or prolonging it. A display of gratitude by black Americans, reflecting the special sentiments they bear towards Lincoln, would undercut white prejudice, by showing blacks capable of “the holiest sentiments of the human heart.”10 Conversely, a college explicitly and exclusively reserved to blacks (whoever foots the bill for it), by accommodating race prejudice, in effect bolsters it. Thus, Douglass accepts all-black institutions only with great reluctance and always with the proviso that, as soon as circumstances permit, blacks must make their way into the majority institutions.11 Douglass is consistent in that he judges instances of racial solidarity and group action by their effects on friendship between the races. His guiding question is always: does the doing of this deed point us toward the overcoming of race prejudice and contribute to an ethos of common citizenship? Acts of black self-reliance, both individual and group-based, can create the conditions for non-racial brotherhood. Douglass understood that before the black man could be recognized as a brother, he must be recognized as a man. Manliness precedes fraternity. Or, to give it a non-gendered formulation: independence precedes friendship. As Douglass had hoped, the monument-college plan was abandoned and, in the end, the memorial took the pure form he had recommended, with Douglass himself delivering the keynote address. Not surprisingly, his first paragraph refers to the “manly pride” with which blacks should view the occasion, while the final paragraph sets forth the black claim to “human brotherhood.” More especially, Douglass informs those whites who seek to “scourge [blacks] beyond the range of human brotherhood” that the Freedmen’s Monument stands as a refutation 5 �of their “blighting slander.” In between the opening invocation of manliness (or independence) and the closing invocation of brotherhood (or friendship), the speech itself demonstrates how a still very divided nation could develop a shared perspective on the achievements of Abraham Lincoln. Any analysis of the speech must take into account not only the uniqueness of the occasion but the rhetorical dilemma posed by the larger historical moment. The speech was given in 1876, as the Reconstruction period was coming to an end. With the federal government increasingly reluctant to enforce the 14th and 15th Amendments, Douglass was rightly worried about the resurgent spirit of the Old South. Douglass worried that reconciliation between Northern whites and Southern whites could end up excluding the freedmen and erasing the real meaning of the Civil War. Thus, he attempts to use the memory of Lincoln to counteract this dangerous tendency, and to revive the “new birth of freedom.” The Oration has a careful structure, being composed of eight distinct sections, each of which begins with what grammarians call a “vocative expression”; in the first two sections he addresses “Friends and Fellow Citizens,” in the subsequent six sections, simply “Fellow Citizens.” Politicians, of course, often rely on direct address of this sort. Sometimes it even becomes a kind of verbal tic, like Lyndon Johnson (in his Texas accent, which I can’t imitate) peppering his speeches with “my fellow Americans.” Douglass’s iterations, however, are more deliberate; they signal new phases of an argument that delineates the different (but not irreconcilable) claims of whites and blacks to the memory of Lincoln. Douglass begins the Oration by addressing his immediate audience: those who assembled that day in Lincoln Park due east of the Capitol building on the 11th anniversary of Lincoln’s assassination. The audience was a large and racially mixed one, composed of 25,000 ordinary 6 �citizens, along with numerous representatives of official Washington. Douglass mentions the presence of members of the House of Representatives and the Senate, the presence of the Chief Justice and Supreme Court, and President Grant himself. These attendees deserved to be called not just “Fellow Citizens,” but “Friends,” whose attendance gave evidence of their sympathies. Interestingly, this first section of the speech makes no mention at all of Lincoln, but instead congratulates “you,” a pronoun that seems to refer, at least initially, only to Douglass’s fellow blacks. Thus, he speaks of “our condition as a people” and the remarkable progress in that condition. The evidence of progress, which Douglass says is a “credit to American civilization,” provides the occasion for a shift to congratulating “all.” Douglass notes that the “new dispensation of freedom”—“has come both to our white fellow-citizens and ourselves.” The second section of the speech acknowledges especially the federal government and its friendly role in this new dispensation. The erection of the memorial received congressional approval; the pedestal for the statue was paid for by congressional appropriation; and the day itself had been declared a federal holiday.12 Douglass, however, highlights the awful sacrifice that lies behind this federal friendship. This section contains Douglass’s first mention of Lincoln, whom he calls “the first martyr President of the United States.” Moreover, Lincoln’s martyrdom is presented as the climax of the larger national sacrifice to which Douglass alludes with his reference to “yonder heights of Arlington.” Arlington Cemetery was visible from Lincoln Park, and 16,000 Civil War soldiers were buried there, including 1500 black troops.13 On the 11th anniversary of Lincoln’s death, what Douglass wanted to remind his audience of was “bloodbought freedom”—“our blood-bought freedom”—in which “we, the colored people” rejoice. While Douglass emphasizes the sentiment of appreciation that gives rise to monuments like the one being unveiled, curiously he says nothing about the actual statue. It is known that he 7 �was not altogether pleased with the design which shows Lincoln, Emancipation Proclamation in one hand, standing over the crouching or half-rising figure of a slave. Dissatisfaction with the sculpture was apparently not limited to Douglass, but was shared by other African-Americans. The official program for the festivities attempted to address these objections, explaining that In the original [design] the kneeling slave [was] represented as perfectly passive, receiving . . . freedom from the hand of the great liberator. But the artist justly changed this, to bring the presentation nearer to the historical fact, by making the emancipated slave an agent in his own deliverance. He is accordingly represented as exerting his own strength with strained muscles in breaking the chain which had bound him.14 The brochure also mentions that there was an alternative design by the female sculptor, Harriet Hosmer, which was rejected as too costly. It would have depicted Lincoln atop a central pillar, flanked by smaller pillars showing, among other figures, black Union soldiers. Douglass would certainly have preferred this design, since it embodied his favorite aphorism: “Hereditary bondsmen! know ye not/ Who would be free themselves must strike the blow?”15 In a sense, Douglass’s speech corrects the submissiveness or paternalism of the statue, by acknowledging both “our loyal, brave, and patriotic soldiers” and “the vast, high, and preeminent services rendered to ourselves, to our race, to our country, and to the whole world by Abraham Lincoln.” In other words, Douglass’s praise of Lincoln is balanced by his recognition of black agency, the invaluable contribution made by black Union troops (by war’s end, there were 180,000 black troops). Having spent the opening two sections proclaiming the generous deed of the moment and commending it to the notice of “men of all parties and opinions,” including “those who despise us,” Douglass in the third section begins to speak to the larger nation-wide audience—an audience of “Fellow-citizens” not all of whom are necessarily “Friends.” Douglass now treads 8 �very carefully. He does not want the black embrace of Lincoln to trigger a white flight from Lincoln. And so, he quite dramatically backs away from the Great Emancipator, insisting that Abraham Lincoln was not, in the fullest sense of the word, either our man or our model. In his interests, in his associations, in his habits of thought, and in his prejudices, he was a white man. He was preeminently the white man’s President, entirely devoted to the welfare of white men. He was ready and willing at any time during the last years of his administration to deny, postpone and sacrifice the rights of humanity in the colored people, to promote the welfare of the white people of his country. . . . The race to which we belong were not the special objects of his consideration. Knowing this, I concede to you, my white fellow-citizens, a pre-eminence in this worship at once full and supreme. . . . You are the children of Abraham Lincoln. Douglass devotes the whole of section 3 to reassuring nervous whites—whites who are patriotic, but probably prejudiced. Basically, he tells them, “Look, don’t worry. Lincoln always loved you best. Take it from me, a Negro, Lincoln was not a Negro-lover.” It’s a rather startling rhetorical gambit, but it allowed Douglass to exhort white Americans to heap high their hosannas of Lincoln. He tells them: To you it especially belongs to sound his praises, to preserve and perpetuate his memory, to multiply his statues, to hang his pictures on your walls, and commend his example, for to you he was a great and glorious friend and benefactor. By the close of this section of the speech, which we might dub the white supremacist section, one might wonder why blacks are bothering to honor Lincoln at all? Douglass’s answer is that while whites are Lincoln’s children, blacks are “his step-children, children by adoption, children by force of circumstances and necessity.” Moreover, what Lincoln did for his step-children, whether it was part of his original intention or not, was deliver them from bondage. Accordingly, Douglass entreats whites “to despise not the humble offering” of former slaves. The separate claims of whites and blacks upon the memory of Lincoln can co-exist. Whites can honor Lincoln for saving the Union; blacks can honor him for Emancipation. Shared homage, if it is ever to develop, must begin with toleration for racially-specific homage. 9 �Frederick Douglass had a gift for metaphor and this image of blacks as Lincoln’s stepchildren is one of his finest. It accords nicely with Lincoln’s own account of the relation between the cause of Union and the cause of Emancipation, as expressed in his famous letter to Horace Greeley. Here is how Lincoln himself explained his duty as president: My paramount object in this struggle is to save the Union, and is not either to save or to destroy slavery. If I could save the Union without freeing any slave I would do it, and if I could save it by freeing all the slaves I would do it; and if I could save it by freeing some and leaving others alone I would also do that.16 Douglass reminds his listeners that Lincoln was a Unionist first and foremost and that he became the Great Emancipator only “by force of circumstances and necessity.” Whites ought to revere Lincoln as the savior of the nation. And indeed, the inscription on the national Lincoln Memorial, built half a century after the Freedmen’s Memorial, reads: “In this temple, as in the hearts of the people for whom he saved the Union, the memory of Abraham Lincoln is enshrined forever.”17 Of course, the Union to which Lincoln was devoted had at its foundation the principle of human equality. The Union was itself a moral project. Because the bond of genuine Union is a teaching about natural right, American patriotism ought to produce citizens who are, as Douglass says, “friendly to the freedom of all men.” In the 4th and central section of the speech, Douglass presents at greater length the step-children’s view of Lincoln, the essential feature of which was faith in Lincoln’s “living and earnest sympathy” with their fate. Again, Douglass doesn’t paper over the disagreements and disappointments that blacks experienced during the war years. “We were,” he admits, “at times stunned, grieved and greatly bewildered.” Douglass provides a litany of reasons why blacks might have doubted Lincoln’s good will: he supported colonization schemes; he refused to enlist black troops; after finally allowing black recruitment, he refused to 10 �retaliate when the Confederates violated the rules of warfare by massacring black prisoners; and he revoked early emancipation decrees by Union generals in the field. Nonetheless, Douglass asserts that “we were able to take a comprehensive view of Abraham Lincoln”—a view that took the measure of the man and, after factoring in the “logic” of events and even “that divinity that shapes our ends,” Douglass says, “we came to the conclusion that the hour and the man of our redemption had met in the person of Abraham Lincoln.” Douglass then gives a counter-litany of the liberationist and racially transformative policies that transpired under Lincoln’s rule. He lists nine achievements, culminating in the Emancipation Proclamation. Each time, he repeats a version of the phrase “under his rule we saw . . . .” The phrase is crucial for both whites and blacks. Blacks—who longed for liberty but who might understandably be suspicious of rule and law, having suffered under generations of misrule—are reminded that their liberty came to them through law and through “wise and beneficent rule.” Conversely, whites are reminded that the actions of Lincoln, which struck not only at slavery but at “prejudice and proscription” as well, were the actions of a dedicated constitutionalist. The closing paragraph of section 4 celebrates Emancipation and, moreover, shows that the celebration can be shared by all. Douglass asks, “Can any colored man, or any white man friendly to the freedom of all men, ever forget the night which followed the first day of January, 1863?” Whites can appreciate black liberation and blacks can appreciate white “statesmanship”—a word that Douglass now uses for the first but not the last time in the address. On this new bi-racial basis of Union and Liberty, Douglass goes on to a reconsideration of Lincoln in sections 5, 6, and 7. He argues that Lincoln’s “great and good” character was transparent to those “who saw him and heard him.” Indeed, direct contact wasn’t even necessary. In a passage with tremendous import for us today, Douglass says “The image of the man went 11 �out with his words, and those who read him knew him.” We are indebted to biographers and historians who have scoured and scavenged for all the bits and pieces of eyewitness testimony and hearsay evidence, and who have laboriously contextualized and hypothesized and speculated, to such a degree that, with the exception of Jesus, there is now no one who ever walked the earth more written about than Abraham Lincoln. Nonetheless, it is reassuring to know that Lincoln’s words alone are enough. In light of this fundamentalist insight, Douglass now revises his earlier “white supremacist” account of Lincoln. He reconsiders Lincoln’s deference to popular prejudice in the appropriate context—the context of democratic statesmanship. Here’s what he says at the close of section 5: I have said that President Lincoln was a white man, and shared the prejudices common to his countrymen towards the colored race. Looking back to his times and to the condition of the country, this unfriendly feeling on his part may safely to set down as one element of his wonderful success in organizing the loyal American people for the tremendous conflict before them, and bringing them safely through that conflict. His great mission was to accomplish two things; first, to save his country from dismemberment and ruin, and second, to free his country from the great crime of slavery. To do one or the other, or both, he must have the earnest sympathy and the powerful cooperation of his loyal fellow-countrymen. Without this primary and essential condition to success, his efforts must have been vain and utterly fruitless. Had he put the abolition of slavery before the salvation of the Union, he would have inevitably driven from him a powerful class of the American people, and rendered resistance to rebellion impossible. Viewed from the genuine abolition ground, Mr. Lincoln seemed tardy, cold, dull, and indifferent; but measuring him by the sentiment of his country, a sentiment he was bound as a statesman to consult, he was swift, zealous, radical, and determined. Frederick Douglass himself always occupied “the genuine abolition ground,” and his speeches and writings, from the early years of the war especially, often manifested great frustration with Lincoln’s caution. In retrospect, however, Douglass generously acknowledges the partiality of his own abolitionist stance and credits Lincoln as the “comprehensive statesman.” I think it is important to note that the final paragraph of this section carefully distinguishes Lincoln’s views on race from his views on slavery. Douglass repeats (for the third 12 �time) that Lincoln was prejudiced, or more precisely that he “shared the prejudices of his white fellow-countrymen against the Negro [italics added].” According to Douglass, racial prejudice is a social construct; there is nothing innate or inevitable about it. It seems that Douglass does not regard Lincoln as particularly progressive on the question of race; he was a follower or a sharer in the dominant opinion of the day. However, in this very same section in which Douglass refers to Lincoln’s prejudices, he explicitly says that “the humblest could approach him and feel at home in his presence.” This statement echoes what Douglass said elsewhere about the experience of being in Lincoln’s personal presence. Speaking of his second meeting with Lincoln, Douglass in his autobiography says: Mr. Lincoln was not only a great President, but a GREAT MAN—too great to be small in anything. In his company I was never in any way reminded of my humble origin, or of my unpopular color. We might wonder whether the presentation of Lincoln’s racial prejudice is compatible with the presentation of his capacious and welcoming humanity. Of course, it might be possible for someone to regard a particular class of people as inferior in certain respects, while still treating individual members of that class with consideration. Lincoln could have been both prejudiced and polite. If so, it would still be necessary to explain why Douglass in the Oration chooses to draw attention to one quality more than the other. Perhaps he wishes to indicate to both blacks and whites that racial prejudice is not an insuperable obstacle to black advancement or bettered race relations. Alternatively, I believe it is possible to interpret Douglass’s remarks in a way consistent with the view that Lincoln deferred to popular prejudice without fully subscribing to popular prejudice. The issue might be elucidated by asking “what was the nature of Lincoln’s ‘sharing’ in white prejudice?” When he describes the relation between Lincoln and “the sentiment of his 13 �country,” Douglass credits Lincoln with being in advance of popular opinion (measured against which he was “swift, zealous, radical, and determined”). Douglass introduces the key verb “consult,” claiming that “the sentiment of his country” was something Lincoln “was bound as a statesman to consult.” To the extent that popular sentiment was unfriendly to blacks, Lincoln’s sharing in it may have been political, rather than personal—deliberately affected rather than deeply held. Douglass here conveys a crucial lesson about the limits within which democratic statesmen operate. Politicians can’t get too far ahead of public opinion if they hope to remain politically viable. More than others perhaps, black citizens must incorporate this insight into their assessment of political figures. A “comprehensive view” must “make reasonable allowance for the circumstances” and not judge on the basis of “stray utterances” or “isolated facts.” In taking the measure of Lincoln, Douglass shows how granting this latitude of maneuver is compatible with respect for the burdens of statesmanship as well as the self-respect of citizens. Douglass tries to model what it looks like to take a comprehensive view of a politician. His people are new voters and there are two dangers they must avoid. Douglass does not want blacks to look to politics for a Moses figure, but he doesn’t want them to fall into the opposite error of cynically seeing only flaws. He shows the possibility of appreciation without idolatry and criticism without rejection. Whichever way one comes down on the question of Lincoln’s views on race, Douglass is emphatic that Lincoln’s attitude toward slavery was above reproach. Douglass quotes from the atonement passage of the Second Inaugural, in which Lincoln interpreted the Civil War as the blood price exacted by a just God for the nation’s sins toward the slave. [You remember the passage: it speaks of the war possibly continuing “until all the wealth piled by the bond-man’s two hundred and fifty years of unrequited toil shall be sunk, and until every drop of blood drawn 14 �with the lash, shall be paid by another drawn with the sword.”] Those were lines that Douglass quoted in nearly every postwar speech he gave that mentioned Lincoln.18 The Second Inaugural’s solemn invocation of divine reparations, Douglass says, “gives all needed proof of [Lincoln’s] feeling on the subject of slavery.” Douglass now revisits an issue he had highlighted earlier. In section 3, when he mentioned Lincoln’s policy of “opposition to the extension of slavery,” he had stressed Lincoln’s willingness to “protect, defend, and perpetuate slavery in the states where it existed.” This tolerance of slavery in the South was there cited as evidence of Lincoln’s pro-white views. Now, however, in section 5, Douglass explains that Lincoln acted as he did not because he was indifferent to the fate of black slaves, but “because he thought that it was so nominated in the bond.” In other words, he acted out of fidelity to the Constitution. Lincoln’s pre-war willingness to leave slavery alone in the Southern states does not in any way disprove or lessen his antislavery convictions. Of course, Douglass himself disagreed with Lincoln about what precisely was “nominated in the bond.” Most notably, Douglass argued that the so-called “fugitive slave” clause of the Constitution did not, in truth, refer to slaves but rather to indentured servants (who had signed contracts and could be held to those legal terms). Nonetheless, even though he is not fully in accord with Lincoln’s reading of the document, Douglass moves his audience toward an appreciation of constitutional devotion. He is acutely aware that racial progress in the future will depend upon the fidelity of both blacks and whites to the Constitution—the Constitution as purified and completed by the 13th, 14th, and 15th Amendments. Fittingly, sections 6 and 7 transcend race altogether. These are the only sections that make no reference to either whites or blacks. Section 6 describes Lincoln’s early years and his preparation, through plain speaking and plain dealing, for the great crisis of civil war. Douglass 15 �emphasizes Lincoln’s humble origins: “A son of toil himself he was linked in brotherly sympathy with the sons of toil in every loyal part of the Republic.” In this section, racial division is overcome and replaced by the class division between the patrician, James Buchanan, who was willing to allow “national dismemberment,” and the plebeian, Abraham Lincoln, who had “an oath in heaven” to preserve, protect, and defend the Constitution of the United States. The division we ought to dwell on, Douglass implies, is that between patriotism and treason. This theme reaches an apotheosis in section 7 which describes the assassination of Abraham Lincoln. Despite the “hell-black spirit of revenge” that motivated the crime, Douglass argues that good has come from it. Dying as a martyr to “union and liberty”—these twin aims now conjoined and equal—Lincoln has become “doubly dear to us.”19 In his autobiography, Douglass noted that one effect of the assassination was to bring him into “close accord” with his white neighbors, feeling, for the first time he said, more like “kin” than “countrymen.” In the final section of the speech, just one paragraph in length, Douglass comes full circle, speaking once more to his largely black audience. He tells them: “In doing honor to the memory of our friend and liberator we have been doing highest honor to ourselves and those who come after us [emphasis added].” Note that despite the “unfriendly feeling” ascribed to Lincoln in sections 3 and 5, Lincoln by the end has become “our friend.”20 Through his interpretation and masterful presentation of Lincoln’s statesmanship, Douglass has knit together the American polity in mutual understanding and appreciation of Lincoln. Douglass has acted as a statesman himself by demonstrating how memory and memorialization, done well, might shape a better American future. In conclusion, let me just say a word about the larger lesson to be drawn from this speech. Frederick Douglass is best known as an activist. Much of his speaking and writing 16 �involved demands for justice: justice toward blacks, justice toward women, justice toward laborers. Approached by a young man asking what he should do for the cause of racial justice, the elderly Douglass is said to have answered “agitate, agitate, agitate.” However, this fabled agitator also devoted a goodly portion of his public speaking to commemorating the past, celebrating the founding ideals of the nation, and praising those citizens and public figures who remained faithful to both the Declaration and the Constitution. In other words, he tried to foster a spirit of friendship and a unified national consciousness. Aristotle (the first political scientist) called this homonoia, or like-mindedness. Likemindedness—or thinking the same—about certain crucial matters, is the form of friendship that should characterize fellow citizens. Aristotle calls this like-mindedness “the greatest of goods for the political order” (Politics 2.4.6). It lessens civic strife among the parts or parties that are always present in any larger collective. Diversity—without this foundation of like-mindedness— is a recipe for growing discord. Like-mindedness allows cooperation and trust to replace contentiousness and suspicion. Aristotle argued that lawmakers should pursue this sort of friendship more than justice even, since civic friendship leads to justice and does so without having to involve the coercive bite of the law (Ethics 8.1). In friendship, what is right and what is pleasing come naturally together. For a model of how to encourage this civic friendship, there are very few who equal Frederick Douglass. Especially in our contemporary moment, as protests have erupted over incidents of racial injustice, as well as over statues and memorials that are thought to symbolize and contribute to ongoing injustice, I can’t think of a better resource than Frederick Douglass. He is one of our nation’s greatest fighters against injustice and he took very seriously the topic of public commemoration. It matters intensely who we memorialize and how we understand the past. 17 �Let me mention just a couple of things that distinguish Douglass from today’s protestors and progressives. While Douglass was a fierce critic of our national transgressions, he also believed deeply in the American project. He considered the principles of the Declaration of Independence to be “saving principles” and he considered the Constitution to be “a great liberty document.” He criticized the nation from the perspective of its own highest ideals, calling us to live up to our professions. Although he could be bitingly satirical, he was never cynical. He was always ready to find and praise what was good and generous and true in the American experiment. I am worried that this spirit of gratitude is being lost. There is a very deep alienation expressed in much contemporary rhetoric and action. Over the last summer, this hostility went so far as to threaten the Freedmen’s Monument itself with destruction. There are many reasons to preserve the monument, including that it was the site of one of Douglass’s most significant speeches and that it marks what Douglass called his people’s first “national act”—the act by which they translated their gratitude for emancipation into an enduring work of art. The controversy over the monument will be salutary if it leads us to revisit its history and reread Douglass’s speech. He can help us toward a more thoughtful and nuanced patriotism. Lerone Bennett, Jr. Forced into Glory: Abraham Lincoln’s White Dream (Chicago: Johnson Publishing Company, 2007). 2 Diana J. Schaub, “Frederick Douglass’s Constitution,” in The American Experiment: Essays on the Theory and Practice of Liberty, ed. Peter Augustine Lawler and Robert Martin Schaefer (Lanham, MD: Rowman & Littlefield, 1994). 3 Lucas Morel has spoken and written insightfully on the “Oration.” See “America’s First Black President?: Lincoln’s Legacy of Political Transcendence” (2001) and “Frederick Douglass’s Emancipation of Abraham Lincoln” (2005). See also the excellent recent article by Peter C. Myers, “‘A Good Work for Our Race To-Day’: Interests, Virtues, and the Achievement of Justice in Frederick Douglass’ Freedmen’s Monument Speech,” American Political Science Review, Vol. 104, No. 2 (May 2010). 4 Frederick Douglass, “To W.J. Wilson,” in The Life and Writings of Frederick Douglass, ed. Philip S. Foner (New York: International Publishers, 1975), 4:173. 5 Ibid., 4:172. 6 Frederick Douglass, “To Harriet Beecher Stowe,” March 8, 1853, in Writings, 2:229-236. 1 18 �See especially “What the Black Man Wants, speech at the Annual Meeting of the Massachusetts Anti-Slavery Society at Boston, April 1865,” in Writings, 4:157-165. 8 Ibid., 4:172. 9 Ibid. 10 Ibid. 11 See especially “The Nation’s Problem: An Address Delivered in Washington, D.C., on 16 April 1889,” in The Frederick Douglass Papers, ed. John W. Blassingame (New Haven: Yale University Press, 1985), 5:414-416. 12 “Inaugural Ceremonies of the Freedmen’s Memorial Monument to Abraham Lincoln Washington City, April 14 th, 1876,” available online through the Frederick Douglass Papers of the Library of Congress. 13 http://www.richardscenter.psu.edu/Documents/ArlingtonNationalCemeteryTour.pdf. 14 “Inaugural Ceremonies of the Freedmen’s Memorial Monument to Abraham Lincoln Washington City, April 14th, 1876.” http://memory.loc.gov/mss/mfd/18/18006/0009.jpg. http://memory.loc.gov/mss/mfd/18/18006/0010.jpg. 15 Douglass cited these lines from Byron’s Childe Harold’s Pilgrimage (Canto II, Stanza LXXVI) often, including in “What Are the Colored People Doing for Themselves?” The North Star, July 14, 1848 in Life and Writings, 1:315. 16 Abraham Lincoln to Horace Greeley, August 22, 1862 in Abraham Lincoln: Speeches and Writings 1859-1865 (NY: The Library of America, 1989), 358. 17 At the dedication of the Lincoln Memorial in 1922, the keynote address was given by Dr. Robert Moton, Booker T. Washington’s successor as president of Tuskegee Institute. Douglass might have been intrigued to learn that Moton spoke not of Union, but of Liberty, fixing Lincoln’s claim to greatness in “the word that gave freedom to a race.” In the draft of his speech, Moton proceeded to transform the Negro’s debt to Lincoln into the nation’s (unpaid) debt to the Negro, a rhetorical move that displeased the organizers and forced Moton to tone down his talk of a “great unfinished work” of “equal opportunity.” Even with the edits, however, the focus of the speech was emancipation. Almost a half-century later, Dr. Martin Luther King, Jr. would sound a very similar theme in his “I Have a Dream” speech on the steps of the Lincoln Memorial. 18 See especially, “The Black Man’s Debt to Abraham Lincoln,” 12 February 1888, and “Abraham Lincoln, the Great Man of Our Century,” 13 February 1893, both in Blassingame, volume 5. 19 Walt Whitman’s lecture, “Death of Abraham Lincoln,” first delivered in 1879, further develops the meaning of Lincoln’s martyrdom. 20 Douglass’s eulogy of his fellow abolitionist Wendell Phillips provides an interesting point of comparison. Douglass asserts that “none have a better right” to honor the memory of Phillips than “the colored people of the United States.” Although he was active for a variety of causes, Phillips “was primarily and pre-eminently the colored man’s friend, . . . The cause of the slave was his first love; and from it he never wavered, but was true and steadfast through life.” “Wendell Phillips Cast his Lot with the Slave: An Address Delivered in Washington, D.C., on 22 February 1884,” in Blassingame, vol. 5, 151-2. 7 19 � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data Word Page numeration Number of pages in the original item. 19 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Learning to love Lincoln : Frederick Douglass's journey from grievance to gratitude Description An account of the resource Transcript of a lecture given on December 4, 2020 by Diana Schaub as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "Having originally been a severe critic of Abraham Lincoln, the radical abolitionist Frederick Douglass grew to appreciate Lincoln’s prudential statesmanship. In his 1876 'Oration in Memory of Lincoln' he recapitulated that intellectual and emotional journey for the benefit of all Americans. Creator An entity primarily responsible for making the resource Schaub, Diana J., 1959- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2020-12-04 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Douglass, Frederick, 1818-1895 Lincoln, Abraham, 1809-1865 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_SchaubD_Learning_to_Love_Lincoln_2020-12-04 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ac4b6f540ba925d1848cb34256cb33cf.pdf 8431c0adbb1b2619139579a74e5a0f49 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 17 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The magnificent pendulums Description An account of the resource Transcript of a lecture given on November 18, 2020 by Howard Fisher as part of the Dean's Lecture and Concert Series. The Dean's Office provided this description of the event: "With Galileo, we learn to understand the two natural motions by reducing change to constancy (constant speed, constant acceleration). But trying to understand the motion of a pendulum by Galileo’s way generates an infinite task. Instead, we look to the Form of pendular motion: the Paradigm Circle. To what extent does the Paradigm Circle tell us which properties of the pendulum are essential to its distinctive motion, and which are merely extrinsic or fortuitous? We will look at several pendulum clock mechanisms, as well as a mechanical device that demonstrates the relation between the pendulum and its Paradigm." Creator An entity primarily responsible for making the resource Fisher, Howard J. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2020-11-18 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Galilei, Galileo, 1564-1642 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_The_Magnificent_Pendulums_2020-11-18 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/23553c1b87bea84f48183634bb5e452c.pdf 5be9420cfda855a79b9eb57a02e540ee Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 23 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Maxwell reads Faraday Description An account of the resource Transcript of a lecture given on April 24, 2020 by Howard Fisher as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Fisher, Howard J. Publisher An entity responsible for making the resource available St. John' s College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2020-04-24 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Maxwell, James Clerk, 1831-1879 Faraday, Michael, 1791-1867 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FisherH_Maxwell_Reads_Faraday_2020-04-24 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4a65bf30753fde968144de5087f5a1a1.pdf 16ab3ce7f64760d83e8ce10568ec5584 PDF Text Text The Phenomenology of Blackness By, Michael E. Sawyer, PhD Assistant Professor of Race, Ethnicity, and Migration Studies and The Department of English Colorado College Delivered at St. John’s College Santa Fe, New Mexico Carol J. Worrell Annual Lecture Series on Literature 22 November 2019 �1. Introduction: My earliest encounter with the academic discipline we generally label as philosophy, occurred during my sophomore year at the Jesuit prep school I attended in Chicago. My teacher, Brother McCabe, was a Franciscan Monk who was a Kantian. In addition to endlessly referencing the categorical imperative, Brother McCabe’s most memorable trope was “always examine your presuppositions”. I vividly recall him asking me a question and when I began to answer he stopped me and demanded I account for the presuppositions that would attend my attempt at an answer. My first inclination was to propose that the central presupposition happened to be that to the extent I didn’t provide any answer, much less a satisfactory one, I would be in danger of failing his class. That seemed insufficient so I spent what seemed the better part of 10 minutes examining the assertions and systems of thinking, understandings, and misunderstandings that marched before the answer I don’t think I ever managed to provide. Looking back now I realize that the genius of Brother McCabe’s pedagogical strategy was to insure we left his class first and foremost inclined to examine our presuppositions. This to me is the foundation of thinking that can best be described as “critical” to the extent it takes stock of the stuff around the thing you find interesting. Here, I have framed the title of this talk as a statement: The Phenomenology of Blackness that includes several important questions and a foundational presupposition. First the questions: What is Phenomenology? We have an academic answer to that question that we can admit into our archive here as axiomatically the manner in which the term is commonly employed in the discipline of philosophy. The Stanford Encyclopedia of Philosophy provides the following definition: Phenomenology is the study of structures of consciousness as experienced from the first-person point of view. The central structure of an experience is its intentionality, its being directed toward something, as it is an experience of or about some object. An experience is directed toward an object by virtue of its content or �meaning (which represents the object) together with appropriate enabling conditions. The next term is perhaps more fraught and will be the subject of the talk: What is Blackness? In referencing that question we identify the central presupposition which feels something like Blackness is a Thing (das Ding) in the Hegelian sense of the term and it is possible to experience this thing in the phenomenological sense. There is quite a bit going on here and we do not have the time to properly account for all of the moving parts so I will be direct in framing our employment of the elements that are commonly understood in the field. This primarily revolves around the naming of Blackness as das Ding or the Thing as the point of inquiry. What I mean here is most productively elaborated in §120 of Hegel’s Phenomenology of Spirit which reads: However, the diverse aspects which consciousness takes upon itself are determinate in that each is regarded as existing on its own within the universal medium. White is only in contrast to black, etc., and the thing is a “one” precisely in virtue of its being contrasted with others. This is not a facile reading of Hegel that extracts this binary opposition of black and white as philosophically significant without context. Here, Hegel is lingering on the implication of “Consciousness” and its marginalization by being negatively implicated by sense certainty and therefore requiring the more robust dialectical process that arrives with his understanding of the relationship between Lord and Bondsman in the well know passages from §190 as an essential element of the torturous journey of mediated self-consciousness to the terminus of Absolute Spirit. Here, Hegel proposes the following: The master is consciousness existing for itself. However, the master is no longer consciousness existing for itself merely as �the concept of such a consciousness. Rather, it is consciousness existing for itself which is mediated with itself through another consciousness, namely, through an other whose essence includes its being synthetically combined with self-sufficient being, that is, with thinghood itself. The intellectual water, so to speak, has become very choppy very quickly. This, in many ways provides the linchpin or hinge to access the preoccupation of this effort: the notion of Blackness as a discernible way of Being. Here, in the elaboration of consciousness, Hegel proposes that the indicator of the need for the move to self-consciousness is the failed employment of sense certainty, in his parlance the apparent separation between black and white, as spectacular. When he arrives at the dialectic of Lord and Bondsman, the existence of the Thing, the master is only able to relate themselves as master by the “thinghood” of the bondsman. What this means for my argument is that the spectacular nature of the color black is the marker of marginal consciousness and with the subjects that will be the preoccupation of this talk that certainty remains by rendering the dialectic of Lord and Bondsman, in this instance moribund. The encounter is already over-determined. The dialectic, when operating properly, does not prefigure the status of the subjects who are its participants. In this instance, because of the spectacular nature of the color Black, the dialectic is reifying rather than determinative. The Black subject is locked as the Bondsman because of the overdetermined nature of the color Black and what we are exposing here as Blackness serves as the prison. I posit that this is not an error on the part of Hegel. No lesser authority than the Socratic Dialog The Phaedrus establishes how the color black arrives in the western imaginary as a lack when compared to its opposite, white. The passage is instructive: “Let us then liken the soul to the natural union of a team of winged horses and their charioteer. The gods have horses and charioteers that are themselves all good and come from good stock besides, while everyone else has a mixture. To begin with, our driver is in charge of a pair of horses; second, one of the horses is beautiful and good and �from stock of the same sort, while the other is the opposite and has the opposite bloodline. This means that the chariot-driving in our case is inevitably a painfully difficult business. (246b)” It is the detail of the description of the horses that concerns our work here in that it speaks to a robust consideration of black and white as oppositional in quality. Socrates delineates the “goodness of the good horse and the badness of the bad.” (253d) “The horse that is on the right, or nobler, side is upright in frame and well jointed, with a high neck and a regal nose; his coat is white, his eyes are coal black, and he is a lover of honor with modesty and selfcontrol; companion to true glory, he needs no whip, and is guided by verbal commands alone. The other horse is a crooked great jumble of limbs with a short bull-neck, a pug nose, black skin, and bloodshot white eyes; companion to wild boasts and indecency, he is shaggy around the ears – deaf as a post – and just barely yields to horsewhip and goad combined. (253d)” Here we witness, at the earliest moments of the western philosophical tradition, explicit reference to the soul as being divided in two with the worst impulses; those that do violence to reason, represented by physical deformity of which “black skin” is but one manifestation of physical and metaphysical disability. Reason and the ability to properly accede to moral authority, in this formulation, are exemplified by beauty, in conforming to a physical standard, and Whiteness. What that means, walking with Hegel on our left and Socrates on our right, is that the visuality of black (Hegel’s sense-certainty and the Socratic “Bad Horse”) arrives at the point of self-consciousness with an externally imposed system of self-knowledge that we will label here as “Blackness”. This understanding is most ably exposed by the canonical formulation of what I have labelled “Tripartite Subaltern Self-Consciousness” by W.E.B. Du Bois in The Souls of Black Folk. In that text, Du Bois establishes the insufficiency of the Cartesian Cogito for a system of knowing for the subject we will label here as existing under conditions of externally imposed Blackness and the presupposition of insufficiency; the figure Du Bois understands as the Negro writing: �After the Egyptian and Indian, the Greek and Roman, the Teuton and Mongolian, the Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world,-a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this doubleconsciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity. One ever feels his two-ness,an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder. (Du Bois, 6) Here, Du Bois understands the fractured nature of the relationship of the self to the self on the part of the figure he understands as the Negro at a discernible remove from being a body that is understood as Black: “two warring ideals in one dark body”, as the fulfillment of the trace that runs from the pronouncement of Black as a lack from Socrates through Hegel’s destabilized dialectic. When I note here that this lack of “true self-consciousness” in the parlance of Du Bois is properly glossed by studying the result of a self that reaches out to touch itself and receives a negative response. Not the “I am” that the Western philosophical tradition situates as the normative response to the self thinking about itself but perhaps an “I am not” in the case of a distorted practice of self-analysis. Where Du Bois describes a subject that is double-conscious, in that it only knows about itself “through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity” we find a subject out of synch with Descartes. Further, and here Jean-Paul Sartre’s text, The Transcendence of the Ego, proves useful, he endeavors to explicate what he calls “consciousness in the first degree” or “unreflected consciousness” as distinct from “consciousness in the second degree” or “reflected consciousness”. Du Bois’ formulation is productively read alongside the theorizing of Sartre. What Du Bois seems to mean is that the subject suffering from a lack of true self-consciousness has, in fact, confused consciousness in the second degree for consciousness in the first degree. Further, the secondary system of consciousness has contempt for the subject so situated and causes the self that reflects on itself here to believe that it is, indeed, aberrant when in fact these are externally imposed �conditions of knowing that fracture the possibility of a way to present the self for recognition by other subjects. This question of mutual recognition is the next step along this continuum that, in important ways, returns to the visual nature of what Hegel understands as consciousness encumbered with sense certainty and what Socrates describes as the physical manifestation of marginalization as exemplified by the Black horse. In many ways, and this is, perhaps, an idiosyncratic methodological point, at this stage of this analysis I believe we reach, what I like to call, Technical Exhaustion. That which has been exhausted, in my understanding is first the utility of prose or discourse in the form of philosophy or theory to describe the phenomenon in question. Second, not only does the visual exceed the technical potentiality of words but Western epistemologies fail here as well. I hope we can discuss this assertion in the Q&A, it is a statement that is meant to be positively provocative. But on this point, taking the last assertion first, the manner in which western epistemology finds itself incapable of properly describing the essence of the subject in question is because this system of knowing in fact is dedicated to the destruction of the subject in question. One need only note the manner in which language has already been stacked against the humanity of the subject understood to be Black. Black, the term itself, is fraught. Our time does not facilitate tracing the tortuous path to resolving this tension but it can broadly be understood as a component in the requirement to decolonize the canon and the reason the academy has derived disciplines that range from Africana studies to Feminist and Gender Studies, etc. One brief methodological point here. I do not believe that one decolonizes the canon through a project of contraction or excision of the central pillars of the western system of knowing. On the contrary, in order to exceed the boundaries of that system one must know what the boundaries happen to be. With that in mind we have employed Socrates, Hegel, Descartes, and at the edges of that tradition Du Bois to approach the phenomenon of Blackness. Beyond that boundary we will explore a relationship that I am in the early stages of exploring: that between visual representations of Black �people being coerced by the police, alternative modes of temporal existence and African-American literature in the form of fiction. This is something of an intellectual bank shot so to continue the analogy to a game of billiards, I will just call the shot. I hypothesize that the bridge forward and backward is the visual. I further hypothesize that the visual representation of police violence represents a temporal shift that speaks to what I have called in other spaces the fractured temporality of the subaltern. And finally, I hypothesize that it is only in fiction that we can grapple effectively with this disorientation. Elements of Roland Barthes’ Camera Lucida speak to this thinking and along with a reference to St. Augustine, will provide the markers of the boundaries we intend to exceed together. Barthes’ writes: The Operator is the Photographer. The Spectator is ourselves, all of us who glance through collections of photographs-in magazines and newspapers, in books, albums, archives…And the person or thing photographed is the target, the referent, a kind of little simulacrum, any eidolon emitted by the object, which I should like to call the Spectrum of the Photograph, because this word retains, through its root, a relation to ‘spectacle’ and adds to it that rather terrible thing which is there in every photograph; the return of the dead. (Barthes, 9) There is an important note here in that I intend to apply Barthes to analyze video even though he proposes that the dynamism of moving pictures is discernably different than the immobility of the photographic image. For our purposes here, we will see that it is the freezing of the subject in space and perhaps consigning them to death in video representations of police violence that the motion of the images is in fact only to memorialize what is the substantively coercively imposed immobility or death. I have a short bit of video that I would like to play at this point that is intended to further buttress my argument. I am interested here in bringing into our conversation the way in which the restriction of mobility memorializes the fractured temporal subjectivity of Blackness. To give some context here, the mechanical process of walking preoccupied the thinking of Ray Bradbury in the �writing of the canonical exposition of the dangers of run-away state power in Fahrenheit 451. Bradbury reveals in the notes preceding his 1951 short story “The Pedestrian” that the arrest of the protagonist for merely walking down the street near his home was informed by the author’s harassment by the Los Angeles Police Department for doing the same. It is this notion of walking down a sidewalk that serves as the opening scene of 451 that is an effect of the causality of police coercive force that restricts the ability to walk that one might note Kant understands as evidence of maturity and I situate here, in its restriction, as a technology for the creation and maintenance of marginalized subjectivity. This is Kant from the essay “What is Enlightenment” for point of reference where he meditates on free movement. Here the progress of the subject is alienated from his humanity and therefore his freedom by the pronouncement of this officer that weaponizes his body and possessions. The shift to spectacle occurs when the officer asserts that the subject is being audio and video taped. The restriction of his mobility occurs through the assertion by the officer that he is not free to leave. There is a way in which that the qualification here, “to leave”. is redundant. The subject is not free. He cannot walk and his immobility is contrasted by the hyper (in comparison) mobility of the humans around him. His possessions are confiscated and inspected. The officer asserts that she will access the videotape to check for evidence of the attack she witnessed. In spite of video evidence that is contrary to the claim that the subject had weaponized his golf club and swung it at the police car he is arrested and charged with a series of crimes. What we have witnessed is the way in which the fractured temporality of the subaltern body, here understood as Blackness places this subject immediately in the clutches of death that has arrived with the presence of the law. Recall the manner in which the officer begins to pronounce his death sentence. One can detect that it is a recitation that is designed to render whatever coercive force she intends to visit upon �this body as necessary. There is much to interrogate with this video but our time together requires us to shift to the next element of my argument, that Western philosophical epistemologies find themselves technically exhausted here and fiction becomes the most efficacious mode of explicating what we are witnessing. Prior to addressing our literary reference, we will briefly linger with St. Augustine as the last stop on the road of the western philosophical canon. Saint Augustine proposes the following, “si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio” roughly translated; “if no one asks me what time is I know but if I have to explain it, I don’t know what time is.” In my reading the Augustine establishes the distinction we need in the front of our minds between time and temporality. In this instance, I am reading Augustine as actually commenting on this distinction. The Saint understands time as a force and as a tool for measurement but when asked to explicate that understanding his relationship to time shifts to the experience of it, or what I am labelling as temporality or the self-referential understanding of the experience of time by the subject or subjects in question. So here we are necessarily taking up the challenge posed by St. Augustine to explicate time and to do so not from the perspective of dealing with it as a force or measure but as a way of being. In the Augustine we find his explanation of the manner in which we experience time instructive. Augustine accomplishes this through his careful analysis of reciting a psalm by memory. Suppose I am about to recite a psalm which I know. Before I begin, my expectation is directed towards the whole. But when I have begun, the verses from it which I take into the past become the object of my memory. The life of this act of mine is stretched two ways, into my memory because of the words I have already said and into my expectation because of those which I am about to say. But my attention is on what is present: by that the future is transferred to the past. As the action advances further and further, the shorter the expectation and the longer the memory, until all expectation is consumed, the entire action is finished, and it has passed into the memory. (11 28:38) �Paul Ricoeur, in Volume 1 of his Time and Narrative series writes the following in commenting on this move by Augustine: “The solution is elegant-but how laborious, how costly, and how fragile!” The fragility of this proposition is on full display when it is applied to the subject under investigation here. Memory, or the past, serves as the sine qua non of this understanding of time. Following Augustine’s analogy, it becomes immediately apparent that there can be no recitation of a psalm from memory unless there has been a time in which the psalm itself was experienced and remembered in some past. This does not attend in the case of the subjects who have had their relationship to genealogy and history fractured via the Middle Passage and its telos, the condition of enslavement. What I mean is that a set of experiences that serve to destroy, fracture, or confuse a coherent relationship to memory, and here think of memory as operating as “culture”, also renders normative temporal existence as established by Augustine, as impossible. Therefore, there can be no past, present, or future from the perspective of Augustinian Time for the figure harmed by the Middle Passage and its echoes we can label here as bigotry and or racism. In spite of this fracture with memory it is empirically “true,” that the subject so harmed has a time in which it exists, it was created, and has an internal time signature, because it has desire. We do not have the time here to for me to fully expose the manner in which I am calling on desire as the foundation of an interior sense of time but let it suffice to say that it is related to the probative power of Terry Pinkerton’s recent re-translation of Hegel’s Phenomenology of Spirit. In that essential document, Prof. Pinkerton revised what had been an error in translating the essential passage, §167 as “self-consciousness is desire itself,” as opposed to the alternative and more traditional “selfconsciousness is desire in general”. This is transformational. §167…But this opposition between its appearance and its truth has only the truth for its essence, namely, the unity of self-consciousness with itself. This unity must become essential to self-consciousness, which is to say self- consciousness is desire itself. As self-consciousness, consciousness henceforth has a doubled object: The first, the �immediate object, the object of sense- certainty and perception, which however is marked for it with the character of the negative; the second, namely itself, which is the true essence and which at the outset is on hand merely in opposition to the first. Self-consciousness exhibits itself therein as the movement within which, in its own eyes, the selfsameness of itself with itself comes to be. It is this movement of the self within the self that I am positing as establishing the internal time signature necessary for being Human. What this means is that the canonical formulation by W.E.B. Du Bois in his Souls of Black Folks must be understood in a manner that asserts that what Du Bois calls a lack of true self-consciousness is a lack of desire which, following this path, amounts to a lack of a coherent internal time signature. What this means here is that we can understand the shattered subject that preoccupies Du Bois as frustrated by an imposed understanding of their lack of historical “situatedness” as an externally imposed subaltern sense of time that we will see Morrison witnesses as the paralysis of “nows” that cannot recede into “thens” which cannot be supersceded by “whens”. It is critically important that we carefully attend to what Du Bois has offered here in that what I am holding up as the nexus of the fracture between the coercive nature of internal time for Blackness and what Jacques Derrida understands as the “self-calling to the self” when he reflects on elements of Kant’s Third Critique. Briefly, I understand the establishment of an internal notion of time, this self-calling or autoaffection, as the foundation of what we understand as being Human, positively and self-referentially aware of the self as a temporal being that then has an individually established time signature that can be presented to other similarly situated beings for purposes of mutual recognition and here we should have Hegel foremost in our minds. Here, operating from an understanding of the Cartesian Cogito, we understand this self-calling as allowing us to witness the self-thinking about the self as a manner to establish the subject in time. What Du Bois describes, the subject with “no true self-consciousness”, is effectively an externally imposed fracturing of the subject forming and reifying power of self-reflection in terms of �Descartes and self-authorizing temporality on the part of thinkers like Derrida through Kant. Hence the paralysis we explore that is exposed by Morrison which, when traced in this way, is what Du Bois understands as the impossibility of the Black Body being able to productively exist as “Negro” in the parlance of the day and “American” at the same time. So here, we should recall the video we observed and witness it as an exemplar of a form of subjective liminality and recursive ways of Being that render him out of time and beyond the explanatory capabilities of tools like Heidegger’s understanding of the relationship of Being to thrownness toward the telos of mortality, death. The subject we observed in the video comes to us as a “rememory” (and we will attend to this shortly) in the way in which Morrison understands that phenomenon. We are witness to his treatment in ways that are at no separation from the treatment of human beings captured and transported against their will through the Middle Passage. His treatment mirrors that of enslaved bodies who found themselves out of place: think here about those who became victims to the empowerment of all white people to arrest any Black person with the passage of the Fugitive Slave Act. His paralysis at the hands of the state is at no remove from that employed in the aftermath of the Civil War by the Contract Labor System. The same goes for Jim Crow and here this man finds himself unwittingly a participant in the present day’s carceral system that is fruit of the poisonous tree of enslavement. So, let us consider our encounter with the video as a form of memory that must be understood as placing bodies so coerced and formed with the notion of Blackness as the way of being, as operating outside of time. Recall here the manner in which I have summoned St. Augustine here. If we view the video as a “rememory” and insert it into the psyche of a similarly situated being that lacks coherent relationship to the past, Augustine’s system of temporal existence authorized by recall that I have noted as fractured through Du Bois’ formulation of the unstable system of consciousness of Blackness, finds itself exhausted and Morrison takes over. What I mean is that the collective system �of coercive threat to the subject that functions like racism creates an interwoven system of awareness that renders it impossible to separate the fate of one Black body from that of another. Further, the fact that the behavior re-memoried here on video is of the same genus and species as the abuse of Black bodies since roughly 1619 on these shores, the ability to fix one’s system of memory along a coherent temporal continuum fails. Morrison, by summoning the ghost of the child Sethe executed to save her from slavery, allows us to render this experience, this Blackness, legible. The point of literary reference here is from Toni Morrison’s masterpiece Beloved that, in my reading, revolves around the fractured temporality of those who are members of a family tree that touches, even tangentially upon the depravity and howling savagery of the Trans-Atlantic slave trade. The section in question here opens with the voice of the haunting figure of the murdered child Beloved pronouncing, “I am Beloved and she is mine.” To read the prose in question is to experience Morrison at her most sublime. Here she lays out a temporal continuum from the wars in West Africa that antecede the Middle Passage to the experience of death by a child at the hands of their mother without pause and without referent to the linear progression of time. What Morrison labels in the same text as “Rememory” and relates the state of being as “All of it is now it is always now”1 I have posited that Morrison’s preoccupation with the Middle Passage as a subject disforming catastrophe is a predictable ramification of the destruction of a coherent relationship to history and culture that is a particularly insidious element of the metaphysical harm done by this transit. 2. Beings Out of Time: I find Morrison’s Beloved to be an enchanted text that causes me deep, and oftentimes unresolvable trauma whenever I visit it. But it is necessary and masterful in that capacity. As I proposed a bit ago, Morrison is about the business of demonstrating that the trauma that is slavery is collective and that trauma reverberates across the generations. What she calls “ramifications of ramifications” �in her text Paradise. In Beloved, Morrison’s character Sethe is explaining this temporal confusion to her daughter Denver. The text reads: “I was talking about time. It’s so hard for me to believe in it. Some things go. Pass on. Some things just stay. I used to think it was my rememory. You know. Some things you forget. Other things you never do. But it’s not. Places, places are still there. If a house burns down, it’s gone, but the place-the picture of it-stays, and not just in my rememory, but out there in the world. What I remember is a picture floating around out there outside my head…” “Can other people see it?” asked Denver. “Oh, yes. Oh, yes, yes, yes. Someday you be walking down the road and you hear something or see something going on. So clear. And you think it’s you thinking it up. A thought picture. But no. It’s when you bump into a rememory that belongs to somebody else.” (Morrison, 43.) I will make the connection here explicit though I hope it might arrive of its own volition from the narrative we are weaving together. Morrison is resolving the tension we have located between the wages of the color black articulated by Plato and Hegel, the imperative of memory on the part of Augustine, and the notion of fractured or compromised self-consciousness as described by Du Bois. When we, whatever our subjectivity, encounter pictures of the coercion of Black bodies we are encountering rememories and our positionality becomes indistinct from that of others similarly situated across time and space. What I mean is that these pictures, these visual representations of bodies in pain in the process of being rendered un-free, are what Sethe referred to as “thought” pictures for two reasons. One might be obvious. They are the disassociated point of view of the bodies in contact with one another, think Hegel’s Lord and Bondsman here and we as observers occupy a third place in the room as spectator. However, I wish here to push Hegel a bit and propose that the relationship between observer and observed is also involved in a dialectical relationship and the Thing between us, Observer and Observed, in the observation of these videos becomes the time and subject destabilizing wage of Blackness. The second is that the behavior we are observing is a thought picture �in that it renders visible the thinking behind systems of white supremacy like Madison’s pronouncement of fractional humanity in Federalist 54. I’ll quote the text here for clarity and context. Madison writes: But we must deny the fact that slaves are considered merely as property, and in no respect whatever as persons. The true state of the case is, that they partake of both those qualities; being considered by our laws; in some respect, as persons, and in other respects, as property. In being compelled to labor not for himself, but for a master; in being vendible by one master to another master, and in being subject at all times to be restrained in his liberty, and chastised in his body, by the capricious will of another, the slave may appear to be degraded from the human rank and classed with the irrational animals, which under the legal domination of property. The Fœderal Constitution therefore, decides with great propriety on the case of our slaves, when it views them in the mixt character of persons and property. Morrison continues and here we are in Book II of the text where she abandons the conventions of punctuation to express what I am framing here as the rememory induced phantasm of Blackness. I AM BELOVED and she is mine. I see her take flowers away from leaves she puts them in a round basket the leaves are not for her she fills the basket she opens the grass I would help her but the clouds are in the way how can I say things that are pictures I am not separate from her there is no place where I stop her face is my own and I want to be there in the place where her face is and to be looking at it too a hot thing All of it is now it is always now there will never be a time when I am not crouching and watching others who are couching too I am always crouching and watching others who are crouching too I am always crouching the man on my face is dead his face is not mine his mouth smells sweet but his eyes are locked. (Morrison. 248) These are complex and fragile passages and worthy of our most diligent efforts at close reading. I have struggled for years with these sections of the text but have found that they yield to my effort when I address them with the understanding we have traced of Blackness. Blackness as an externally imposed system of subjective disorientation that mires the subject in the impossible task of �achieving the form of self-consciousness required to achieve forward progress. Further, I have been aided in understanding Morrison’s notion of rememory as “thought pictures” by the overwhelming presence of videos of Black bodies under conditions of coercion. All of this creates the very system of disorientation that the spectral presence called Beloved experiences that disallows her from being able to separate the experiences of those in her genealogy who have suffered coercive force. There is a great deal going on here in these passages which represent the third in a series of four of this form of narrative where Morrison takes up the challenge that she has embedded in the passage in question, “how can I say things that are pictures”. In fact, the implicit question here is how the subject experiencing these visions, these rememories, might process them and situate herself in time and space and resist the coercion that she is experiencing as a ramification of those ramifications. The passage opens with a decentering of the notion of internal mind/body separation as well as the separation between discreet subjects. “I am Beloved and she is mine.” It is important to note here that this is the only declarative sentence with the employment of a period to eliminate ambiguity. With this understanding we can read the next passages as if the observer is also the actor. “I see her take flowers away from leaves she puts them in a round basket the leaves are not for her she fills the basket she opens the grass” What is important to note here is that Beloved has bumped into rememories of some other subject in her genealogy. It both is and is not her mother. In this system of perception, it is her mother and hers and everyone in between, starting with a time before the middle passage. The beauty of the images she bumps into are necessary as the counterpoint to the depravity of Atlantic World Slavery. This metaphysical impossibility of separation is experienced by this subject as physical inseparability. “I am not separate from her there is no place where I stop her face is my own and I want to be there in the place where her face is and to be looking at it too a hot thing” If one explores this text and these sections in particular one will encounter this refrain “a hot thing”, over and over again. Things indeed become hot and this subjective immersion in the wretched horror of �the Middle Passage freezes time: “All of it is now it is always now there will never be a time when I am not crouching and watching others who are couching too I am always crouching and watching others who are crouching too I am always crouching”. Beloved, the spirit come to haunt her mother for the act of killing her as a technique to emancipate her from slavery, has never picked flowers in Africa nor directly experienced the Middle Passage but the presentism of these thought pictures is a result of the terrible power of this regime of coercion. For the specter and for the reader it is now, always now. The video we experienced a bit ago in the parlance of Morrison., is a hot thing. 3. Conclusion: The challenge now is to tie this up in a manner that allows us to discuss it and view this thinking as a point of departure that points in many directions at once and sweeps through and across multiple systems of knowing and archives past, present, and to come. There are several points of inflection here that we should mark: the translation of the existence of the visual encounter with the Other yields to the gloss put on that experience by the Socratic dialog that is absorbed by Hegel whose thinking then becomes the target of intellectual challenge by Du Bois. This system must again yield to the somatic or the corporeal encounter with the wages of Blackness as experienced through the employment of video. That experience confuses and disorients us all independent of subject position, yields as well and is gathered together, in the parlance of Morrison, by saying things that are pictures. The real question, the foundational presupposition, is why should we care? I use “we” here advisedly. I don’t mean the “we” of those who are understood to be living under the experience of what we have labelled here as “Blackness”. As a practical matter I also do not mean the life of the mind that privileges this kind of thought experiment as valuable for the sake of the effort. I mean we as a question of humanity and to be succinct we have to care because we are all participants, perpetrators, or observers in what we can frame as the wages of the sin of establishing a societal order that builds its demos on the imperative of exclusion and the notion of freedom as valuable only in the �presence of the possibility or actuality of its opposite. Blackness, as a master signifier here, can be understood to, in its abstraction, speak for the plight of all the aggrieved and the maligned. Speaking for violence against trans bodies in the same way it shouts the despair of children separated from their guardians and caged because they are seeking safety. The same goes for the mosque, synagogue, or bar that is attacked for the presence of what are coercively framed as transgressive bodies or systems of thinking. The same goes for a future that has the potential to erase human existence, in the way we understand it, based upon the poor stewardship of the earth that is the wretched refuse of rabid capitalism. I have gestured, perhaps obliquely at the central problematic of how the figure we have examined here has been called into existence as the unwitting oppositional way of being that allows something like democracy to exist. This is based upon the notion that the value of freedom is only discernible and measurable in its dialectical relationship against its opposite way of being: un-freedom. With that in mind the challenge before us is to imagine and bring into being a type of humanism and in its aggregation, in the form of a societal order that forms itself outside of the logic of seeking the middle point between two extremes. The reason for this effort is not to create something like the debunked and reductive notion of colorblindness. If nothing else remains in our collective minds at the close of this talk it must be this. Blackness is not a race or a color nor an ethnicity. It is an externally imposed system of marginalization that renders its victims and purveyors locked in an unnecessary system of subjective destruction. What this means is that “Blackness” is only related to being a Black person in that this political epoch has assembled that figure and allowed for it to stand as a master signifier for the Muslim, the Queer, the Native, the immigrant, the Jew, the Trans, it is endless, which means that systems of power will always seek a figure to cloak in the subjective disability of Blackness. In his recent text Necropolitics, Achille Mbembe proposes the following and with its recitation I will close our time together. Mbembe writes: The colonial world, as an offspring of democracy, was not the antithesis of the democratic order. It has always been its double or, �again, its nocturnal face. No democracy exists without its double, without its colony – little matter the name and the structure. The colony is not external to democracy and it is not necessarily located outside its walls. Democracy bears the colony within it, just as colonialism bears democracy, often in the guise of a mask…In other terms, the cost of the mythological logics required for modern democracies to function and survive is the exteriorization of their originary violence to third places, to nonplaces, of which the plantation, the colony, or today, the camp and the prison, are emblematic figures. (Mbembe) This quotation, perhaps in some measure, explains the way in which the videos we encounter serve to memorialize and resist the erasure that might allow us the luxury of believing that these excesses are either a thing of the past or not meant for us. So long as anyone suffers under the refined technology of Othering, the result of which we have labelled here as Blackness, we all suffer and more to the point, are necessarily at risk. The methodological question for all of us that have, in one way or another, chosen the life of the mind is to focus our attention, across the canon we study and through the scholarship we create, on requiring that the proper attention be paid to the presence of the tail of the dragon of hatred that weaves its way through our consciousness. Thank you. 1 Ibid. 248. � https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4461b8c6adcb75ab9ba142a9fb596772.pdf 7e23480032f607a5ea7605fb9b37796a PDF Text Text The Phenomenology of Blackness Presented by, Michael E. Sawyer, PhD Colorado College St. John’s College Santa Fe, New Mexico Carol J. Worrell Annual Lecture Series on Literature 22 November 2019 �“However, the diverse aspects which consciousness takes upon itself are determinate in that each is regarded as existing on its own within the universal medium. White is only in contrast to black, etc., and the thing is a “one” precisely in virtue of its being contrasted with others.” G.W. F. Hegel Phenomenology of Spirit §120 �“The master is consciousness existing for itself. However, the master is no longer consciousness existing for itself merely as the concept of such a consciousness. Rather, it is consciousness existing for itself which is mediated with itself through another consciousness, namely, through another whose essence includes its being synthetically combined with self-sufficient being, that is, with thinghood itself.” Ibid. §190 �“‘Let us then liken the soul to the natural union of a team of winged horses and their charioteer. The gods have horses and charioteers that are themselves all good and come from good stock besides, while everyone else has a mixture. To begin with, our driver is in charge of a pair of horses; second, one of the horses is beautiful and good and from stock of the same sort, while the other is the opposite and has the opposite bloodline. This means that the chariot-driving in our case is inevitably a painfully difficult business.’” The Phaedrus (246b) �“ ‘The horse that is on the right, or nobler, side is upright in frame and well jointed, with a high neck and a regal nose; his coat is white, his eyes are coal black, and he is a lover of honor with modesty and self-control; companion to true glory, he needs no whip, and is guided by verbal commands alone. The other horse is a crooked great jumble of limbs with a short bull-neck, a pug nose, black skin, and bloodshot white eyes; companion to wild boasts and indecency, he is shaggy around the ears – deaf as a post – and just barely yields to horsewhip and goad combined.’ ” Ibid. (253d) �“After the Egyptian and Indian, the Greek and Roman, the Teuton and Mongolian, the Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world, - a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity. One ever feels his two-ness,- an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder.” W.E.B. Du Bois The Souls of Black Folk ��““The Operator is the Photographer. The Spectator is ourselves, all of us who glance through collections of photographs-in magazines and newspapers, in books, albums, archives…And the person or thing photographed is the target, the referent, a kind of little simulacrum, any eidolon emitted by the object, which I should like to call the Spectrum of the Photograph, because this word retains, through its root, a relation to ‘spectacle’ and adds to it that rather terrible thig which is there in every photograph; the return of the dead.” Roland Barthes Camera Lucida �������“Suppose I am about to recite a psalm which I know. Before I begin, my expectation is directed towards the whole. But when I have begun, the verses from it which I take into the past become the object of my memory. The life of this act of mine is stretched two ways, into my memory because of the words I have already said and into my expectation because of those which I am about to say. But my attention is on what is present: by that the future is transferred to the past. As the action advances further and further, the shorter the expectation and the longer the memory, until all expectation is consumed, the entire action is finished, and it has passed into the memory.” St. Augustine Confessions 28:38 �“…But this opposition between its appearance and its truth has only the truth for its essence, namely, the unity of selfconsciousness with itself. This unity must become essential to selfconsciousness, which is to say self- consciousness is desire itself. As self-consciousness, consciousness henceforth has a doubled object: The first, the immediate object, the object of sense- certainty and perception, which however is marked for it with the character of the negative; the second, namely itself, which is the true essence and which at the outset is on hand merely in opposition to the first. Selfconsciousness exhibits itself therein as the movement within which, in its own eyes, the selfsameness of itself with itself comes to be.” Hegel §167 �“I was talking about time. It’s so hard for me to believe in it. Some things go. Pass on. Some things just stay. I used to think it was my rememory. You know. Some things you forget. Other things you never do. But it’s not. Places, places are still there. If a house burns down, it’s gone, but the place-the picture of it-stays, and not just in my rememory, but out there in the world. What I remember is a picture floating around out there outside my head…” �“Can other people see it?” asked Denver. “Oh, yes. Oh, yes, yes, yes. Someday you be walking down the road and you hear something or see something going on. So clear. And you think it’s you thinking it up. A thought picture. But no. It’s when you bump into a rememory that belongs to somebody else.” Toni Morrison Beloved �“But we must deny the fact that slaves are considered merely as property, and in no respect whatever as persons. The true state of the case is, that they partake of both those qualities; being considered by our laws; in some respect, as persons, and in other respects, as property. In being compelled to labor not for himself, but for a master; in being vendible by one master to another master, and in being subject at all times to be restrained in his liberty, and chastised in his body, by the capricious will of another, the slave may appear to be degraded from the human rank and classed with the irrational animals, which under the legal domination of property. The Fœderal Constitution therefore, decides with great propriety on the case of our slaves, when it views them in the mixt character of persons and property.” James Madison Federalist 54 �I AM BELOVED and she is mine. I see her take flowers away from leaves she puts them in a round basket the leaves are not for her she fills the basket she opens the grass I would help her but the clouds are in the way how can I say things that are pictures I am not separate from her there is no place where I stop her face is my own and I want to be there in the place where her face is and to be looking at it too a hot thing All of it is now it is always now there will never be a time when I am not crouching and watching others who are couching too I am always crouching and watching others who are crouching too I am always crouching the man on my face is dead his face is not mine his mouth smells sweet but his eyes are locked. Morrison Beloved � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 20 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The phenomenology of blackness Description An account of the resource Transcript of a lecture given on November 22, 2019 by Michael Sawyer as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Sawyer, Michael E. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-11-22 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Relation A related resource <p>The following video is referenced in this lecture:</p> <p><a href="https://youtu.be/n92zfXkRdnk">https://youtu.be/n92zfXkRdnk</a></p> Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_SawyerM_The_Phenomenology_of_Blackness_2019-11-22 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4076cf9a600178cf678478821f331c65.pdf cc69968c45c5be06d9f9a13dede8202e PDF Text Text The Heptadecagon Grant Franks October 8, 2019 Review We’ve looked at the first encounters with the -1 , the development of complex arithmetic, the roots of unity, constructibility of points on the complex plane (considered algebraically) and followed the construction of the pentagon. These points will be reviewed a final time next Wednesday in the final lecture. (Remember: Wednesday, October 16, Junior Common Room, 3:15 pm.) Let’s move on to the Heptadecagon. Remember the Pentagon? We’ve seen how to approach the algebraic construction of the pentagon. The process begins with the equation: x5 - 1 = 0 One factors out the one real solution that all “roots of unity” equations share, namely (x - 1): 1 + x + x2 + x3 + x4 = 0 This equation is irreducible so long as one allows only rational numbers, Q, as solutions. (In mathjargon, it is “irreducible over the rationals.”) But if one constructs a finite quadratic field extension Q( 5 ), it can be factored into two quadratics. A second finite quadratic field extension allows it to be factored fully into four linear factors from which you can read off the solutions readily. �2 ��� 5 The Heptadecagon.nb 0.31 + 0.95 ⅈ -0.81 + 0.59 ⅈ -0.81 - 0.59 ⅈ 0.31 - 0.95 ⅈ x4 + x3 + x2 + x + 1 = 0 Irreducible over Q Then adjoin 5 x2 - η2 x +1 x2 - η1 x +1 (x - ζ1 ) Irreducible over F1 Irreducible over F1 Then adjoin ζ1 Then adjoin ζ2 (x - ζ4 ) Do the Same Thing, But More O�en (x - ζ2 ) (x - ζ3 ) �5 The Heptadecagon.nb ��� 3 We’ll follow the same basic plan to construct the heptadecagon. However, the procedure has a few additional complications due to the greater number of steps. The Equation For the heptadecagon, we start with the equation: x 17 - 1 = 0. Again we factor out the one real solution (x - 1) to obtain: 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = 0 This 16th degree equation has sixteen roots, all complex, that we designate: ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 , ζ11 , ζ12 , ζ13 , ζ14 , ζ15 , ζ16 Graphically, these roots appear on the complex plane as vertices of a regular 17-gon, making equal angles at the center: ζ5 ζ4 0+1i ζ3 ζ6 ζ2 ζ7 ζ ζ8 -1 + 0 i 1+0i ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 0 - 13 1i ζ ζ 14 Unsurprisingly, these occur in eight pairs of complex conjugates. (In the diagram above, complex conjugates are joined by orange dotted lines). Arrangement of the Sixteen Roots: the Eight-Periods �4 ��� 5 The Heptadecagon.nb Following the general procedure seen with the pentagon, we are going to split these roots up into -two groups of eight, then four groups of four, then eight groups of two, then sixteen individuals. At each stage, we will make numbers by taking the sums of the members in each group. With the pentagon, we found that even though we didn’t know the values of any of the roots (the ζ’s), we could figure out a quadratic formula for the values of the two intermediate sums: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 because we could figure out their sum and the product, η1 + η2 and η1 ×η2 . When working on the pentagon, the way in which to subdivide the four roots presented little trouble. We had reason to believe that the complex conjugates had to stay together, so there was only one possible subdivision of the four roots into two pairs. The second division separated the two pairs roots from their conjugate mates. Now, however, we have sixteen roots and eight pairs of conjugates. For the first subdivision, there are 8 x 7 x 6 x 5 = 1,680 possible ways to separate the eight pairs into two groups of four. We don’t know a priori whether some or all, or not all or possibly only one will work. On the surface, it seems that trial and error might not work. Sorting out the roots properly is more than half the battle in doing this construction. It will require a small detour. Half the Problem is Pretty Easy First, some good news. We’re looking for a sorting of the roots that will allow us to find the sum and the product of the two groups. In that quest, the sum of the two groups will pose no problem. No matter how we divide the sixteen roots into two bunches, we will be able to get their total sum. Say we just sort out the first eight and the last eight: A = ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 B = ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 Now, when we add A + B, we get the sum of all sixteen roots. And we know that to be equal to negative one. In fact, no matter what subdivision we make, the sum of the two divisions will be negative one. Remember: �5 The Heptadecagon.nb ��� 5 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = 0 which is to say: x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = -1 Just put in ζ for x: ζ + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 So the issue that has to be addressed is η1 × η2 , the product of two eight-term sums. Modular Arithmetic Sorting out the other half of the problem will involve us in modular arithmetic. It’s not at all difficult for anyone who has read clock. � ζ6 ζ ζ7 ζ2 ζ5 ζ3 ζ8 ζ4 Remember that multiplying roots of unity by themselves -- that is, raising them to powers -- will move the solution around the unit circle in the complex plane like a clock hand. (In this case the clock hand goes counterclockwise; all analogies have problems!). For example, there are five fi�h roots of unity. If I square the first one, then cube it and so forth, the result moves around the unit circle. Also, when the �6 ��� 5 The Heptadecagon.nb hand has gone completely around, all later solutions are equivalent to one or another of the first five solutions. Thus, ζ 6 is equivalent to ζ 1 . The technical term for this sort of equivalence is “congruence”; we write ζ 6 ≡ (ζ 1 )Mod 5 , “zeta to the sixth is congruent with zeta one, modulo 5.” Congruence of this sort will be very useful for us, for Gauss’s solution to the sorting problem involves some very high powers of the 17th roots of unity. Primitive Roots One observation about modular arithmetic before we go on. Suppose we are working in modulo 17 -as we will be doing. Take some number, a, and raise it to successive powers. In ordinary arithmetic, it will grow continually. In modular arithmetic, it will go around the cycle of available numbers. Some numbers in doing to touch all the values available; some do not. Take 2, for instance: TableFormTablen, 2n , Mod2n , 17, {n, 1, 16}, TableHeadings → None, "n", "2n ", "(2n )mod 17 " n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2n 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16 384 32 768 65 536 (2n )mod 17 2 4 8 16 15 13 9 1 2 4 8 16 15 13 9 1 If I use “n” to designate the power to which one raises the root, notice that 2 cycles through seven values before coming to n=1 and repeating itself. On the other hand, 3 cycles through all the possible values before repeating itself �5 The Heptadecagon.nb ��� 7 TableFormTablen, 3n , Mod3n , 17, {n, 1, 16}, TableHeadings → None, "n", "3n ", "(3n )mod 17 " n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 3n 3 9 27 81 243 729 2187 6561 19 683 59 049 177 147 531 441 1 594 323 4 782 969 14 348 907 43 046 721 (3n )mod 17 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1 It can be shown that, for prime numbers, there is always at least one such value. For our purposes, with the 17-gon, we only need one. The number three will work for us. (There are others; 2, 4, 8, 9, 13 and 15 don’t work; 3, 5, 6, 7, 10, 11, 12, and 14 do.) Ordering of the Sixteen Roots Gauss ordered the sixteen roots in accordance with the expression: n ζ (3 ) Since 3n modulo 17 cycles through all values from 1 to 16 before repeating, this ordering will encompass all sixteen complex roots. The ordering looks like this: n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 3n 1 3 9 27 81 243 729 2187 6561 19 683 59 049 177 147 531 441 1 594 323 4 782 969 14 348 907 43 046 721 (3n )mod 17 1 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1 ζ3 ζ ζ3 ζ9 ζ10 ζ13 ζ5 ζ15 ζ11 ζ16 ζ14 ζ8 ζ7 ζ4 ζ12 ζ2 ζ6 ζ n mod 17 �8 ��� 5 The Heptadecagon.nb Notice that each term is obtained from the previous one by successive powers of three. Notice also, that if one takes every other term, one has a succession by powers of nine: 1, 9, 81, 729 … or 3, 27 = 3 x9, 243 = 3 x 81, 2187 = 3 x 729 … This will be useful in what follows. Two Eight Periods The 16-period is divided into two 8-periods by taking alternate members of the series and summing the. η1 = ζ + ζ9 + ζ13 + ζ15 + ζ16 + ζ8 + ζ4 + ζ2 η2 = ζ3 + ζ10 + ζ5 + ζ11 + ζ14 + ζ7 + ζ12 + ζ6 Notice that each period contains four pairs of complex conjugates. ζ5 ζ ζ4 ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 ζ 14 ζ 13 Complex 17th roots of unity - Two Eight-Periods Sum of the 8-periods η1 + η2 = -1, as explained above. Product of the 8-periods The product η1 η2 requires some calculation. We have the multiplication of two eight-term sums, which will yield sixty four terms: �5 The Heptadecagon.nb ζ + ζ9 + ζ13 + ζ15 + ζ16 + ζ8 + ζ4 + ζ2  ζ3 + ζ10 + ζ5 + ζ11 + ζ14 + ζ7 + ζ12 + ζ6  = … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 exponent1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 13 13 13 13 13 13 13 13 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 8 8 8 8 8 8 8 8 4 4 4 4 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + exponent 2 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = sum 4 11 6 12 15 8 13 7 12 19 14 20 23 16 21 15 16 23 18 24 27 20 25 19 18 25 20 26 29 22 27 21 19 26 21 27 30 23 28 22 11 18 13 19 22 15 20 14 7 14 9 15 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ summod 17 4 11 6 12 15 8 13 7 12 2 14 3 6 16 4 15 16 6 1 7 10 3 8 2 1 8 3 9 12 5 10 4 2 9 4 10 13 6 11 5 11 1 13 2 5 15 3 14 7 14 9 15 ��� 9 �10 ��� 5 The Heptadecagon.nb 53 54 55 56 57 58 59 60 61 62 63 64 4 4 4 4 2 2 2 2 2 2 2 2 + + + + + + + + + + + + 14 7 12 6 3 10 5 11 14 7 12 6 = = = = = = = = = = = = 18 11 16 10 5 12 7 13 16 9 14 8 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 1 11 16 10 5 12 7 13 16 9 14 8 This is a little hard to digest. However, there are some patterns and repetitions. Looking just at the exponents of the terms: 1 + 3 = 4 ; 9 + 10 = 19 ; 13 + 5 = 18 ; 1 + 10 = 11 ; 9 + 5 = 14 ; 1 + 5 = 6 ; 15 + 11 = 26 ; 16 + 14 = 30 ; 8 + 7 = 15 ; 13 + 11 = 24 ; 15 + 14 = 29 ; 16 + 7 = 23 ; 9 + 11 = 20 ; 13 + 14 = 27 ; 15 + 7 = 22 ; 1 + 11 = 12 ; 9 + 14 = 23 ; 13 + 7 = 20 ; 1 + 14 = 15 ; 9 + 7 = 16 ; 1 + 7 = 8 ; 13 + 12 = 25 ; 15 + 6 = 21 ; 9 + 12 = 21 ; 13 + 6 = 19 ; 15 + 3 = 18 ; 1 + 12 = 13 ; 9 + 6 = 15 ; 13 + 3 = 16 ; 1 + 6 = 7 ; 13 + 10 = 23 ; 15 + 5 = 20 ; 9 + 3 = 12 ; 8 + 12 = 20 ; 4 + 6 = 10 ; 16 + 12 = 28 ; 8 + 6 = 14 ; 15 + 12 = 27 ; 16 + 6 = 22 ; 16 + 3 = 19 ; 8 + 3 = 11 ; 4 + 3 = 7 ; 2 + 3 = 5 ; 2 + 10 = 12 ; 4 + 10 = 14 ; 2 + 5 = 7 ; 8 + 10 = 18 ; 4 + 5 = 9 ; 16 + 10 = 26 ; 8 + 5 = 13 ; 15 + 10 = 25 ; 16 + 5 = 21 ; 4 + 12 = 16 ; 2 + 6 = 8 ; 2 + 11 = 13 ; 4 + 11 = 15 ; 2 + 14 = 16 ; 8 + 11 = 19 ; 4 + 14 = 18 ; 2 + 7 = 9 ; 16 + 11 = 27 ; 8 + 14 = 22 ; 4 + 7 = 11 ; 2 + 12 = 14 ; Or, with the sums reduced modulo 17: 1 + 3 = 4 ; 9 + 10 = 2 ; 13 + 5 = 1 ; 1 + 10 = 11 ; 9 + 5 = 14 ; 13 + 11 = 7 ; 1 + 5 = 6 ; 15 + 11 = 9 ; 16 + 14 = 13 ; 8 + 7 = 15 ; 4 + 12 = 16 ; 2 + 6 = 8 ; 15 + 14 = 12 ; 16 + 7 = 6 ; 9 + 11 = 3 ; 13 + 14 = 10 ; 15 + 7 = 5 ; 8 + 12 = 3 ; 4 + 6 = 10 ; 16 + 12 = 11 ; 8 + 6 = 14 ; 4 + 3 = 7 ; 2 + 3 = 5 ; 2 + 10 = 12 ; 1 + 11 = 12 ; 9 + 14 = 6 ; 13 + 7 = 3 ; 15 + 12 = 10 ; 16 + 6 = 5 ; 8 + 3 = 11 ; 4 + 10 = 14 ; 2 + 5 = 7 ; 1 + 14 = 15 ; 9 + 7 = 16 ; 13 + 12 = 8 ; 15 + 6 = 4 ; 16 + 3 = 2 ; 8 + 10 = 1 ; 4 + 5 = 9 ; 1 + 7 = 8 ; 15 + 3 = 1 ; 16 + 10 = 9 ; 8 + 5 = 13 ; 4 + 11 = 15 ; 2 + 14 = 16 ; 1 + 12 = 13 ; 9 + 6 = 15 ; 13 + 3 = 16 ; 15 + 10 = 8 ; 16 + 5 = 4 ; 8 + 11 = 2 ; 4 + 14 = 1 ; 1 + 6 = 7 ; 15 + 5 = 3 ; 16 + 11 = 10 ; 8 + 14 = 5 ; 4 + 7 = 11 ; 9 + 12 = 4 ; 13 + 6 = 2 ; 9 + 3 = 12 ; 13 + 10 = 6 ; Or, again, more graphically: 2 + 11 = 13 ; 2 + 7 = 9 ; 2 + 12 = 14 ; �5 The Heptadecagon.nb ζ4 ζ 11 ζ6 ζ 12 ζ 15 ζ8 ζ 13 ζ7 ζ2 ζ 14 ζ3 ζ6 ζ 16 ζ4 ζ 15 ζ 12 ζ ζ7 ζ 10 ζ3 ζ8 ζ2 ζ 16 ζ6 ζ9 ζ 12 ζ5 ζ 10 ζ4 ζ ζ8 ζ3 ζ 13 ζ6 ζ 11 ζ5 ζ2 ζ9 ζ4 ζ 10 ζ 15 ζ3 ζ 14 ζ 11 ζ ζ 13 ζ2 ζ5 ζ 16 ζ 10 ζ7 ζ 14 ζ9 ζ 15 ζ ζ 11 ��� 11 ζ8 ζ5 ζ 12 ζ7 ζ 13 ζ 16 ζ9 ζ 14 Notice that the yellow rows have all the same members; so do the green rows. Also, that the yellow rows and the green rows have all different members, so that a yellow row and a green row together have all sixteen elements. And all sixteen elements together equal negative one, so that the total of all sixtyfour terms is … (drum roll, please) … negative four. This is not an accident. It was carefully orchestrated by Gauss’s arrangement of the two eight-groups. In fact, the necessity that leads to this arrangement can be seen through examination and analysis of the patterns of the terms entering into the multiplication together with about a week of practice with modular multiplication. I can’t undertake that dissection in more detail here; any one interested can pursue a more complete presentation in the texts referred to in the handout. For our purposes, what is essential is that we know the sum and the product of the eight-periods, η1 and η2 . Constructing and Solving an Appropriate Quadratic We now know that the sum η1 + η2 = -1 , and the product η1 η2 = -4. We can therefore make a quadratic equation with these two numbers as its solutions. Using y as a variable, we have: y2 -  (-1) y +  ( -4) = 0 η1 + η 2 η1 η2 whose solutions will be η1 and η2 . This equation can be solved with the quadratic formula. y= -1 ± 1 + 16 2 = -1 ± 2 17 Beautiful. So I know values of the η’s, which are the sums of the 8-periods. The two values are: �12 ��� 5 The Heptadecagon.nb η1, 2 = -1 ± 2 17 The approximate values for η1 and η2 are 1.56155 and -2.56155. The Four Periods Next we make four periods of four by taking every fourth root from the original series, starting with the first, second, third and fourth, respectively: ζ, ζ3 , ζ9 , ζ10 , ζ13 , ζ5 , ζ15 , ζ11 , ζ16 , ζ14 , ζ8 , ζ7 , ζ4 , ζ12 , ζ2 , ζ6 ζ13 , Period 1 = ζ, Period 2 = Period 3 = ζ3 , ζ16 , ζ5 , ζ9 , Period 4 = ζ4 , ζ14 , ζ15 , ζ10 , ζ12 ζ8 , ζ11 , ζ2 ζ7 , ζ6 Sums of the Four Periods We make the sums of each of the four-periods: μ1 = ζ + ζ4 + ζ13 + ζ16 μ2 = ζ3 + ζ5 + ζ12 + ζ14 μ3 = ζ2 + ζ8 + ζ9 + ζ15 μ4 = ζ6 + ζ7 + ζ10 + ζ11 Again, the sums present no difficuly: μ1 + μ3 = η1 and μ2 + μ4 = η2 , since the the four-periods are gotten by segregating elements of the two eight-periods. Graphically, the four periods are pictured below. Notice that the 8-periods have been subdivided: the red 8-period into red and green 4-periods; the blue 8-period into blue and orange 4-periods. Once again, notice that each four-period includes two pairs of complex conjugates: �5 The Heptadecagon.nb ζ5 ζ ��� ζ4 ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 ζ 14 ζ 13 Inner dots = 8 periods; Outer dots = 4 periods Products of the 4-Periods As for the products of the μ’s, we can work out the terms directly. First, take μ1 times μ3 : Expand[μ1 μ3] ζ3 + ζ6 + ζ9 + ζ10 + ζ12 + ζ13 + ζ15 + ζ16 + ζ18 + ζ19 + ζ21 + ζ22 + ζ24 + ζ25 + ζ28 + ζ31 Which, when simplified by re-expressing the exponents modulo 17: ζ3 + ζ6 + ζ9 + ζ10 + ζ12 + ζ13 + ζ15 + ζ16 + ζ1 + ζ2 + ζ4 + ζ5 + ζ7 + ζ8 + ζ11 + ζ14 Put in numerical order of the exponents: ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 And for μ2 times μ4 : Expand[μ2 μ4] ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 + ζ18 + ζ19 + ζ20 + ζ21 + ζ22 + ζ23 + ζ24 + ζ25 Again, reduced by re-expressing the exponents modulo 17: ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 + ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 Put in numerical order of the exponents ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 13 �14 ��� 5 The Heptadecagon.nb Cool. Solving for the μ’s Once again, we know the sums and products of pairs of variables, in this case μ1 and μ3 and also μ2 and μ4 : μ1 + μ3 = η1 μ1 × μ3 = -1 μ2 + μ 4 = η 2 μ2 × μ4 = -1 With these sums-and-products, we can make two quadratic equations; we use v and w as variables: v 2 - η1 v - 1 = 0 μ1 , μ 3 = η1 ± whose solutions are μ1 and μ3 η1 2 + 4 2 w2 - η 2 w - 1 = 0 And μ2 , μ 4 = η2 ± whose solutions are μ2 and μ4 η2 2 + 4 2 Just to show where we are at this point, we can identify the values of the μ’s: μ1 = η1 + η1 2 + 4 2 = 1 2 1 2 -1 + 17  + 4 + 14 -1 + 17  μ2 = η2 + η2 2 + 4 2 = 1 2 1 2 -1 - 17  + 4 + 14 -1 - 17  μ3 = η1 - η1 2 + 4 2 = 1 2 1 2 -1 + 17  - 4 + 14 -1 + μ1 = η2 - η2 2 + 4 2 = 1 2 1 2 -1 - 17  - 4 + 14 -1 - 2 ≈ 2.04948 2 ≈ 0.344151 17  2 ≈ 0.487928 17  2 ≈ -2.9057 The Two Periods With this in hand, we look at the two-periods, obtained as before but this time taking every eighth root from the original list: β1 = ζ + ζ 16 β2 = ζ 3 + ζ 14 β3 = ζ 8 + ζ 9 �5 The Heptadecagon.nb β4 β5 β6 β7 β8 = = = = = ��� ζ 7 + ζ 10 ζ 4 + ζ 13 ζ 5 + ζ 12 ζ 2 + ζ 15 ζ 6 + ζ 11 It may be worth noting that each pair of roots that make up a β is a complex conjugate pair. This is importanT, although its special importance won’t appear until the next stage. As in previous steps, these two-periods come about by separating elements of the four-periods. Their sums thus lead us back to the variables of the previous step: β1 β2 β3 β4 + + + + β5 = ζ + ζ 16 + ζ 4 + ζ 13 = μ1 β6 = ζ 3 + ζ 14 + ζ 5 + ζ 12 = μ2 β7 = ζ 8 + ζ 9 + ζ 2 + ζ 15 = μ3 β8 = ζ 7 + ζ 10 + ζ 6 + ζ 11 = μ4 We thus have sums of pairs of the β’s. It remains to figure out the products of the same pairs. Products of the 2-Periods With a little labor, we can figure out the products of the 2-periods paired in way given above. The products are given below, including the reduction of the exponents modulo 17: β1 β5 = ζ + ζ16  ζ4 + ζ13  = ζ5 + ζ14 + ζ20 + ζ29 = ζ5 + ζ14 + ζ3 + ζ12 = μ2 β2 β6 = ζ5 + ζ12  ζ3 + ζ14  = ζ8 + ζ15 + ζ19 + ζ26 = ζ8 + ζ15 + ζ2 + ζ9 = μ3 β3 β7 = ζ2 + ζ15  ζ8 + ζ9  = ζ10 + ζ11 + ζ23 + ζ24 = ζ10 + ζ11 + ζ6 + ζ7 = μ4 β4 β8 = ζ6 + ζ11  ζ7 + ζ10  = ζ13 + ζ16 + ζ18 + ζ21 = ζ13 + ζ16 + ζ1 + ζ4 = μ1 Now we have defined the products as well as the sums of the four pairs of 2-periods. Again, the way in which this multiplication works out is not an accident; it follows from Gauss’s original ordering of the roots that combinations taken by twos, by fours, and so forth will always multiply so as to produce these intermediate periods. Constructing Four Quadratic Equations With that in mind, we can construct four quadratic equations with the β’s as roots: q2 - μ1 q + μ2 = 0 whose roots are β1 and β5 15 �16 ��� 5 The Heptadecagon.nb r2 - μ2 r + μ3 = 0 whose roots are β2 and β6 s2 - μ3 s + μ4 = 0 whose roots are β3 and β7 t2 - μ4 t + μ1 = 0 whose roots are β4 and β8 Their solutions can be obtained with the quadratic formula. Since we know the values of the μ’s, we can calculate the values of the β’s: β1 = μ1 + μ1 2 - 4 μ2 2 = 1.86494 β5 = μ1 - μ1 2 - 4 μ2 2 = 0.184537 β2 = μ2 + μ2 2 - 4 μ3 2 = 0.891477 β6 = μ2 - μ2 2 - 4 μ3 2 = -0.547326 β3 = μ3 + μ3 2 - 4 μ4 2 = 1.47802 β7 = μ3 - μ3 2 - 4 μ4 2 = -1.96595 β4 = μ4 + μ4 2 - 4 μ1 2 = -1.20527 β8 = μ4 - μ4 2 - 4 μ1 2 = -1.70043 The Singletons One more step remains: dividing the 2-periods into individual roots. This is in some ways the easiest step of all. Their Sums There are sixteen individual roots: ζ, ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 , ζ11 , ζ12 , ζ13 , ζ14 , ζ15 , ζ16 These, take pairwise in a particular order, constitute the β’s: β1 β2 β3 β4 β5 β6 β7 β8 = = = = = = = = ζ + ζ 16 ζ 3 + ζ 14 ζ8 + ζ9 ζ 7 + ζ 10 ζ 4 + ζ 13 ζ 5 + ζ 12 ζ 2 + ζ 15 ζ 6 + ζ 11 Here we see that we already have the sums of the sixteen ζ’s, taken pairwise. Their Products �5 The Heptadecagon.nb ��� This time, the product of the roots just as simple as the sums. Since, as already noted above, each β pair constitutes a pair of complex conjugates, their product -- obtained by adding their exponents -- is always seventeen or, on the unit circle in the complex plane, +1. ζ ζ16 = ζ17 = ζ3 ζ14 = ζ17 = ζ8 ζ9 = ζ17 = ζ7 ζ10 = ζ17 = ζ4 ζ13 = ζ17 = ζ5 ζ12 = ζ17 = ζ2 ζ15 = ζ17 = ζ6 ζ11 = ζ17 = 1 1 1 1 1 1 1 1 WIth this information, we have the sum and the products of the roots (taken in this special order) we can construct eight quadratic equations whose solutions are the ζ’s. r 2 - β1 r + 1 = 0 s2 - β 2 s + 1 = 0 t2 - β3 t + 1 = 0 v 2 - β4 v + 1 = 0 w2 - β 5 w + 1 = 0 x 2 - β6 x + 1 = 0 y 2 - β7 y + 1 = 0 z2 - β8 z + 1 = 0 whose solutions are ζ and ζ16 whose solutions are ζ3 and ζ14 whose solutions are ζ8 and ζ9 whose solutions are ζ7 and ζ10 whose solutions are ζ4 and ζ13 whose solutions are ζ5 and ζ12 whose solutions are ζ2 and ζ15 whose solutions are ζ6 and ζ11 We can apply the quadratic formula to find the solutions. Since we have the values for the β’s, we can obtain values for the ζ’s: ζ and ζ16 = β1 ± β1 2 - 4 2 = 0.932472 ± 0.361242 ⅈ ζ3 and ζ14 = β2 ± β2 2 - 4 2 = 0.445738 ± 0.895163 ⅈ ζ8 and ζ9 = β3 ± β3 2 - 4 2 = -0.982973 ± 0.18375 ⅈ ζ7 and ζ10 = β4 ± β4 2 - 4 2 = -0.850217 ± 0.526432 ⅈ ζ4 and ζ13 = β5 ± β5 2 - 4 2 = 0.0922684 ± 0.995734 ⅈ ζ5 and ζ12 = β6 ± β6 2 - 4 2 = -0.273663 ± 0.961826 ⅈ ζ2 and ζ15 = β7 ± β7 2 - 4 2 = 0.739009 ± 0.673696 ⅈ ζ6 and ζ11 = β8 ± β8 2 - 4 2 = -0.602635 ± 0.798017 ⅈ Shown graphically: 17 �18 ��� 5 The Heptadecagon.nb ζ4 ζ5 ζ ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 14 ζ 12 ζ 13 As advertised. Full Algebraic Presentation of the Sixteen Complex Roots of Unity The stack of quadratic equations involved in calculating the heptadecagon vertices is difficult to grasp when they are all assembled into a single formula. One of the roots is represented as: 1 2 1 4 1 2 17 - 17  + 1 8 -1 + 1 17  + 34+6  -4 + 1 64 -1 + 17 + 34 - 2 + 2 4 17 + 578-34 17 +  2 34 + 6 17 - 34-2 17 -8 2 17+ 17 17 + 2 578 - 34 17 - 34 - 2 17 - 8 2 17 + 17  It’s a challenge to grasp such a thing, but even a casual inspection shows that it consists exclusively of stacks of rational numbers and square roots combined with rational functions (addition, subtraction, multiplication and division). That alone is enough to guarantee its constructibility. �5 The Heptadecagon.nb ��� 19 An abbreviated graphic representation of the process of constructing the heptadecagon might look like this: ζ1 … 16 = β1, 2, 3, 4, 5, 6, 7, 8 = βn 2 - 4 βn ± 2 μ1, 2, 3, 4 2 - 4 μ2, 3, 4, 1 μ1, 2, 3, 4 ± 2 μ1, 3, 2, 4 = η1,2 = η1,2 ± η1,2 2 + 4 2 17 -1 ± 2 The sixteen roots were subdivided successively as follows. ζ , ζ 3 , ζ 9 , ζ 10 , ζ 13 , ζ 5 , ζ 15 , ζ 11 , ζ 16 , ζ 14 , ζ 8 , ζ 7 , ζ 4 , ζ 12 , ζ 2 , ζ 6 The whole t The μ's The ζ's ζ ζ 16 ζ 13 , ζ 4 ζ 13 ζ 3 , ζ 5 , ζ 14 , ζ 12 ζ 9 , ζ 15 , ζ 8 , ζ 2 ζ , ζ 13 , ζ 16 , ζ 4 ζ , ζ 16 The β's ζ 3 , ζ 10 , ζ 5 , ζ 11 , ζ 14 , ζ 7 , ζ 12 , ζ 6 ζ , ζ 9 , ζ 13 , ζ 15 , ζ 16 , ζ 8 , ζ 4 , ζ 2 The η's ζ4 ζ 9, ζ 8 ζ9 ζ8 ζ 15 , ζ 2 ζ 15 ζ 3 , ζ 14 ζ2 ζ3 ζ 14 ζ 10 , ζ 11 , ζ 7 , ζ 6 ζ 5 , ζ 12 ζ5 ζ 12 ζ 10 , ζ 7 ζ 10 ζ7 ζ 11 , ζ 6 ζ 11 ζ6 The sum of the whole set was -1. The sum of the η’s required a finite quadratic field extension to include -1 ± 17 2 . Each successive subdivision required another finite quadratic field extension of what went before. But (and?) that is the sine qua non of constructibility: a point is constructible if (and only �20 ��� 5 The Heptadecagon.nb if) it is defined by numbers that are either rational or the result of a succession of finite, quadratic field extensions from the rationals. Extension This is a pretty remarkable result: the 17-gon is constructible using the techniques available in Euclid’s Elements. In fact, Gauss’s result is even more remarkable than that. It has both a positive and a negative side, which I can report although we haven’t done quite enough work to demonstrate both sides fully. The Positive Side The positive side is that any figure is constructible if it the number of its sides is either, (a) a prime number equal to 2n + 1, or (b) some multiple of 2p times such a number, or (c) the sum of two of the primes in (a), or 2p times that sum. So, 20 = 1, 1 + 1 is 2, which is prime. You can’t make a 2-gon, really, but you can make polygons that are multiples of 2 times it: a 4-gon (square), and 8-gon (octagon), etc. Next, 21 = 2. 2 + 1 is 3, which is prime. You can make a 3-gon (triangle), or a 6-gon (hexagon), etc. A�er that, 22 = 4. 4 + 1 is 5, which is prime. You can make a 5-gon (pentagon). Euclid could do this. These are all the constructible prime n-gons that Euclid knew. He doesn’t say so much, but if he had known the construction of another it is hard to believe he would not have given it. A�er that, 24 = 16. 16 + 1 is 17, which is prime. You can make a 17-gon (heptadecagon). This was Gauss’s great discovery. I cannot believe that Euclid knew that the 17-gon was constructible. But there are more! Consider: 28 = 256. 256 + 1 = 257, which is prime. The 257-gon is constructible. Consider: 216 = 64 536. 64,536 + 1 = 64,537, which is prime. The 64,537-gon is constructible. In fact, if you can find another number of the form 2n + 1 which is prime, it too will be constructible. These are the so-called “Fermat numbers,” named for Pierre Fermat who conjectured that all numbers of the form 2n + 1 were prime, provided that n itself is a power of 2. Such numbers are: �5 The Heptadecagon.nb ��� 21 3, 5, 17, 257, 64537, … The next one would be 4,294,967,297 … but this on turns out not to be prime. (It is 641 times 6,700,417). As of last year, only the first eleven such numbers have been fully tested. The last one, 11 22 + 1, has 617 digits; it has two prime factors. The next one has 1,234 digits; whether it is prime or not is still undetermined. Las Vegas is not giving odds. In fact, now it is conjectured that apart from the first five, no other Fermat numbers are prime, although as far as I know that guess hasn’t been proven or disproven. The Negative Side The negative claim is that only these polygons are constructible. Gauss did not demonstrate that his construction method was the only one possible. The negative claim was demonstrated later by Pierre Wantzel in 1837. The seven-gon, which Euclid just skips, cannot be constructed. Neither can the 9-gon, the 11-gon or the 13-gon. Euclid constructs the (non-prime) 15-gon by the combination of the triangle and the pentagon. His leap from the hexagon to the 15-gon is completely unexplained in the Elements, and to my knowledge few students remark on it. (They should.) So What? What follows from this exercise? That is the subject of next week’s talk. Further Reading For further explication of Gauss’s sorting of the roots and modular congruences, you may wish to consult Hadlock, Field Theory and Its Classical Problems (Mathematical Association of America, 1978) � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 21 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The heptadecagon Description An account of the resource Transcript of a lecture given on October 8, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-08 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Polygons Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_The_Heptadecagon_2019-10-08 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c243d0e09b752feef01b9ac5cc9990b8.pdf 40ff74d84f2a2b7d84d208491d3d62f3 PDF Text Text The Pentagon and the Heptagon Recapitulation Two weeks ago, we saw how Rafael Bombelli began to suspect that imaginary numbers might be meaningful as he worked on the cubic equation x 3 - 15 x - 4 = 0 Using the formula Cardano stole from Tartaglia, got x= 3 2 + 11 -1 + 3 2 - 11 -1 which he was then able to solve by intuiting that 2 + 11 -1 = 2 + 3 -1  . The second lecture described Caspar Wessel’s graphic presentation of the arithmetic of complex numbers. On the complex number plane -(i) complex numbers can be expressed in polar coordinates by giving a distance (modulus) and an angle (argument); (ii) multiplication of complex numbers amounts to (a) multiplication of their distances (moduli) and (b) adding their angles (arguments); and (iii) the solutions of equations of the form x n - 1 = 0, known as the “Roots of Unity,” appear graphically as the vertices of an equilateral n-gon in the unit circle on the complex plane. Last week, we encountered the idea of “Constructible Numbers.” We showed that Euclid’s postulates allowed construction of lengths that correspond to the field of rational numbers, a collection of numbers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to irrational numbers). However, Euclid’s postulates do not permit construction of all incommensurable magnitudes. We can only construct those that correspond to numbers that can be found in towers of finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the countable infinity of the algebraic numbers. 3 �2 bers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to irrational numbers). However, Euclid’s postulates do not permit construction of all incommensurable 4 The Pentagon and the Heptagon.nb magnitudes. We can only construct those that correspond to numbers that can be found in towers of finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the countable infinity of the algebraic numbers. 3 We now turn to an application of what we have seen so far: construction of a regular pentagon in a given circle. The Lesser-Know Impossibility Problem Ancient geometry knew several classical problems that seemed impossible; the three most famous were trisecting the angle, doubling the cube and squaring the circle. These were daunting challenges. No one had found how to accomplish any of them, but the ancients did not know whether they were really impossible or only difficulties awaiting clever solutions Only in the late 18th and 19th centuries did we learn that these three problems really are impossible, at least with the tools of Euclidean geometry. In addition to these three, lesser-known but equally interesting problem is that of the heptagon, the regular seven-sided polygon. In book IV of the Elements, Euclid shows how to construct in a given circle an equilateral triangle (IV.2), a square (IV. 6), a pentagon (IV. 11) and a hexagon (IV. 15), regular figures with three, four, five and six sides. He then shows how to construct a regular 15-gon (IV. 16). Then he stops. The reader might be expected to wonder, why? Why jump from 6 to 15? Euclid, in his customary laconic way, says nothing. Some of the figures he skips over were easily constructible. The octagon is easily made by bisecting the angles of the square. The 10-gon can be gotten similarly from the pentagon and the 12-gon, from the hexagon. But the orderliness of Euclid’s sequence really falls apart with the 7-gon. With what we learned last week, it is easy to see that the 7-gon is impossible to construct. It is obtained from the polynomial: x 7 - 1 = ( x - 1)  x 6 + x 5 + x 4 + x 3 + x 2 + x 1 + 1 = 0 That sixth-degree polynomial is irreducible and, since its degree over the rationals is not a power of two, we can see right away that these complex roots are not constructible. Did Euclid know that the 7-gon was impossible? He probably suspected it. He surely knew that he couldn’t do it, which is not quite the same thing. Beyond the Impossible: the Unsuspected Possible In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what is mor, he realized at the same time that there are other polygons that can be constructed. Looking only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what Euclid knew. To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic �4 The Pentagon and the Heptagon.nb 3 In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what is mor, he realized at the same time that there are other polygons that can be constructed. Looking only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what Euclid knew. To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic construction of the pentagon. Euclid’s Construction of the Pentagon Of course, Euclid knew how to construct a regular pentagon in a given circle. To begin, let’s review how Euclid’s construction works. First, a Special Triangle He begins with construction of a very special triangle, one that is isosceles and whose base angles are both twice as big as its vertex angle. θ 2θ 2θ A little reflection shows why this triangle might be important to the construction of a regular pentagon: the three angles of the triangle total up to 180°, of course, but they also add up to five times the vertex angle. This triangle creates one angle that is one-fifth of 180°, and two that are one-fifth of 360°. If this triangle can be made, it will be the key to constructing the pentagon. However, constructing this triangle is no simple matter. To make it, Euclid recalls that back in book II, proposition 11, he had shown how to cut a line at a point so that the square on one portion of the line is equal to the rectangle contained by the whole line and the remaining portion of the line. �4 4 The Pentagon and the Heptagon.nb Digression This kind of division is known as one into "mean and extreme ratio," sometimes also referred to as the "Golden Ratio." It has many cool features, including connection to Fibonacci numbers and logarithmic spirals, but we haven' t time to get into all these things right now. If we take the whole AB to be “1” and the distance AC to be “x”, then finding this ratio can be understood as analogous to solving the equation: x 2 = (1 - x ) x2 + x - 1 = 0 or whose solutions are: 1± 1 - 4 (-1) 2 = 1± 5 2 You may note that these values are not rational, since they contains the square root of five. They are, of course, constructible, which we know because (a) we are dealing with a quadratic extension of the rationals and (b) because Euclid in fact constructs one of them. (No surprise there.) �1± 1 - 4 (-1) 2 = 1± 5 4 The Pentagon and the Heptagon.nb 2 5 You may note that these values are not rational, since they contains the square root of five. They are, of course, constructible, which we know because (a) we are dealing with a quadratic extension of the rationals and (b) because Euclid in fact constructs one of them. (No surprise there.) Returning to the Construction Euclid takes a line divided in this way and, using one end as a center, draws a circle with the whole line as a radius: A C B He then makes a chord in the circle equal to the larger segment of the divided line: A C B D He completes the triangle ABD, and joins CD: �6 4 The Pentagon and the Heptagon.nb A C B D Finally, he draws a circle that goes through points A, C and D: A C B D Thanks to a proposition from earlier in Book III, he knows that when from a point outside a circle (like point B) a line cuts a circle (as line BCA), and another line is draw to the circumference of the circle (as line BD), and when the rectangle on AB, AC is equal to the square on BD, then the line (BD) is tangent to the circle (ACD). With that established, another proposition of Book III allows him to say that the angle CDB (angle 1) is equal to the angle CAD (angle 2): �4 The Pentagon and the Heptagon.nb A C 2 B 4 5 3 1 D Add angle CDA to both. Thus angles 2 + 3 are equal to angles 1 + 3. But because AB = AD (in the circle around A), angles 1 + 3 are equal to angle 5 . So: angle 5 = angles 1 + angle 3 = angle 2 +angle 3 and because of exterior angles in triangle CBD angle 4 = angle 2 + angle 3 Therefore, triangle ABD is isosceles and line DB = line DC. And line DB = line AC. Therefore, angle 3 = angle 2 = angle 1. This, then, is the isosceles triangle with its base angles equal to twice the vertex angle. The Pentagon With the isosceles triangle having the base angles equal to the vertex angle now available, the rest is easy. 7 �8 4 The Pentagon and the Heptagon.nb Simply bisect the arcs standing on the longer sides, which are each twice the arc on the shorter side. Now you have five equal sides and your pentagon is complete. Join the vertices and you have not only a pentagon, but a pentangle (a regular five-pointed star). �4 The Pentagon and the Heptagon.nb 9 This construction is completely rigorous and very clever. However, it offers no clues at all about how to pursue construction of other such prime-sided polygons, such as the 7-gon, the 11-gon, the 13-gon, etc. Preliminary: the Pentagon The algebraic construction of the pentagon amounts to finding the roots of the fifth degree cyclotomic polynomial. That is, we begin with the equation: x5 = 1 or x 5 - 1 = 0. The number 1 is evidently a solution to this equation. It is, in fact, the only rational solution. Therefore, the equation can be factored by removing the factor (x - 1): x 5 - 1 = ( x - 1)  x 4 + x 3 + x 2 + x + 1 = 0 The Fifth Order Cyclotomic Polynomial The second expression,  x 4 + x 3 + x 2 + x + 1, is irreducible “over the rationals”; that is, it can’t be simplified by showing it to be the product of factors of lower degree among the rationals. We can be completely sure that this expression is irreducible because we know that the four roots of the polynomial x 4 + x 3 + x 2 + x + 1 = 0 are complex with imaginary components. They are the four non-real fifth roots of unity. �10 4 The Pentagon and the Heptagon.nb ζ1 ζ2 ζ3 ζ4 But being irreducible over the rationals doesn’t mean that this thing can’t be factored in an extended field. In fact, it has been shown that every polynomial of nth degree can be factored into n linear factors in the full complex number field. Our challenge is to find which factors need to be appended to the rationals in order to factor or “split” this fourth degree polynomial. Complex Conjugates We haven’t discussed complex conjugates, but this diagram presents the idea nicely. Notice that the complex roots of this polynomial appear as two pairs of complex numbers, symmetrically arranged above and below the real number axis. Root ζ1 is paired this way with root ζ4 and root ζ2 is paired with root ζ3 . Being so arranged, these roots are written in this form: a+bi and a - bi The expressions are the same except for the positive and negative signs attached to the imaginary portions. Complex conjugates have this handy feature: when a pair of complex conjugates are added, their sum is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a real number. This feature is handy because we are often looking for roots of polynomials whose coefficients are rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots: (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = … But if you multiply this trio out, you will have some imaginary coefficients. �is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a real number. 4 The Pentagon and the Heptagon.nb 11 This feature is handy because we are often looking for roots of polynomials whose coefficients are rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots: (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = … But if you multiply this trio out, you will have some imaginary coefficients. (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = (813 + 699 ⅈ) - (142 - 41 ⅈ) x + (4 - 8 ⅈ) x2 + x3 In fact, the only way to eliminate the imaginary components from the expanded polynomial is if the coefficients occur in pairs of complex conjugates. That way, when the conjugates are multiplied, the imaginary components disappear. Return to the Problem To solve our fourth-degree cyclotomic polynomial: 1 + x + x2 + x3 + x4 = 0 We will proceed in the usual, brash algebraic way: we will pretend that we already have the solutions. Then we’ll work to discover what they are. The Fundamental Theorem of Algebra tells us that this fourth degree equation has four solutions, which we will designate (as in the picture) ζ1 , ζ2 , ζ3 and ζ4 . Roots ζ1 and ζ4 are one pair of complex conjugates; ζ2 and ζ3 are another pair. Two-Stage Solution Take the sums of ζ1 , ζ4 and of ζ2 , ζ3 , like this: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 When added together, η1 and η2 sum up to -1 (because all the fifth roots of unity together sum to zero, and η1 and η2 include all the roots except (+1 + 0 i): η1 + η2 = ζ 1 + ζ 4 + ζ 2 + ζ 3 = -1 Also, the product of η1 and η2 works out like this: ζ 1 + ζ 4  ζ 2 + ζ 3  = ζ 3 + ζ 4 + ζ 6 + ζ 7 Restate this result with the exponents taken Modulo 5, because, on the unit circle in the complex plane, ζ 5 = ζ 0 = 1. Thus, we have ζ6 = ζ5 ζ1 = ζ1 ζ7 = ζ5 ζ2 = ζ2 Substitute: �12 Restate thisand result with the exponents 4 The Pentagon the Heptagon.nb 5 taken Modulo 5, because, on the unit circle in the complex 0 plane, ζ = ζ = 1. Thus, we have ζ6 = ζ5 ζ1 = ζ1 ζ7 = ζ5 ζ2 = ζ2 Substitute: ζ3 + ζ4 + ζ6 + ζ7 = ζ3 + ζ4 + ζ1 + ζ2 = -1 Presto! We have the sum of the four non-real roots of the equations x 5 - 1 = 0. We know that these sum to -1. Building a Quadratic Equation for η1, η2 Great! We have two terms, η1 and η2 . We don’t know what they are, but we do know that their sum is -1 and their product is also -1. Does that sound like a familiar situation? When we know that when we know the sum and product of two terms, we can construct a quadratic equation that has these terms as roots. In this case, we have: x2 + x - 1 = 0 whose roots are given by the quadratic formula: η1 and η2 = -1 ± 1+4 2 = 1 2 -1 + 5  and 1 2 -1 - 5 . (The approximate values of these are 0.61803 and -1.61803.) Behold! Now It Factors! Remember that we said that the expression x 4 + x 3 + x 2 + x + 1 = 0 could not be factored over the rationals? Now it can be factored in an extended field when we append 12 -1 + append 5  -- or even if we just 5 -- to the rationals. We have: ( x - ζ1 ) ( x - ζ4 ) = x 2 - ζ1 x - ζ4 x + ζ1 ζ4 ( x - ζ1 ) ( x - ζ4 ) = x 2 - (ζ1 + ζ4 ) x + ζ1 ζ4 = x 2 - (ζ1 + ζ4 ) x + 1 = η1 This expression, x 2 - (ζ1 + ζ4 ) x + ζ1 ζ4 , has coefficients that are in the extended field. The = η1 coefficient of x is the sum of the two roots ζ1 + ζ4 ; we don’t know them individually yet, but we know that they sum to η1 , which is in the extended field. The constant term is ζ1 ζ4 ; we know right away that the product of these two is 1 (product of their moduli, sum of their arguments). �4 The Pentagon and the Heptagon.nb 13 The Four Singletons Now look at the four roots individually: ζ1 , ζ2 , ζ3 , ζ4 We know how they sum in pairs: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 We also know the products of the same pairs : ζ1 ζ4 = ζ5 = 1 ζ2 ζ3 = ζ5 = 1 So we can make two more quadratic equations: w2 - η1 w + 1 = 0 whose roots are ζ1 and ζ4 , which are solved as w = η1 ± y 2 - η2 y + 1 = 0 whose roots are ζ2 and ζ3 which are solved as y = η2 ± η1 2 - 4 2 η2 2 - 4 2 We now have enough information to solve for the four roots: ζ 1 ζ 4 ζ 2 ζ 3 = = = = η1 + η1 2 - 4 2 η1 - η1 2 - 4 2 η2 + η2 2 - 4 2 η2 - η2 2 - 4 2 1 = 2 2 2 -1+ 5  - 1 -1- 5  + 1 2 -1- 5  - = -0.809017 + 0.587785 ⅈ 2  2 -1- 5  - 4 2 = 0.309017 - 0.951057 ⅈ 2  2 -1- 5  - 4 1 = 0.309017 + 0.951057 ⅈ 2  2 -1+ 5  - 4 2 1 = 2 2 1 = 1  2 -1+ 5  - 4 2 1 = -1+ 5  + = -0.809017 - 0.587785 ⅈ You can see that these solutions contain radicals of radicals. These expressions are not in the first extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that it includes these four solutions. �14 4 The Pentagon and the Heptagon.nb You can see that these solutions contain radicals of radicals. These expressions are not in the first extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that it includes these four solutions. These can be plotted on the complex plane: 0.31 + 0.95 ⅈ -0.81 + 0.59 ⅈ -0.81 - 0.59 ⅈ 0.31 - 0.95 ⅈ Voila. More Important Than the Answer To summarize and review. More important that getting the answer or than drawing the pentagon is to notice how the field extensions were built. Beginning with the rationals, which are all constructible, we first got the values for η1 and η2 , which were the sums of ζ 1 + ζ 4 and ζ 2 + ζ 3 respectively, the two pairs of complex conjugates. These values were 1 2 -1 ± 5 , and thus required that we move into an extended field: Q ⟶ Q(η1, 2 ) This is a quadratic extension and is thus constructible. Then, getting the four roots themselves required another field extension. The four roots are η1,2 ± η1,2 2 - 4 2 , and each will require one more quadratic field extension. Q ⟶ Q(η1, 2 ) ⟶ Q(η1, 2 , η1,2 ± η1,2 2 - 4 2 ) Sequences of quadratic field extensions are constructible. Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots. 1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs �4 The Pentagon and the Heptagon.nb 15 Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots. 1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs : 1 + x + x2 + x3 + x4 = {(x - ζ1 ) (x - ζ4 )} × {(x - ζ2 ) (x - ζ3 )} 1 + x + x2 + x3 + x4 = x2 - (ζ4 + ζ1 ) x + ζ1 ζ4  × x2 - (ζ2 + ζ3 ) x + ζ2 ζ3  1 + x + x2 + x3 + x4 = x2 - η1 x + 1 × x2 - η2 x + 1 Is this interesting? Yes! If we confine ourselves to rational numbers, then our original equation could not be factored. If we admit η1 and η2 , it could be factored into two factors. If we admit all the complex numbers -- really, we needed go so far; a finite field extension adding η1,2 ± η1,2 2 - 4 2 to the mix would be enough -- then it factors into four factors: In Q 1 + x + x2 + x3 + x4 In Q(η1, 2 ) "" factors to x2 - η1 x + 1 × x2 - η2 x + 1 In Q(η1, 2 , η1,2 ± η1,2 2 - 4 2 ) “” is irreducible factors to (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) The procedure we have followed does exactly what is required for specifying constructible figures: it has made a sequence of finite field extensions, starting with the rationals, Q, and proceeding by quadratic field extensions until the polynomial with our desired points as roots is completely factored. This stepwise factorization works for the pentagon because at each step it was possible to subdivide the roots into two groups, each of which could be shown to be a quadratic expression of the preceding group. That is not always possible. Conclusion We have seen here an application of the technique of algebraic decomposition. The equation we are trying to solve is broken into simpler and simpler parts as the field in which we operate is expanded step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors. Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which polygons are constructible and which are not. We will see this method play out on a larger stage next week with the construction of the hep- �16 We have seen here an application of the technique of algebraic decomposition. The equation we are trying to solve is broken into simpler and simpler parts as the field in which we operate is expanded 4 The Pentagon and the Heptagon.nb step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors. Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which polygons are constructible and which are not. We will see this method play out on a larger stage next week with the construction of the heptadecagon. Thank you. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 16 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The pentagon and the heptagon Description An account of the resource Transcript of a lecture given on October 1, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-01 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Polygons Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_The Pentagon_and_the_Heptagon https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/475e77346471eb8b8d17111a59ad06cb.pdf e7795fa0d668289e15620ac4c85c0693 PDF Text Text Constructible Numbers Grant Franks June 3, 2019, revised September 21, 2019 Introduction Recapitulation Two weeks ago, we saw that Rafael Bombelli confronted the possibility that the square root of negative one might have some positive significance when he solved the cubic equation x 3 - 15 x - 4 = 0 and, using the formula Cardano stole from Tartaglia, got x= 3 2 + 11 -1 + 3 2 - 11 -1 which he was then able to solve by intuiting that 2 + 11 -1 = 2 + 3 -1  . Last week, we followed Caspar Wessel in developing a graphic understanding of the arithmetic of complex numbers. In particular, we saw that (i) complex numbers can be expressed in polar coordinates by giving a distance (modulus) and an angle (argument); (ii) multiplication of complex numbers amounts to (a) multiplication of their distances (moduli) and (b) adding their angles (arguments); and (iii) the solutions of equations of the form x n - 1 = 0, known as the “Roots of Unity,” appear graphically as the vertices of an equilateral n-gon in the unit circle on the complex plane. So far, so good. Today’s Agenda �2 ��� 3 Constructible Numbers.nb Today, we’re going in different direction. Today, we will talk about constructing numbers and the emptiness of the Euclidean plane. The Geometrical Plane The Awkward Question “Are there holes in the geometric plane?” This is a question that rarely gets asked when doing geometry. There is no reason to raise such a question until some awkward questions get asked, and there is no way to answer the question without being able to step back from intuitively given geometrical space by thinking algebraically. Construction In Euclid’s geometry, one begins with five postulates, three of which authorize constructions: Postulate One: to draw a line between any point and any other point; Postulate Two: to continue a line indefinitely; and Postulate Three: to draw a circle with any center and radius. The references to “any point,” “any center” and “any radius” give the impression that the Euclidean plane contains all possible points. But that isn’t quite true or, to be more precises, there are many points in a plane to which Euclidean geometry provides no access. For instance, suppose you wanted to “square the circle.” This is a classical problem that amounts to drawing a rectangle with an area equal to that of a given circle. Archimedes shows that such a rectangle would have a height equal to the radius of the circle and base equal to half its circumference. If you can draw a circle with any center and radius, could you draw one whose radius is equal to the semicircumference of the circle you are trying to square? If you could do that just by saying it, your problem would be solved. That would not satisfy a mathematician. She or he would want to know how to get that radius length, that is, how to construct it. Beginning with the one length that is given -- the radius of the circle to be squared -- and using only permissible manipulations, how can we construct the desired straight line with a length equal to the semi-circumference of the given circle? The Meno Problem This problem sounds a little like the geometrical problem of the Meno: Socrates asks the slave boy, “If you are given this square with these sides, can you show the side of a square with double the area?” The slave boy is stumped. Some modern students think they have a better answer than the slave boy and say, “That’s easy! It’s the square root of two!” They don’t realize that they are not giving an �3 Constructible Numbers.nb ��� 3 answer at all. The phrase “square root of two” is just a slightly shortened version of “the magnitude which, when multiplied by itself, gives two.” So, to Socrates’ question, “What is the magnitude which, when multiplied by itself, yields two?” they have answered, “The magnitude which, when multiplied by itself, gives two.” True, no doubt, as any tautology is true, but it doesn’t really advance our knowledge of, well, anything. Socrates provides what is really needed: a geometrical construction for finding the square root of two by looking at the diagonal of a square with sides equal to one. (Really, he gives the construction for 2 2 because his original square had sides equal to two, but that is minor detail.) For the squaring of the circle, we want a construction for a straight line whose length is equal to the semidiameter of the given circle. Alas, not all that humankind desires does it obtain! There is no Euclidean construction for that length. The story that leads to that result culminates in the late nineteenth century. It involves Ferdinand Lindemann’s demonstration of that π is not just an irrational magnitude -- that had been known for over a century (Johann Lambert 1761) -- but that it is a particular kind of irrational magnitude. Getting to that result, if anyone here is interested in it, would require work that goes beyond what this series of talks will cover. If it is any consolation, however, the ideas we will cover tonight are necessary preliminaries to that work. The Delian Problem Once upon a time, a long time ago, a great plague afflicted the city of Delos. The citizens consulted Apollo’s oracle at Delphi who told them that the god was dissatisfied with the altar of his temple. The altar was made in the shape of a cube, and the oracle said that the god wanted an altar twice as big. The citizens, eager to be rid of the plague, got a great piece of marble and built an altar twice as long, twice as deep and twice as tall as the one that was there. The plague continued. The priest of the oracle corrected the people saying the god wanted an altar with twice the volume of the present one. The newly built altar had eight times the volume, and was not what was wanted. The people were understandably annoyed, but everyone in the ancient world knew that gods love to mess with people by issuing weirdly misleading oracular pronouncements. Gods are cruel, that’s all there is to it. Geometers realized immediately that what was needed was, in effect, an altar whose side was 2 times bigger than the present one. When they set out to design it, however, they found that the god had been even crueler than expected. They couldn’t figure out how to construct the altar. Finding a construction for 2 was easy, but finding one for the 2 was surprisingly difficult. 3 3 To get to the bottom of the problem that they faced, we have to sketch out a new kind of algebraic operation, finite field extensions. Ordinarily this would be a semester-long study, but since this is St. John’s College, I will try to compress it into about fi�een minutes. �4 ��� 3 Constructible Numbers.nb Rational Operations What is a Field? To begin, we need to define “a field.” For our purposes, a “field” is a collection of objects -- we’re going to be talking about numbers -- that are closed under the operations of addition (and its inverse, subtraction) and multiplication (and its inverse, division). Consider first, then, the whole numbers: 1, 2, 3, …. These are closed under addition, that is, if you add any two whole numbers you get a whole number. But they are not closed under subtraction. Although you can subtract 5 from 7 to get 2, you cannot subtract 7 from 5. So set aside the whole numbers and take up the integers: … -3, -2, -1, 0, +1, +2, +3, …. Now you have closure under addition and subtraction. You also have closure under multiplication but not division. 10 divided by 5 is 2, but 10 divided by 3 is not among the integers. So set aside the integers and take up the rational numbers: … -1 -1 -2 , 5 , 11 , 3 …0… 1 , 2, 5 10 7 3 …. That is, all the numbers made up by ratios of integers with one another (forbidding division by zero). Now we have it: this is a field. It is closed under addition and subtraction, it is closed under multiplication and division. For convenience’ sake, we will give the rational numbers a symbolic name, Q. (Why Q and not R? Because R is reserved for the real numbers. Alas.) Euclid’s Operations Allow Us to Form a Field The operations of Euclid’s geometry allow us to construct lengths on a line that correspond to the field of rational numbers. If we begin with a given length that we will call the “unit,” we can with straightedge and compass easily make a double length, a triple length, etc. If we define “negative” to be mean motion in one direction from an arbitrary starting point and “positive” to mean going in the opposite direction, we can construct lengths corresponding to all integers. By an easy construction, we can also divide our given unit length into equal parts corresponding to any whole number. Thus we can make lengths corresponding to any positive proper fraction; by multiplying these we can make any proper or improper fraction, and by directing them toward the negative side of our arbitrary zero point, we can identify places corresponding to any rational number. The lengths from zero to these points can be added, subtracted, multiplied and divided at will and the result will always be another rational length. We have a field. If we erect two such lines at right angles to one another, we can locate and label any point on a plane �3 Constructible Numbers.nb ��� 5 that corresponds to (a, b), where a and b are rational numbers. As my grandfather used to say, “Now we’re cookin’ with gas!” Other Lengths All points with rational coordinates is a lot of points, but we know that there are other lengths that can be found in Euclidean geometry. There is, for example, 2 , which is the diagonal of the square with sides of unit length. In fact, we can construct lengths equal to the square roots of any lengths we can find through other means. D A C B If you want to find a length equal to r , draw line AB in length equal to r + 1. Here, let Ac = r and let CB = 1. Erect a semi-circle on line AB and a perpendicular at C meeting the semicircle at D. Join AD and DB. Triangle ADC is similar to triangle DCB and to the combined triangle ADB. Therefore: AC : CD :: CD : CB AC × CB = CD2 But AC = r and CB = 1; therefore: r × 1 = 2 = CD2 r = CD. Combinations A little examination will show that these are all the operations that are available to us. We have addition, subtraction, multiplication, division -- these are sufficient to find any rational lengths. In addition to this, we can take the square root of any length that we can find. Not only that: we can do so as many times as we please. So, can construct 2 , or 5 , or 17 3 , or of any rational length. And that’s not �6 ��� 3 Constructible Numbers.nb all! We can construct 2 . Or 2+ 2 . Or 17 3 + 2+ 7 3 … or any sequence or combina- tion of the rational operations and repeated extraction of square roots. With these techniques in hand, someone might easily jump to the conclusion that these Euclidean operations can construct any length whatsoever. That’s probably what I would have said if anyone had asked me back when I was a Johnnie Freshman more years ago than I care to think about. But I would have been wrong. Jumping to Conclusions When I learned about the Pythagorean theorem and irrational numbers (or their equivalents, incommensurable lengths), I didn’t take time to think about these new numbers as carefully as, in retrospect, I should have. Looking back, I think my understanding ran something like this: “We had the whole numbers, but they weren’t enough to do subtraction so we added the negative numbers and got the integers. But the integers weren’t enough to do division, so we added the fractions and got the rational numbers. But even the rationals weren’t enough to account for all the lengths we could find in geometry -- the 2 is irrational! (Hey! I was just as surprised by this as the Greeks were!) -- so we added the irrational numbers to the rational numbers and now we have all the real numbers, which is all that there are!” That understanding didn’t get challenged for decades until I began working on a preceptorial on Galois Theory and Professor Charles Hadlock, author of Field Theory and Its Classical Problems, introduced me to finite field extensions. It was here that I learned the humbling lesson that not all irrational numbers are the same. Some numbers are more irrational than others, and lumping them all together blurred together distinctions that are best kept separate. Baby Steps Let’s start over. Go back to when we had just the rational lengths and could find any point with rational coordinates. Now, we read the Meno and find out about 2 . Instead of pretending that we are now able to generate all possible irrational lengths, look carefully at what we have. We can make any rational length, and we can make the square root of two. If we continue now to use just with our rational operations (+, -, x, ÷) on the two lengths we have at hand, 1 and 2 , we can make any number that looks like this: a+b 2 where a and b are rational numbers. Notice something important about these numbers: we can add them, subtract them, multiply them, and divide them any way we please and we always get other �3 Constructible Numbers.nb ��� 7 numbers of this same kind. So, if we have: 3 + 5 2  + -1 + 7 2  = 2 + 12 3 + 5 2  × -1 + 7 2  = -3 + 21 2. or 2 -5 2 2 + 35  2  = 67 + 16 2 Division is a bit more complicated, but it works as well. The upshot is this: the numbers a + b 2 form a field of their own. This new field is called an extension field. Because we added a finite number of elements to form it (in this case, just one), it is called a finite field extension. And because the element we added was a solution of a polynomial with elements of the original field as coefficients -- in this case, x 2 - 2 = 0 -- it is called a finite algebraic field extension. And because it was made by adding an element that is the square-root of a member of the original field, it is called a quadratic finite algebraic field extension. Let’s call it F1 and write F = Q( 2 ) to signify that F was formed by appending 2 to Q and making all the numbers of the form a + b 2 where a and b are elements of Q. F is big. It’s bigger than Q, the rational numbers. It includes Q as a subset, so we write: F1 ⊃ Q. But F does not include all the numbers (lengths) that we can construct because we can adjoin other elements if we wish. We can even take the square root of some squirrely element of F1 that already has a square root of two, say: 3+7 2 We can append this element to F1 and form the numbers: c+d 3+7 2 where c and d are elements of F. This is a quadratic finite algebraic field extension of F1 . Let’s call it F2 and write F2 = F1 ( 3+7 2 ) to signify that F2 was formed by appending ing all the numbers of the form c + d 3+7 3+7 2 to F1 and mak- 2 where c and d are elements of F1 . F2 is big. It’s bigger than F1 and much bigger than Q. It includes F1 as a subset, so we write: F2 ⊃ F1 ⊃ Q. Do you see where this is going? We can continue this process as long as we wish. �8 ��� 3 Constructible Numbers.nb … F5 ⊃ F 4 ⊃ F 3 ⊃ F 2 ⊃ F 1 ⊃ Q When I put them all together, I have a tower of finite quadratic field extensions. Every number that corresponds to every possible constructible length is somewhere in that tower. Altogether, they are called the constructible numbers. The set of constructible numbers is very big. But it’s not everything. There are Non-Constructible Numbers There are, as it turns out, non-constructible numbers, as can be shown in several ways. For instance, 2 (the real cube root of two) is not a constructible number. It does not belong to any tower of quadratic field extensions over the rationals. 3 For, proceeding in the time honored way of reductio ad absurdum, suppose that 2 were constructible. Also remember that, since in the real numbers y = x 3 is strictly increasing, there is only one real cube root of two. Now, if 2 were constructible, it would belong to some quadratic field extension of a field that was itself part of a tower of quadratic field extensions leading back to the rationals. 3 3 2 ∈ Fn 3 That means that include 2 . 3 where 2 =a+b Fn ⊃ Fn-1 ⊃ Fn-2 ⊃ Fn-3 ⊃ … F1 ⊃ Q c , where a, b and c are all parts of Fn-1 , but where Fn-1 does not itself 3 Cube both sides of this equation. 3  2  = a + b 3 3 c  = a3 + 3 a 2 b 2 = (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c ) c + 3 a b 2 c + b3 c c c The number 2 is a part of Fn - 1 ; we know this because it is a member of Q. Therefore, it has no component multiplied by c , which means that ( 3 a2 b + b3 c ) = 0. Next consider (a3 + 3 a b2 c ) - ( 3 a2 b + b3 c ) c (notice the minus sign). Since 3 a2 b + b3 c = 0, this 3 has the same value as (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c ) c . But it unpacks into a - b c  . So we have two cube roots of 2: a+b c a-b c But there is only one value ; therefore b = 0. That means that a + b c is really just a, and 3 2 is in �3 Constructible Numbers.nb ��� 9 Fn - 1 . By the same reasoning, it is a member of Fn - 2 and Fn - 3 and so on until we discover it is a member of Q, that is, that it is rational. But 3 2 is NOT rational. (Those who doubt this can see Appendix 1.) Thus, 2 is NOT in ANY tower of quadratic field extensions beginning with the rationals. It is not constructible. 3 The Delian Problem is Thus Solved At this point, the architects and engineers at Delos should despair: if 2 is not constructible, then they cannot double the size of the altar of Apollo with Euclidean mathematics. The gods are cruel, but they are are mathematically well-informed. 3 There are Lots of Non-Constructible Numbers. The demonstration that the 2 is not constructible is all well and good, but it is rather ad hoc. It doesn’t immediately produce any broad conclusions about constructible vs. non-constructible numbers. 3 A slightly more detailed investigation of field theory allows broader conclusions. It is possible to characterize the size or “degree” of one algebraic field extension over another. Compare, for example, the quadratic extension that results from adjoining 2 to the rationals with what would be called the “cubic” extension that occurs when you adjoin 2 . In the first case, we can express any number in the extended field by an expression that looks like this: 3 2. a+b These numbers form a field. You can add, subtract, multiply and divide to your heart’s content and never leave the field. If you try this with 2 , however, a problem arises. Form a number like: 3 a+b 3 2. You can add and subtract alright, but as soon as you start multiplying, you’ll find yourself running into: 3 2 × 3 2 = 3 4 The cube root of four is not the same as the cube root of two. It can’t be expressed by combinations of rational numbers and the cube root of two. It is outside of the (purported) field. This problem did not arise with 2 because 2 × 2 =2 �10 ��� 3 Constructible Numbers.nb which is within the field defined by a + b 2 . With the cube root, it is not enough to add one term; you need to add two. To get a field that includes 2 , you need numbers of this form: 3 a+b 3 2 +c 3 4. A little experimentation will persuade you that these numbers do form a field. Notice that when you formed a extended field with 2 , your new numbers had two terms, a + b 2 . With 2 of two, your new numbers have three terms. The quadratic extension is of “degree two,” while the cubic extension is of “degree three.” Field extensions can be compounded: an extension of degree two followed by an extension of degree three will yield an extended field of degree six over the original field. Field extensions can get remarkably complex, but for our purposes it will be enough to focus on relatively simple extensions of relatively small degrees. 3 3 The degree of an extension is measured by complexity of the minimal polynomial needed to produce the new elements whose addition to the original field leads to the extension. Determining whether a polynomial is “minimal” poses some problems, but this approach can produce sweeping knowledge about whole classes of extensions. So, for instance, every quadratic field extension over the one before it, so that a tower of quadratic field extensions -- that is, the collection leading to any constructible numbers -- will have powers 2, 4, 8, 16 … 2n over the rationals. At the same time, it can be shown that for equations of the form: xn - 2 = 0 the number n gives the degree of the extension resulting from appending one of the solutions of the equation to a field. This result allows is to know that Solutions of Appended to Q are of degree And thus are generally 2 2 Constructible 2 3 Not Constructible - 2 4 2 4 Constructible - 2 5 2 5 Not Constructible x6 - 2 6 2 6 Not Constructible x7 7 2 7 Not Constructible x2 I.e. - 2 2 x3 - 2 3 x4 x5 - 2 Also, it can be show that the solutions of the equations for the nth roots of unity, a�er factoring out (x 1), are minimal when n is prime. So Roots of structible Roots of structible x3 - 1 = 0 appended to Q are of degree 2 over the rationals, so con- x5 - 1 = 0 appended to Q are of degree 4 over the rationals, so con- �3 Constructible Numbers.nb Roots of structible Roots of structible Roots of structible Roots of structible Roots of structible etc. ��� x7 - 1 = 0 appended to Q are of degree 6 over the rationals, so NOT con- x 11 - 1 = 0 appended to Q are of degree 10 over the rationals, so NOT con- x 13 - 1 = 0 appended to Q are of degree 12 over the rationals, so NOT con- x 17 - 1 = 0 appended to Q are of degree 16 over the rationals, so con- x 19 - 1 = 0 appended to Q are of degree 18 over the rationals, so NOT con- 11 Many -- indeed, most -- of these create extensions whose degrees are not powers of two over the rationals. Thus they create field extensions filled with numbers that are not constructible. Literally infinite fields of non-constructible numbers emerge. Hierarchy of Irrationals Viewing all the irrationals as an undifferentiated mob is a mistake. We can distinguish between those irrationals that are constructible and those that are not. The constructible numbers are built by successive quadratic field extensions starting from the rationals. Non-Constructibles Constructibles Rationals The distinction between the constructible and non-constructible numbers is interesting enough, but situation is even stranger than that. As we have seen, the Constructibles are made from towers of field extensions all of degree two. We �12 ��� 3 Constructible Numbers.nb have just seen that many polynomials have solutions which, when appended to a field, give an extension of degree other than two; the equation x 3 - 2 = 0, for instance, has solutions of degree 3, which takes us away from the powers-of-two towers of constructible numbers. Suppose we toss away that restriction. Suppose we consider all algebraic field extensions of any degree. What if we allow ourselves to build towers in which each step can be of any degree -- that is, to append to a field the solutions of a polynomial of any degree. In this way, we could make towers of fields that include the solutions of any finite polynomial equations. That would include all the constructible numbers and much, much more. This immense collection is known as the algebraic numbers. It includes the rational numbers and the constructible numbers and much, much more. It does not, however, include everything. There are numbers that are not included among the algebraic numbers. You know a few: π is not an algebraic number. Neither is e, the base of the natural logarithm system. Leibniz and later Euler called these non-algebraic numbers “transcendental numbers,” a wonderfully mystical “woo-woo” name that stuck and is in common use today. Transcendentals Algebraics Constructibles Rationals There are LOTS of Transcendental Numbers When I name π and e as transcendental numbers, you may be misled into thinking that there are only a few such numbers and that each of them is a precious rarity, much treasured by mathematicians like these two specimens. Au contraire! Far from being scarce, the transcendental numbers not only surpass all other numbers in �3 Constructible Numbers.nb ��� 13 quantity, they do so by an infinite amount. Of course, to characterize one infinity as greater or lesser than another is a controversial project first pioneered by nineteenth century mathematician Georg Cantor. According to Cantor, the smallest sort of infinity is like that of the natural numbers which can be ordered in such a way that one can count off the members of an infinite set sequentially and be sure eventually to encounter every member. The natural numbers are obviously countable in this way, as are the integers if we number them like this: 0 +1 -1 +2 -2 +3 -3 … 1st 2nd 3rd 4th 5th 6th 7th … It takes a little more work to see that the rationals can be placed in countable order (they can), and a bit more still to figure out that the algebraic numbers can also be ordered and counted. But they can. Cantor designates this infinity by the symbol ℵ0 . The complete collection of real numbers -- and also the collection of all complex number a + b i where a and b can be any real number -- cannot be so ordered. These numbers, according to Canto, form a higher degree of infinity, the infinity of the continuum, ℵ1 which is widely take to be equivalent to 2 ℵ0 , a quantity distinctly different, and distinctly bigger than ℵ0 -- insofar as “bigger” is a concept applicable to infinities. From this perspective, the relation of the algebraic numbers to the entire set of complex numbers is pretty much what Ptolemy would call “the ratio of a point to a line.” The whole realm of all our manipulations, geometric and algebraic, occur in a vanishingly small subset of the totality of real numbers. Yet though we speak of infinities, do not imagine that these transcendentals are far away. They are not far away in heaven, so that you have to ask, “Who will ascend into heaven to get them?” Nor are they beyond the sea, so that you have to ask, “Who will cross the sea to get them?” No, they are very near to you always on every side, crowding about with incredible density. And the net of constructible numbers seems now to spread across the Euclidean plane like ever-thinning gossamer network of barely perceptible points, each separated from the next by gulfs teeming full with inaccessible points. While we all contemplate the miserable smallness of all our endeavors, let us take a brief break and then there will be time for questions. Appendix 1: Irrationality of 2 3 This result follows from Euclid, Book X, proposition 9. It can also be shown from arithmetic principles as follows: Suppose that 3 2 is rational. Then it can be expressed as a fraction numbers. We may assume also that the fraction a b a b where a and b are finite whole is expressed in lowest terms, so that a and b have no �14 ��� 3 Constructible Numbers.nb factors in common. We have: 3 a b 2 = 2= a3 b3 2 b3 = a 3 That means that a3 is even, which means that a is even; thus a3 is divisible by 8. Let it be 8 c3 . 2 b3 = 8 c 3 b3 = 4 c 3 That means that b3 is divisible by 4, which means that b is even (and that b3 is in fact divisible by 64). But we began with the hypothesis ab was a fraction where a and b have no factors in common. Therefore, 3 2 cannot be expressed as a fraction ab . � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 14 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Constructible numbers Description An account of the resource Transcript of a lecture given on September 24, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-24 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Number theory Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_Constructible_Numbers_2019-09-24 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ce789f61b6997180ca1dc76624bae159.pdf 96136207810f761155e2fa715e4dd4fa PDF Text Text Trigonometric Interpretation of Complex Numbers Grant Franks June 3, 2019, revised September 16, 2019 Dedication Caspar Wessel (1745 - 1818) Let us pause for a moment to remember and give thanks for Caspar Wessel, Norwegian mathematician and cartographer, who conceived the idea that complex numbers might be usefully portrayed on a map. Introduction We ended the last talk with Rafael Bombelli staring at Cardano’s Formula as applied to the cubic equation x 3 - 15 x - 4 = 0 The formula gives for a solution: x= 3 2 + 11 -1 + 3 2 - 11 -1 which at first glance appears to be nonsensical since it consists of two terms both containing the cuberoot of expressions involving the square-root of negative one, which mathematicians in other contexts �2 ��� 2 Trignometry and Complex Numbers.nb agreed meant that no solution was possible. But Bombelli knew that there was a solution to this equation. In fact, he knew what at least one of the solutions was: the integer + 4 solves the equation. 43 - (15) (4) - 4 = 64 - 60 - 4 = 0 Bombelli figured out by some combination of guesswork, deduction and just inspired staring that there are expressions which, when cubed, give 2 + 11 -1 and 2 - 11 -1 . They are 2 + -1 and 2 + -1 , respectively. Hard as it is to find them, it is easy to confirm that they work. All you need do is to multiply them by themselves three times, keeping in mind the one rule we know about -1 , namely, that when multiplied by itself it gives -1. -1  = 4 + 4 2 + -1  2 + 2 + -1  = 2 + 2 - -1  2 - 2 - -1  = 2 - 3 -1 - 1 = 3 + 4 2 -1  2 + -1  = 3 + 4 -1 -1  2 + -1  = 2 + 11 -1 -1  = 2 - 11 -1 And -1  = 4 - 4 3 2 -1  2 - Knowing the cube roots of 2 ± 11 x= 3 2 + 11 -1 -1 - 1 = 3 - 4 + 3 -1  = 3 - 4 -1 -1  2 - -1 allowed Bombelli to solve the particular problem facing him: 2 - 11 -1 = 2 + -1 + 2 - -1 = 4. However knowing the answer to this problem doesn’t show us how to deal more generally with other numbers involving -1 . The answer to that problem is the subject of tonight’s talk. Arithmetic of Complex Numbers Their Real and Imaginary Parts of a Complex Number The most evident problem with -1 is that it doesn’t stand in a relation of “more” or “less” with regard to other numbers we have come across. That feature more than any other makes -1 seem especially weird. When one takes the step from the whole numbers to fractions (that is, to positive rational numbers), things like “one-half” or “five and a quarter” could be related as greater or less than whole numbers we were already familiar with. Later, for all its undefinable strangeness, an irrational like 2 at least sat snugly between rational numbers, greater than some and less than others. (That, in fact, is how Dedekind defined irrational numbers, namely, by identifying which rationals each was �2 Trignometry and Complex Numbers.nb ��� 3 greater than and which it was less than.) Even the very strange negative numbers aren’t as peculiar as -1 . If your idea of a number is that it should respond to counting something or measuring something, negative numbers are nonsense because there is less than nothing there to count or to measure. But if you can get over that problem, at least negative numbers still stand in greater-and-lesser relations to one another. Not so -1 . It is neither greater than nor less than any real number. That much is pretty clear: the square of every real number is positive, or at least “non-negative.” The -1 is not anywhere on the real number line that stretches from enormous negatives to enormous positives. So, if we are to imagine it at all, we have to picture it being “somewhere else.” Caspar Wessel set the imaginary numbers apart from the reals on an axis of their own at right angles to the real number line. He thus established a complex number plane. One axis represents the real numbers, the other the numbers that include -1 . A real number and an imaginary number together form a two-part entity called a “complex number.” Each point on the complex number plane represents a single complex number. A complex number can look like: 2+ -1 or -3 + 9 -1 or 0 +5 -1 or -4 + 0 -1 . The first and second examples have both a real and imaginary part. The third has only an imaginary part; the real part is zero. The last example has only a real part; the imaginary part has a zero coefficient. At the risk of seeming overly pedantic, I want to note here that “having an imaginary part with a zero coefficient” is not quite the same thing as “being a real number.” The complex number “-4 + 0 -1 ” is not quite the same thing as the real number “-4.” The reason for this hyper-technicality and squeamishness about nomenclature is not at all clear at this point and it won’t become clear until the last lecture. It’s not unusual to overlook this distinction and, for now, doing so won’t cause any problems. It is common, even convenient, to skip over the zero terms and to write “-4 + 0 i” as just “-4” I mention this not-yet-developed distinction only so that, when it comes back again in the final lecture, I can say “As I have already said …”, and you will all nod sagely in agreement. The real and imaginary parts of a complex number stand in different orders and, when they are added or subtracted, they act independently of one other. In modern parlance, one might say that a complex number can be represented as a vector on a plane with a real axis in one direction (generally, le�-right) and an imaginary axis orthogonal to it (up-down). That’s not how Caspar Wessel spoke because the term of a “vector” wasn’t introduced until the middle of the 19th century, decades a�er Wessel died. But the fundamental idea is there: a complex number is a two-part object whose parts add independently of one another. That idea had been around for years, at least since Isaac Newton had analysed motions into components towards and parallel to the sides of a parallelogram. �4 ��� 2 Trignometry and Complex Numbers.nb 5i 4i 3+4i 3i 2i -3 + 2 i i -5 -4 -3 -2 -1 0 1 2 3 4 5 -i -4 - i -2 i -3 i -4 i 2-4i -5 i Here, then, is the representation of four complex numbers on a complex number plane: 3 + 4 i, 2 - 4 i, -4 - i and -3 + 2 i. So far, this is just a picture. Its value appears as we see how it is used. Addition and Subtraction of Complex Numbers In addition, the real and imaginary parts act separately. So, if one adds -3 + 2 i to 4+2i one gets (-3 + 4) + (2 + 2) i The real parts add ordinarily, and the imaginary parts do too, thanks to the (formerly implicit, now explicit) understanding that “distribution of multiplication over addition” works for the number i as it does for other numbers, so that we have: 2 i + 2 i = (2 + 2) i = 4 i. This procedure is just what one would do with components of a vector or of a decomposed Newtonian force or velocity. Graphically, as Wessel proposes envisioning complex numbers, the result looks like this: �2 Trignometry and Complex Numbers.nb ��� 5i 4i 1 + 4i 4+2i 3i 2i -3 + 2 i i -4 -3 -2 -1 0 1 2 3 4 -i Multiplication by a real number (or real part of a complex number) is like ordinary multiplication Multiplying a complex number by a real number amounts to multiplying each of the real and complex parts of the complex number as you would expect. For multiplication by positive integers, the result is just like repeated addition of the vector representing the complex number. Multiplication by negative numbers is like repeated subtraction. 5 �6 ��� 2 Trignometry and Complex Numbers.nb 5i 4i Multiplication of 3 + i by 3 + 0 i 3i 2i i -2 -1 3+i 0 1 2 3 4 5 6 7 8 9 10 -i -2 i So far, so good. The graphic representation hasn’t yet shown us anything novel about complex numbers or given us new, but there is more and better yet to come. The Crux of the Problem: Imaginary Multiplication Next we have to deal with complex numbers times other complex numbers. This is where things get interesting. It’s not immediately clear what that means graphically, but we do have an algebraic understanding. The one thing we know for sure about -1 is that when you multiply it by itself, it gives -1. Let’s go back to the example we have already seen: Bombelli’s discovery that 2 + i is the cube root of 2 + 11 i. As we showed already, we can multiply 2 + i times itself: (2 + i) (2 + i) = (2 ⨯ 2 )+ (2 ⨯ i )+ (2 i ⨯ 2) + ( i ⨯ i) = 4 + 2 i + 2 i - 1 = 3 + 4 i. �2 Trignometry and Complex Numbers.nb 5i (2 + i)(2 + i) = 3 + i 4i 3i 2i i -2 -1 2+ i 0 1 2 3 4 -i -2 i So far, this is not too revealing. Multiply the product by 2 + i again: 5 ��� 7 �8 ��� 2 Trignometry and Complex Numbers.nb 12 i (2 + i)3 = 2 + 11 i 11 i 10 i 9i 8i 7i 6i 5i 4i (2 + i)(2 + i) = 3 + i 3i 2i i -2 -1 2+ i 0 1 2 3 4 5 -i -2 i What sense does that make? The meaning appears more easily with Polar Coordinates So far, we’ve been writing complex numbers like points on a plane using Cartesian coordinates. For some purposes, it is a LOT easier to understand what is going on if you use polar coordinates. (Trust me.) To start, consider a circle with a radius of one centered on the origin. This is the “unit circle in the complex plane.” �2 Trignometry and Complex Numbers.nb ��� � ������ 1.51 = A 1.11841 = θ A (cos θ + i sin θ) Now if you choose any angle θ, the point (Cosine(θ) + i Sine (θ)) will necessarily fall on the unit circle. As the angle θ goes through the complete cycle from 0 to 2 π -- we measure angles in radians, which is easier for all sorts of reasons once you get used to it; if you are thinking in degrees, say “0° to 360°” -the point (Cosine(θ) + i Sine (θ)) goes around the circle. If the angle continues to grow, the point spins endlessly around the unit circle. If you want a point, that is to say “a complex number,” inside or outside the unit circle, multiply the result by some constant A. If A is greater than one, the corresponding point (complex number) will be outside the unit circle; if it is between zero and one, the point (complex number) will be inside the circle. Any point on the complex plane can be designated with a pair of numbers A (for length) and θ (for angle). In complex-number-speak, the angle of the complex number expressed in polar coordinate form is called the “argument”; the length is called the “modulus” of the number. 9 �10 ��� 2 Trignometry and Complex Numbers.nb Multiplication of Two Arbitrary Complex Numbers Try multiplication again with two arbitrary complex numbers, this time expressed in polar form. Let the two numbers be: A B (cos(θ) cos(ϕ) - sin(θ) sin(ϕ) + i (cos(θ) sin(ϕ) + cos(ϕ) sin(θ)) “Okay,” you say. “How has this helped me?” The answer to that would be clear if you had been careful about memorizing trigonometric identities, in particular, the identities for the sine and cosine of the sum of two angles. On the off chance that you don’t have those identities burned into the forefront of your minds, let me show you what you need to “remember” or, as Socrates might say, “recollect.” Digression: Trigonometric Identities for Sine and Cosine of the Sum of Two Angles. Consider a portion of a unit circle with center at O. From center, draw a line OA at any (acute) angle; call the angle ϕ. Drop a perpendicular AF to the horizontal diameter of the circle. The right triangle formed as lengths that represent cos ϕ (horizontal OF) and sin ϕ (vertical AF). Now draw a line OB, creating another angle, θ, on top of the first one. Drop a perpendicular BC to OA, the hypotenuse of the first triangle. The segments OC and BC represent cos θ and sin θ, respectively. Drop perpendicular CH to the original diameter OA. Also, drop a perpendicular from BD at the top of angle θ down onto the original diameter. The segments thus created, OD and BD, represent cos (ϕ + θ) and sin (ϕ + θ) respectively. Note draw a horizontal CE from C to the line BD. In triangle BEC notice that angle EBC is equal to ϕ. Since segment BC is equal to sin ϕ, we conclude that BE = sin θ cos ϕ and that EC = sin θ sin ϕ. Meanwhile, since OC = cos ϕ, we conclude that CH = cos θ sin ϕ and that OH = cos θ cos ϕ. Examination will show that: BD = sin (θ + ϕ) = BE+ EC = sin θ cos ϕ + cos θ sin ϕ ; and CD = cos (θ + ϕ) = OH - DH = OH - EC = cos θ cos ϕ - sin θ sin ϕ. �2 Trignometry and Complex Numbers.nb ��� 11 ������ ����� ϕ ��� ����� θ ������ ���-��� ������ �������� ���� ��� ���������� B Sin θ Sin θ Cos ϕ ϕ A G θ C Sin ϕ sθ Co Sin θ Sin ϕ Cos θ Sin ϕ E ϕ O D Cos θ Cos ϕ Cos ϕ H F Now look back at the product that we just obtained in multiplying two complex numbers. A B (cos (θ) cos (ϕ) - sin (θ) sin (ϕ) + i (cos (θ) sin (ϕ) + cos (ϕ) sin (θ)) cos (θ + ϕ ) sin(θ + ϕ ) The collection of trigonometric terms associated with the real portion of the expression is cos (θ + ϕ). The collection of trigonometric terms associated with the imaginary portion of the expression is sin (θ + ϕ). The numbers associated with the lengths (modulus) are multiplied; the angles (arguments) are added. The significance of the imaginary multiplication is now visible: In multiplying two complex numbers, whether written as A (cos(θ) + i sin(θ)) and B (cos(ϕ) + i sin(ϕ)) or as a + b i and c + d i, graphically speaking what happens is that one �12 ��� 2 Trignometry and Complex Numbers.nb (i) multiplies the distances of each number from the origin of the plane (the moduli), and (ii) add the angles (arguments) made between the positive real axis and the line from the origin to the point representing the number. In short, again: in complex multiplication, distances from the center (moduli) multiply; angles from the center add. All sorts of neat things follow from this observation. Raising Complex Numbers to Powers Causes Them to Spin! If you raise a complex number to a (real) power, the argument (angle) of the result will grow continually as the distance from the center grows (if it begins outside the unit circle) or shrinks (if it begins inside the unit circle). Raising complex numbers to real powers therefore causes the results to trace spirals in the complex plane. Here is the exponentiation of a complex number represented by a point a little bit outside the unit circle: ������� {Modulus =, 1.0435} If we reduce the modulus (the “length”) so that the point falls inside the unit circle, the spiral will go inwards because increasing powers of a length (modulus) less than one will shrink. �2 Trignometry and Complex Numbers.nb ��� 13 Between these two cases is the balanced point, where the modulus is one and the point lies on the unit circle. Then, increasing powers of the complex numbers will result in a representative point that spins forever around the circumference of the unit circle. The investigation of complex numbers is a vast field. Thick textbooks are devoted to “functions of a complex variable.” The Mandelbrot set, which lies at the beginning of complexity studies, exists in the complex field. (It is defined as the set of complex numbers c that do not diverge when the function fc (z) = z2 + c is iterated from z = 0.) All this would be subject matter for an immense study. However, for the present , I want only to point to two results that are relevant to the particular path that these talks are taking toward their goal, constructing the heptadecagon. Taking Integral Roots First, now that we understand how complex numbers are multiplied and raised to powers, we can easily find how to find integral roots of any complex number and thereby develop a general solution to the problem that faced Rafael Bombelli. His great triumph, recall, was finding the cube root of one complex number, 2 + 11 i, which he did by a combination of great genius, immense labor and fabulous luck. (Almost any other complex number would have been much harder for him to deal with.) However now we can see how easily to take the cube root of any complex number. Remember, to cube a complex number, you cube the real number that is its modulus and triple the angle (argument). So, �14 ��� 2 Trignometry and Complex Numbers.nb to take the cube root of a number, all you need do is to (i) take the cube root of the length (the “modulus”) and (ii) and divide the angle (the “argument”) by three. The Cube Root of 2 + 11 i The particular problem that Bombelli faced was finding the cube root of 2 + 11 i. To take its cube root the new way, first calculate its modulus (length) and argument (angle). The length of the vector from the origin to (2 + 11i) we can get with the Pythagorean Theorem: length (modulus) = 22 + 112 = 4 + 121 = 125 If we allow ourselves some trigonometry, the angle is easy enough, too:  = 1.39094 radians (79.7 degrees). angle (argument) = ArcTan 11 2 To take the cube root, take the cube root of the length (modulus). In this case, we are assisted because 125 = 53 : 3 125 = 3 125 = Take the angle and divide by three: 5. 1.39094 3 = 0.463648 radians. So we get: 5 (Cos(0.463648) + i Sin (0.463648) ) = (2.236) (0.894427 + i 0.447214) =2+i Just the result that Bombelli arrived at by genius, sweat and divine guesswork. Here, for comparison, are the values Bombelli worked on plotted atop the graph of the spiral z = (2 + i)n �2 Trignometry and Complex Numbers.nb ��� 15 12 i (2 + i)3 = 2 + 11 i 11 i 10 i 9i 8i 7i 6i (2 + i)n 5i 4i (2 + i)(2 + i) = 3 + i 3i 2i i -2 -1 2+ i 0 1 2 3 4 5 -i -2 i The Roots of Unity When the Modulus Equals One We have seen that when complex numbers whose representative points lie outside the unit circle spiral outward when squared, cubed, or generally raised to powers greater than one. Those that lie inside the unit circle spiral inward. Those that lie on the unit circle -- those with a modulus that is exactly equal to one -- spin around the unit circle with out moving inward or outward. These are very interesting, very handy numbers. Because the cosine of a given angle and the sine of the same angle can form the sides of a right triangle whose hypotenuse is equal to one, we can write these complex numbers with modulus one in the form: z = cos θ + i sin θ �16 ��� 2 Trignometry and Complex Numbers.nb We have seen that multiplying two complex numbers adds their angles (arguments) and multiplies their lengths (moduli). In the case of these numbers, the modulus is one, so multiplying it any number of times leaves it unchanged. For these numbers, multiplying means just adding the angles. So, if we take a number and multiply it by itself, we get: z2 = (cos θ + i sin θ) (cos θ + i sin θ) = (cos 2 θ + i sin 2 θ) If we do it again, we get: z3 = (cos θ + i sin θ) (cos θ + i sin θ) (cos θ + i sin θ) = (cos 3 θ + i sin 3 θ) And in general, zn = (cos θ + i sin θ)n = (cos n θ + i sin n θ). If two different modulus one numbers are multiplied, we get: z1 z2 = (cos θ + i sin θ) (cos ϕ + i sin ϕ) = cos (θ + ϕ) + i sin (θ + ϕ) �2 Trignometry and Complex Numbers.nb ��� θ ϕ Cos ϕ + i Sin ϕ Cos θ + i Sin θ Cos (θ + ϕ) + i Sin (θ + ϕ) In this operation, multiplication of the complex numbers is tightly bound up with addition of the angles. Such tight linkage of multiplication and addition is characteristic of exponentiation and logarithms, and in fact it is a very short step from what we have seen here to a formula expounded by Leonhard Euler in his work Introduction to the Analysis of the Infinite that identifies the two: ei θ = cos θ + i sin θ. (A few years ago I gave a whole lecture on this identity; I’ll see about having it available on the library web-site alongside this one.) For now, we will be especially interested in a subset of these numbers that bear the intriguing and evocative name, the “Roots of Unity.” Roots of Unity The “Roots of Unity” sounds like a New Age metaphysical treatise or the name of a theologically inclined folk-rock ensemble, but in our present context it means something rather different and more precise. It refers to numbers that, when raised to integral powers come to the result 1. Numbers like: 17 �18 ��� 2 Trignometry and Complex Numbers.nb 2 1, 3 1, 4 1, 5 1 … etc. To put the matter slightly differently, we are talking about numbers that are the solutions of equations like: x2 x3 x4 x5 - 1 1 1 1 = = = = 0 0 0 0 or in general, xn - 1 = 0 Based on what I learned in high school, these equations are not hard to solve. For x 2 - 1 = 0, I know that there are two solutions, + 1 and -1. For x 3 - 1 = 0, there is only one solution, +1, because (-1)3 = -1. That pattern continues down the line, with even numbered powers having two solutions and odd numbered powers having only one. That understanding works so long as one considers only the real numbers. But in the complex number field the answer is more complete, more interesting and in some ways more satisfying. Take x 3 - 1 = 0 for example. We are looking here for a number which, when cubed, is equal to one, that is, the cubed root of one. Easy! One, when cubed, is equal to one. That’s fine, but it’s not the full story. Consider the number on the unit circle whose angle is 120°: when squared it is still on the unit circle and its angle is 120° × 2 = 240°; when cubed, it is still on the unit circle and its angle is 120° × 2 = 360° = 0°. That number is +1 + 0 i. Thus, the complex number at 120° on the unit circle is also a cubed root of one! So, for that matter, is the number on the unit circle at 240°: squared, its angle is 480° = 120°; cubed, its angle is 360° = 0°. There are, in fact, three cube roots of one, and the points that represent them form an equilateral triangle in the unit circle. �2 Trignometry and Complex Numbers.nb ��� 2i The Cube Roots of Unity i 1 - , + i , 2 -2 3  2 0 -1 1 3 - , + i ,  2 2 {1, + i , 0} 1 2 -i -2 i The Algebraic Approach The graphical approach to the cube root of unity is simple: take the 360° of the circle and divide them by three. One can also take a strictly algebraic approach which is a little more intricate but which reaches the same result. Begin with the equation: x 3 - 1 = 0. As you noticed at first, the integer 1 (or, better, the complex number, 1 + 0 i) is solution. Therefore, we expect that this polynomial will be divisible by the linear factor (x - 1), as indeed it is: x 3 - 1 = (x - 1 ) (x 2 + x + 1) = 0. The new factor, (x 2 + x + 1), can easily be broken down into two linear factors by applying the quadratic formula to the equation x 2 + x + 1 = 0 : x= -1 ± 1 - 4 (1) 2 = -1 2 ±i 3 2 . So, the complete breakdown of the equation x 3 - 1 = 0 into linear factors is: (x - 1), x -  -12 + i 3 2 , x -  -12 - i 3 2  19 �20 ��� 2 Trignometry and Complex Numbers.nb You can verify this result by multiplying any of the solutions -- 1,  -12 3 2 + i  or  -12 3 2 - i  -- by itself three times and seeing that you get the result 1 + 0 i. More Roots of Unity It should not surprise you to learn that the equation x 4 - 1 = 0 gives four fourth roots of unity: +1, -1, +i and- i. And the equation x 5 - 1 = 0 gives five fi�h roots of unity, like this: 2i 2i i -2 -1 i 0 1 2 -2 -i -2 i -1 0 1 2 -i -2 i And so forth. Generally speaking, there are always n nth roots of unity. This tidy fact is a special case of a more general result proved by Gauss and called the “Fundamental Theorem of Algebra” which states that in the complex number field a polynomial equation of the nth degree always has n solutions. That is a wonderful result, but we don’t need its full generality for our task-at-hand. Look again at the polynomials that define the n roots of unity. We can see from the graphic representations that the number 1 + 0 i is a solution of each of these “roots of unity” equations. Consequently, we can divide any of them by the factor (x - 1), just as we did with the cube-root of unity equation: x 4 - 1 = (x - 1) (x 3 + x 2 + x + 1) = 0 x 5 - 1 = (x - 1) (x 4 + x 3 + x 2 + x + 1) = 0 and generally: x n - 1 = (x - 1) (x n - 1 + x n - 2 + … + x 2 + x + 1) = 0 The increasingly lengthy remainder terms are of special interest to us. The solutions to the corresponding polynomial equations �2 Trignometry and Complex Numbers.nb ��� 21 x3 + x2 + x + 1 = 0 x4 + x3 + x2 + x + 1 = 0 and generally: xn - 1 + xn - 2 + … + x2 + x + 1 = 0 are precisely what we need in order to find the vertices of regular polygons inside a unit circle. For fairly evident reasons, these equations are called collectively the cyclotomic (that is, “circle-cutting”) polynomials. They will be the subject, not of the next talk, but the one a�er that. Oh, By the Way … One More Thing to Note About Roots of Unity! Before closing, I want to note one more feature about the roots of unity that will show up in a later talk. It is this: for any whole number n, the sum of the nth roots of unity comes to zero. This can be seen pretty easily by looking at the case of the four fourth roots of unity: 2i i -2 -1 0 1 2 -i -2 i The four roots are +1, +i, -1 and -i. It is evident (isn’t it?) that when these four are added together, the sum is zero. A�er all, the pair + 1 and -1 add to zero, as do the pair +i and -i. �22 ��� 2 Trignometry and Complex Numbers.nb Only a little less evident is what happens with the three third roots of unity: 2i i -2 -1 0 1 2 -i -2 i The two red vectors are parallel and equal to the blue vectors to the two complex third roots of unity. Placing the three vectors end-to-end in the usual way for vector addition gives a closed triangle, beginning and ending at (0, 0). Similarly for all nth roots of unity: their sum always comes out to zero. As a quick corollary, if one takes all nth the roots of unity for any n, the whole collection excluding the number +1 sum up to -1. This follows easily from the fact that all the roots of unity sum to zero; if one excludes +1, the rest must sum to -1 so that all of them together come to zero. These facts will be used repeatedly in what follows. If you don’t remember them, I’ll remind you of them when they come up again. Conclusion So these are the fundamentals of the arithmetic of complex numbers. The next talk will concern itself with another topic altogether, the algebraic difference between points that can be constructed and those that can’t. In the fourth lecture, these two topic will come together to demonstrate how algebra can decide whether a construction is possible or not; we’ll look at two classical problems -- the trisection of an angle and the doubling of the cube -- and then at a new problem: the construction of the seven-gon. The fourth Tuesday lecture will bring all that has been said to bear on Gauss’s surprise, the construction of the seventeen-gon. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 22 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Trigonometric interpretation of complex numbers Description An account of the resource Transcript of a lecture given on September 17, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-17 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Numbers, Complex Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_Trigonometric_Interpretation_of_Complex_Numbers_2019-09-17