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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:
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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
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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
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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.
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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
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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
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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.
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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.
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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.
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“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).
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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.
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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
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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
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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.
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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
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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
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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.
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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.
�
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St. John's College Lecture Transcripts—Santa Fe
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21 pages
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What is the Measure of Electricity?
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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?"
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Fisher, Howard J., 1942-
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2024-02-23
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text
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Electricity
Faraday, Michael, 1791-1867
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English
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SF_FisherH_What_is_the_Measure_of_Electricity_2024-02-23
Friday night lecture
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https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/b227db32913978575ac398583d8bd9cb.pdf
da0c883f6eacef07a660e0b643760a2f
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St. John's College Lecture Transcripts—Santa Fe
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24 pages
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Alternating Current
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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."
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Fisher, Howard J., 1942-
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2023-05
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Electric currents, Alternating
Maxwell, James Clerk, 1831-1879
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https://digitalarchives.sjc.edu/items/show/7751
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English
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SF_FisherH_Alternating_Current_2023-04-26_text
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https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a6b0e707f0e7b6e1e9935ba88756f3b4.pdf
3db26f32f3d7a5bb28d89ba2438c6768
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
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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.
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26 pages
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Faraday's Galvanometer
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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."
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Fisher, Howard J., 1942-
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St. John's College
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Santa Fe, NM
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2023-02-22
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text
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Faraday, Michael, 1791-1867
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English
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SF_FisherH_Faraday's_Galvanometer_2023-02-22
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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
�
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What would Kepler say to Einstein?
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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."
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2021-11-05
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Kepler, Johannes, 1571-1630
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SF_DonahueW_What_would_Kepler_say_to_Einstein_2021-11-05
Friday night lecture
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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?
�
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Incommensurability : Meno and the diagonal
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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.'"
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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
�
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St. John's College Lecture Transcripts—Santa Fe
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The Qur'an : an introduction for Johnnies
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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."
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2021-06-16
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Qur'an
Islam
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English
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SF_WolfeK_The_Qur'an--An_Introduction_for_Johnnies_2021-06-16
Graduate Institute
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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
�
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St. John's College Lecture Transcripts—Santa Fe
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Learning to love Lincoln : Frederick Douglass's journey from grievance to gratitude
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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.
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Schaub, Diana J., 1959-
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2020-12-04
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Douglass, Frederick, 1818-1895
Lincoln, Abraham, 1809-1865
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English
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SF_SchaubD_Learning_to_Love_Lincoln_2020-12-04
Friday night lecture
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St. John's College Lecture Transcripts—Santa Fe
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The magnificent pendulums
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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."
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2020-11-18
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Galilei, Galileo, 1564-1642
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SF_FisherH_The_Magnificent_Pendulums_2020-11-18
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Maxwell reads Faraday
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Transcript of a lecture given on April 24, 2020 by Howard Fisher as part of the Dean's Lecture and Concert Series.
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2020-04-24
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Maxwell, James Clerk, 1831-1879
Faraday, Michael, 1791-1867
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English
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SF_FisherH_Maxwell_Reads_Faraday_2020-04-24
Friday night lecture
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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.
�
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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
�
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The phenomenology of blackness
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Transcript of a lecture given on November 22, 2019 by Michael Sawyer as part of the Dean's Lecture and Concert Series.
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<p>The following video is referenced in this lecture:</p>
<p><a href="https://youtu.be/n92zfXkRdnk">https://youtu.be/n92zfXkRdnk</a></p>
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SF_SawyerM_The_Phenomenology_of_Blackness_2019-11-22
Friday night lecture
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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:
TableFormTablen, 2n , Mod2n , 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
TableFormTablen, 3n , Mod3n , 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
ζ3
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
μ1, 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)
�
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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.
�
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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 .
�
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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.
�
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Text
Lecture 1: Meet Your New Best Friend: the Square Root of
Negative One
Grant Franks
June 19, 2019, rev Sept. 9, 2019
Introduction
This is the first of a series of six lectures on algebra. A�er today’s lecture, there will be four more talks
on successive Tuesday evenings, each unpacking a step along the way to the final result that I want to
share with you, Gauss’s demonstration of the Constructibility of the Seventeen-Sided Polygon or
“Heptadecagon.” (The schedule is available on the handouts.) A�erwards, on October 16, there will be
a final Wednesday a�ernoon lecture that very briefly recaps the contents of the Tuesday evening
technical talks and concludes with some reflections on the significance of the square root of negative
one to the foundations of arithmetic and the relation of ordinary experience to mathematics.
This series of talks grew out of a remark that a former dean of the College made to me years ago. This
person, whom I respect and admire greatly, said something I thought was seriously questionable.
“Algebra is boring,” he opined. “Our students love geometry,” he said, “because its beauty strikes
them immediately. But algebra is just a tool, a technique. Nobody wants to spend any class time
studying it. It’s just dull.”
I took those words as a challenge. It didn’t seem plausible that sane people would devote countless
hours of intense intellectual effort to something that is inherently dull, at least without being paid a lot
of money. (Algebraists generally are not highly paid.) Do they really enjoy tedium?
No, they don’t. Algebraic structures have a real beauty, even if it takes a little work to notice and
appreciate it.
For better or for worse, algebraic beauty is invisible beauty, and the taste for it is an acquired taste, like
that for single malt scotch, twelve-tone music or the word-play of Finnegans Wake. My on-again, offagain quest, therefore, for many years has therefore been to find the some entryway into what algebra
has to offer, something more appealing than exercises in factoring polynomials. I wanted to find the
algebraic equivalent of the Pythagorean Theorem, some result that would make someone stand still in
wonder and say, “Whoa! Really?,” as Thomas Hobbes reportedly did when he saw a copy of Euclid
open to proposition 47 of book one. Legend has it that he stood transfixed at a library table for hours
reading the entire first book of the Elements BACKWARD until he arrived at the postulates. That
encounter reportedly made him “in love with geometrie.” (John Aubrey, Brief Lives (c. 1700))
What, then, would make someone “in love with algebra?” If anything could do it, I think it would be
Carl F. Gauss’s construction of the heptadecagon. It’s a beautiful result, and the pathway to it, while
�2 ���
1 Square Root of Negative one.nb
not perfectly smooth, requires only a few hours of preliminary work, not years. Also, for this audience,
it speaks directly to geometrical demonstrations that all St. John’s students have encountered by the
middle of the first semester of their freshman year, namely, the propositions of Book Four of Euclid’s
Elements where we see the construction of regular-sided polygons in circles. Euclid shows how to
construct the equilateral triangle, the square, the pentagon -- that’s a hard one! -- and the hexagon
inside a given circle. Then, without explanation, he skips ahead to the fi�een-gon (Elements IV, 16).
Then he stops. Why? Euclid, characteristically laconic, says nothing.
Two thousand years later, a very young Carl Friedrich Gauss provided the answer.
However, in order reach that result, you need to come to terms with [horror suspense sound effect] the
square root of negative one. [mad scientist laugh sound effect]
ζ5
ζ4
ζ6
ζ3
ζ2
ζ7
ζ
ζ8
-1
ζ9
ζ 10
ζ0
ζ 16
ζ 15
ζ 11
ζ 12
ζ 13
ζ
14
Some people have difficulty accepting this number. They say, for instance, that they are put off by the
fact that it doesn’t exist. Which is ridiculous! You shouldn’t let so trivial a problem prevent you from
embracing this concept, for the square root of negative one, also denoted by the single letter i, is the
gateway to an algebraic realm of amazing results. Through it we come to the Complex Number Field.
This is a spectacular realm where we see hidden machinery that links algebra and trigonometry,
wonderfully expressed in DeMoivre’s formula
(cos θ + i sin θ)n = cos n θ + i sin n θ ;
where we find Euler’s famous identity “ei π = -1,” where all polynomial equations can be completely
decomposed into linear factors and where we hear
the buzzin’ of the bees in the cigarette trees
‘round the soda water fountains!
No, wait. That’s the Big Rock Candy Mountains. Never mind. The Complex Number Field is still an
algebraic paradise. It is here that you must go to find the result that I most want to show you, the
construction of the heptadecagon.
Gauss’s result is not overly complicated, but it does involve several separate stages, so I thought it best
not to try to jam all of it into a single talk. Doing that would generate more confusion than understand-
�1 Square Root of Negative one.nb
���
3
ing. So, all I want to do today is to set up the issue of the square root of negative one by describing how
it first appeared, how it was at first dismissed, and how an algebraic triumph led an Italian algebraist to
reconsider the possibility that it might not be gibberish but an important numerical results.
Start Simple: The Linear Equation
We will be concerned with polynomial formulas and their solutions. The simplest polynomial formula
is an equation in the first degree, that is, one where the variable -- we’ll call it x -- appears only to the
first power:
x - a = 0,
Here “x” is an unknown quantity and “a” is something given. To be more particular, we might have:
x - 3 = 0.
What is x? In this case, x is 3. Is that obvious? You don’t need to do the explicit manipulation of adding
3 to both sides of the equation, although you could:
x-3+3=0+3
x = 3.
Already there are, in fact, subtleties and complexities that could be discussed at length. What are these
quantities “x” and “a”? Are they lengths? areas? numbers? magnitudes? Do we know? Do we care?
Are they particular lengths or numbers? It is possible to imagine a “general” quantity that is nothing in
particular? Even if we cannot imagine it, can we conceive of it? What is a “variable” like x? How is a
variable similar to or different from a “constant term” like a? In a situation in which we don’t know
what “x” and “a” are, are there differences in the manner in which we “don’t know x” and in which we
“don’t know a?” Can we “add the same thing to both sides of an equation” if we have no idea what
those things are?
All these questions are interesting and important. However, I’m going to pass by all of them because
the real concern of this lecture lies further down the road in the direction of more complex equations.
The Next Step: the quadratic, x2 + b x + c = 0
I hope you found the linear equation x - 3 = 0 easy to solve. Things get more complicated quickly.
In the realm of polynomials, the next step in complexity takes us to the quadratic equation:
x2 + b x + c = 0
�4 ���
1 Square Root of Negative one.nb
where “b” and “c” are rational numbers. (Why start with “b” and not “a”? Because the really simple
form is “a x 2 + b x + c = 0” where “a,” “b,” and “c” are all integers. But it is convenient to divide out
the lead term “a” and re-define b and c so that one has a polynomial whose leading coefficient is 1.
Such an expression is called a “monic polynomial.”)
There is a path to solving this equation quickly and reliably. It’s called “completing the square.” I
suspect it may be familiar to many of you. In case it isn’t, I will review it here. Stepping through the
derivation of the quadratic formula will be useful when we turn in a moment to the next step, the cubic.
Consider: what does a “perfect square” polynomial look like in algebra? That is, what do we get when
we multiply a single factor by itself. Try making one:
(x - b)(x - b) = x 2 - 2 b x + b2
That’s what a “perfect square” polynomial looks like, one whose two solutions are both b. If someone
posed that polynomial for us to solve,
x 2 - 2 b x + b2 = 0
life would be easy! We would just take the square root of both sides:
x 2 - 2 b x + b2 =
0 =0
(x - b)2 = 0
x - b =0
x = b.
Sadly, x 2 + b x + c = 0 is not perfect square. Happily, however, we can make it into a perfect square, or
at least get close enough. Look again at the general equation for a perfect square:
x 2 - 2 b x + b2 = 0
Notice the relation between the coefficient of x (- 2 b) and the constant term (b2 ). If you take the
coefficient of x, divide by 2 then square it, you get the constant term. Now look at the equation we
actually have:
x2 + b x + c = 0
�1 Square Root of Negative one.nb
By brute force, let’s make a perfect square. First, subtract c from both sides of the equation, just to
“clear the decks.”
x2 + b x = - c
Take half the middle term ( b2 ) and square it: you get
x2 + b x +
b2
4
b2
4
=
-c =
b2
.
4
Add that to both sides of our equation:
b2 - 4 c
4
Now, on the le�, you have a perfect square: x 2 + b x +
b2
4
2
= x - b2 . This is great! Take the square
root of both sides:
x2 + b x +
b2
4
b2 - 4 c
4
=
On the le�, we have the square root of a perfect square, a situation that we deliberately contrived:
x + b2
2
± x + b2 =
b2 - 4 c
4
=
b2 - 4 c
2
(We need to say “±” because both x +
b
2
and -x +
b
when
2
squared give the same result.)
x =
-b
2
±
b2 - 4 c
2
This is the standard form of the quadratic formula for a monic, quadratic polynomial. Plug in the
coefficients b and c, turn the crank and out pop two values for x.
A Cute Quadratic Trick
Before we go on, however, there’s a clever little trick involving quadratics that I want to show you. It
will be used many times in what follows.
Suppose you have two different factors, (x - a) and (x - b) of a polynomial equation.
(x - a) (x - b) = 0
Multiply and expand:
���
5
�6 ���
1 Square Root of Negative one.nb
x 2 - (a + b) x + a b = 0
Look at the coefficient of x and at the constant term: a + b and a times b. The first is the sum of a and b,
the second is the product, where a and b are the two roots of the polynomial. This observation can
easily be generalized. If we had three terms:
(x - a) (x - b)(x - c)= 0
The expanded version would look like this:
x 3 - (a + b + c) x 2 + (a b + a c + b c) x + a b c = 0
For the fourth degree:
(x - a) (x - b)(x - c)(x - d)= 0
The expanded version would look like this:
x 4 - (a + b + c + d) x 3 + (a b + a c + a d + b c + b d + c d) x 2 + (a b c + a b d + b c d) x + a b c d = 0
Maybe you see where this is going: the first non-zero coefficient is always the sum of all the roots. The
second is the sum of all the roots taken two at a time. The third is the sum of all the roots taken three at a
time. The constant term, when you get to it, is always the product of all the roots. These patterns are
very interesting and very useful. The patterns that you see here can be described by saying that the
coefficients are “symmetric functions” of the roots because, as you can see, each of a, b, c play the
same role in each expression. Exploration of symmetric functions is fascinating, but for us right now it
is beside the point.
Look back at the second degree equation, and particularly at the coefficients of the equation: the
coefficient of x is the sum of the solutions a and b. The constant term, is the product of the solutions a
and b. Pause over that for a second: just looking at the polynomial may not tell us the two solutions
immediately, but even a glance at the coefficients give us the sum and the product of the solutions.
Now, turn that observation inside out. Suppose you have two unknown numbers, call them r1 and r2 .
Suppose further that you don’t know what these two numbers are, but you DO know their sum and their
product. In that case, you can make a quadratic equation that has these two numbers as its solutions:
x 2 - (r1 + r2 ) x + r1 r2 = 0
The solutions of this equation can be found with the quadratic formula:
�1 Square Root of Negative one.nb
(r1 + r2 ) ±
(r1 + r2 )2 - 4 r1 r2
2
���
7
=x
You may say to yourself, “That’s great, Mr. Franks. But how o�en, really, does it happen that I come to
know the sum and the product of two numbers without knowing what those numbers are individually?”
Well, in your daily life, maybe not so o�en. However, in the algebraic journey that lies before us, this
little trick is going to show up more o�en than you may imagine.
What Happens When Things Go Wrong
So far, we have constructed the Quadratic Formula, which solves a general quadratic equation:
x2 + b x + c = 0
x =
⟹
-b
2
±
b2 - 4 c
2
You notice that the solution contains a radical, b2 - 4 c . If the quantity under this radical, which is
called “the discriminant,” is positive, all is well. But if that quantity becomes negative, that is, if 4c is
greater than b2 , then we have a problem.
Here is a graph of the equation y = x 2 + 2 x + c. We have arranged matters so that we can vary the
value of the constant term c:
�
y = x2 + 2 x + c,
b2 - 4 c = , -9.16
10
5
-3
-2
1
-1
2
3
-5
The “solutions” -- that is, the x values where the function equals zero, are given by:
x=
-2 ±
4-4c
2
=
-2 ± 2
2
1- c
= -1 ±
1- c
�8 ���
1 Square Root of Negative one.nb
When c is less than one, the expression is positive and there are two solutions. When c is equal to 1, the
expression under the radical is zero and there is one solution. When it is greater than one, the expression under the radical is less than one and, as you can see graphically, there appears to be no solution.
From this example, we might conclude that the square root of a negative number means “impossible.”
Another Example
Descartes came to this conclusion looking at a slightly different example. He considered the expression:
x=
1 - y2
He could picture this by an illustration like the one below. Consider a semi-circle with radius 1. Draw a
line parallel to the diameter with a variable height, y.
�
������
y
x
So long as y lies between zero and one, the quantity under the radical is positive and there are two
solutions, a negative one and a positive one, indicated by the intersection of the horizontal blue line
with the semicircle. When y reaches the value 1, then the horizontal line is tangent to the semicircle
and there is one solution only. When y is greater than 1, the horizontal line misses the semicircle. It
looks as if then there are no solutions at all.
Tentative conclusion: when a radical contains a negative sign, the formula is meaningless. There is no
solution to the equation, and the expression should be rejected as absurd.
That conclusion was generally accepted before Rafael Bombelli began to work with the cubic equation.
�1 Square Root of Negative one.nb
���
9
The Cubic
The cubic equation poses greater challenges than the quadratic. The quadratic equation is not exactly
simple, but its solution has been known for a long time. Babylonians had techniques that were more or
less equivalent of solving a quadratic equation. Some of Euclid’s geometrical manipulations in Book II
of the Elements also answer questions that are closely analogous to finding the solution of the general
quadratic equation.
By contrast, the solution for the cubic equation was not published until the sixteenth century. The
formula for the cubic is known as “Cardano’s Formula,” named a�er Girolamo Cardano who succeeded
where many others had failed … by stealing it from the man who invented it, Niccolo Tartaglia. Cardano published this formula in his [Cardano’s] book, The Great Art or the Rules of Algebra (1545). Let
this be a lesson to you: if you want to make a name for yourself in mathematics, steal freely and publish early. (This is an example of “Stigler’s Law of Eponomy” which states that no result in science or
mathematics is named a�er the person who first discovered it. See, Tom Lehrer’s song, Lobachevsky.)
Because Cardano’s formula is important, I propose to walk through its derivation here. I am aware,
however, that it is hard to follow a sequence of algebraic steps in a lecture format; that’s why I have
printed copies for anyone who wishes to review the derivation later at her or his leisure. The lecture
will also be posted on some part of the College’s web page. Of course, you are also free (if you wish)
just to accept the result on faith.
One begins, not with the full form of the cubic, but with the “depressed cubic,” a formula that has been
manipulated so that the x 2 term disappears. There is a routine procedure for making this happen, so it
does not limit the generality of the demonstration. (The procedure for “depressing the cubic” is
included as an appendix to the printed version of this lecture for anyone to examine at leisure.) Thus,
we begin with:
x3 + c x + d = 0
We want to know what x is. To solve this equation, Cardano -- or Tartaglia, really -- came up with a
special trick: re-conceive x as divided into two parts, p and q so that
x = p + q.
Imagine, then, a cube whose whole side is x, a length that has been divided into two parts, p and q:
�10 ���
1 Square Root of Negative one.nb
�
We can see that the cube is broken up into
(i) a cube of side p (red);
(ii) a cube of side q (blue); and
(iii) three “slabs” of volume p q (p + q) (yellow, green and purple).
�1 Square Root of Negative one.nb
�
�
Symbolically, we have:
(p + q)3 = p3 + 3 p2 q + 3 p q2 + q3 = 3 p q (p + q) + (p3 + q3 )
or
(p + q)3 - 3 p q (p + q) - p3 - q3 = 0
which gives us:
(p + q)3 + (-3 p q) (p + q) + (- p3 - q3 ) = 0
x3
c
x
Compare this to our depressed cubic:
x3 + c x + d = 0
d
���
11
�12 ���
1 Square Root of Negative one.nb
The two are the same PROVIDED:
-3pq=c
pq=
c
-3
p3 q3 =
and
- (p3 + q3 ) = d
or
and
- (p3 + q3 ) = d
or, cubing the first expression --
- c3
27
(p3 + q3 ) = - d
and
Now look: we have expressions for the the sum and the product of the two quantities, p3 and q3 .
Therefore, we can construct the quadratic equation. (I told you this procedure would be helpful!) I’ll
use “w” as a variable:
w2 - (p3 + q3 ) w + p3 q3 = 0
values we just determined:
which will have p3 and q3 as it solutions. Substitute the
3
w2 + d w - 3c = 0
whose two solutions are:
w=
-d±
c 3
(d)2 + 4 3
2
=
-d±
d 2
c 3
4 2 + 4 3
2
=
-d± 2
d 2
c 3
2 + 3
2
=
-d
2
±
2
d2 + 3c
3
The two solutions of this are p3 and q3 . If we take the cube root of each and add them, we get p + q = x:
x=
3
-d
2
+
2
d2 + 3c
p
3
+
3
-d
2
-
2
d2 + 3c
3
q
That is Cardano’s formula -- the one he stole from Tartaglia -- for solving the cubic equation.
What Happens When Things Go Wrong?
For the Quadratic: Apparent Impossibility
If you have followed so far, we have two formulas, one for second degree (quadratic) equations and
one for third degree (cubic) equations.
The quadratic formula:
�1 Square Root of Negative one.nb
For the equation:
x2 + b x + c = 0
x is given by:
x =
-b
2
���
13
b2 - 4 c
2
±
The cubic formula (“Cardano’s Formula”):
For the equation:
x3 + c x + d = 0
x is given by:
x=
-d
2
3
2
d2 + 3c
+
3
+
-d
2
3
-
2
d2 + 3c
3
This is all very well … except if the coefficients are such that the quantities under the square-root signs
become negative.
We have already seen what happens to the quadratic when the quantity under the radical is negative:
the formula seems to give no answer at all and the expression appears to be meaningless.
A priori, we seem to have no reason to suspect that the cubic will act differently.
But it does.
You Can Use Cardano’s Formula to Solve Cubics
Sometimes, Cardano’s formula works just fine. Consider, just as an example:
0 = x 3 + 6 x + 20
where c = 6 and d = 20
Apply the formula to get:
x=
x=
x=
x=
3
-d
2
3
- 20
2
3
3
+
+
2
d2 + 3c
3
2
20
+ 63
2
-10 +
100 + (2)3
-10 +
108
+
3
+
3
-d
2
3
+
+
-10 -
3
3
2
d2 + 3c
-
- 20
2
-
-10 -
108
3
2
20
+ 63
2
100 + (2)3
3
�14 ���
1 Square Root of Negative one.nb
x=
-10 + 12
3
3 +
3
-10 - 12
3
x = 0.73205 + (-2.73205)
x = -2
… which really is one of the solutions, as you can check. (The other two are imaginary, which you
can find a�er you have factored out (x + 2) from 0 = x 3 + 6 x + 20.
x 3 + 6 x + 20 = (x + 2)(x 2 - 2 x + 10)
You can solve x 2 - 2 x + 10 = 0 with the quadratic formula.)
However, Sometimes Things are Very Weird
But things can also go terribly, terribly wrong. Take another example:
x 3 - 15 x - 4 = 0
Here, c = - 15 and d = - 4. Put the values into Cardano’s Formula:
x=
x=
3
-d
2
3
4
2
2
d2 + 3c
+
3
2
-4
+ -15
2
3
+
x=
3
2+
4 + -125
x=
3
2+
-121
x=
3
2 + 11
-1
+
+
+
3
3
+
3
3
+
3
2-
2-
2
3
-d
2
-
d2 + 3c
4
2
-
-4
+ -15
2
3
3
2
3
4 + -125
-121
2 - 11
-1 .
You have an imaginary quantity in each part of the solution. Does result this mean that the equation
has no solution?
No. It obviously does have a solution. Every cubic has at least one real solution. Just look at the graph
of this formula:
�1 Square Root of Negative one.nb
Plotx3 - 15 x - 4, {x, - 5, 5}, PlotLabel → "y = x3 - 15 x - 4"
y = x3 - 15 x - 4
40
20
-4
2
-2
4
-20
-40
This formula should have three solutions. The solution farthest to the right seems to be +4. If there
were any doubt, you can check and see that one of the solutions is +4.
43 - (15) 4 - 4 = 64 - 60 - 4 = 0 .
If the solution of the equation is +4, why did Cardano’s formula give a result with two expressions
involving the square roots of negative numbers? What went wrong with Cardano’s Formula?
This is the question that confronted sixteenth century student of algebra, Rafael Bombelli.
(Rafael Bombelli, 1525 - 1572)
The short answer is that nothing went wrong. The slightly longer answer is that
2 + 11 -1 + 2 - 11 -1 is in fact equal to four. That is far from obvious, which is why we
should be grateful that Rafael Bombelli was a genius.
3
3
���
15
�16 ���
1 Square Root of Negative one.nb
For it was Bombelli who stared at this formula, aware that one of the answers of the equation should
be 4, and finally figured out how to make some sense of what he saw. It occurred to him -- we do not
know how (although mathematician have reconstructed some plausible guesses) -- that the quantities
2 + 11 -1 and 2 - 11 -1 were in fact cubes of other complex quantities:
2+
and 2 -
-1
-1 .
Try it!
2
-1 = 2 +
2 +
-1 2 +
-1 = 4 + 2
-1 + 2
-1 - 1 = 3 + 4
-1
3
2 +
-1 =
2
2 +
-1 2 +
-1 = 3 + 4
-1 2 +
-1 = 6 + 8
-1 + 3
-1 - 4 = 2 + 11
-1
And:
2
-1 = 2 -
2 -
-1 2 -
-1 = 4 - 2
-1 - 2
-1 - 1 = 3 - 4
-1
3
2 -
-1 =
2 -
2
-1 2 -
-1 = 3 - 4
-1 2 -
-1 = 6 - 8
-1 - 3
-1 - 4 = 2 - 11
-1
With this in hand, the strange result of Cardano’s Formula reduces like this:
x=
3
3
2 + 11
-1
2 +
-1
+
3
+
2 - 11
3
3
2 -
-1 =
3
-1 = 2 +
-1 + 2 -
-1 = 4
The square roots of negative one cancel out and the simple answer appears. This looks like magic,
especially if you have spent hours staring hopelessly at the problem, increasingly convinced that it was
created by the devil to torment incautious humans.
This strange result seems to suggest that the square root of a negative number can form part of a
meaningful formula, provided the imaginary quantities in the expression cancel one another out
before the final solution appears. We still don’t know quite what -1 means, but its appearance no
longer implies instantly that a formula is meaningless. It can participate in the strange morrice-dance
of algebra and lead ultimately to verifiable solutions.
Bombelli’s result, wonderful as it is, is far from fully enlightening. Pleased as we might be that Bombelli
could guess what complex number, when cubed, would give the particular quantity he was looking for,
�1 Square Root of Negative one.nb
���
17
his success doesn’t give us much help in solving other problems. Moreover, it doesn’t give us much
help in figuring out what the square root of negative one is. Do the imaginary numbers have any real
meaning? Or are they bizarre brambles that need to be cleared away by ad hoc trickery? Was Bombelli’s result a lucky accident? Or does it represent something valuable?
All that is the subject of the next lecture.
Appendix 1: How to Depress a Cubic Equation
Grant Franks
August 28, 2019
Begin with the general cubic:
x3 + b x2 + c x + d = 0
For x substitute y - b3 :
3
2
y - b3 + b y - b3 + c y - b3 + d = 0
y3 - b y2 +
b2
3
y-
b3
27
+ b y2 -
2 b2
3
y+
y3 + ( b - b) y2 + c -
b2
y
3
+ 227b -
y3 +
b2
y
3
+ 227b -
…
+c-
b3
9
+ cy -
3
cb
3
+ d = 0
3
cb
3
+ d = 0
cb
3
+d=0
Et voila, a depressed cubic.
For example, suppose you had x 3 + 9 x 2 - 3 x + 2 = 0. Substitute x = y - 93 :
(y - 3)3 + 9 (y - 3)2 - 3 (y - 3) + 2 = 0
(y3 - 9 y2 + 27 t - 27) + (9 y2 - 54 y + 81 ) - (3 y + 9) + 2 = 0
y3 - 24 y + 47 = 0
You can solve this (depressed) equation using Cardano’s formula. Then from the values of y, find the
values of x from the relation:
x = y - b3
which means
x+
b
3
=y
�18 ���
1 Square Root of Negative one.nb
Appendix 2: Lectures in the Series
Wednesday a�ernoon lectures, 3:15 pm in the Junior Common Room
Tuesday evening lectures, 7:30 pm in Room FAB 109
1.
Say Hello to Your New Best Friend: - 1
(Wednesday, September 11)
2.
The Bridge between Algebra and Trigonometry
(Tuesday, September 17)
3.
Constructible Numbers
(Tuesday, September 24)
4.
Building the Pentagon with Algebra
(Tuesday, October 1)
5.
The Heptadecagon (17-gon)
(Tuesday, October 8)
6.
Numbers and Meaning
(Wednesday, October 16)
�
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Text
Reconciling Subjectivity and Substance: Hegel’s Critique of Pure
Personhood
Jonathan Hand, St. John’s College, Santa Fe
A while ago, a friend told me how excited she was, when she got to college, in
first learning about ancient Greece. For her, the Greek polis seemed to embody a world
that had those very things which modern life lacked: a sense of purpose, and connection
to a larger whole. The difficulty of translating “polis” flags the issue. The usual
translation, “city-state,” attempts to capture the fact that political units around the size of
Santa Fe were sovereign countries, with their own armies and foreign policy: citizens
assembled to discuss, not property tax rates or zoning, but issues that bore on the city’s
very existence. The “state” in “city-state” however, is misleading, because it implicitly
presupposes the modern liberal distinction between “state” and “society.” Unlike the
quasi-autonomy domains such art, theatre, the economy, and religion have from politics
and each other in our world, in the ancient world these were all mutually reinforcing parts
of civic life. Socrates was accused of not believing in the city’s gods—but in his defense
he did not, as we would, argue that he had a “right” to his own opinions, that there was
some “private sphere,” such as religion, out of which the government should stay. The
fate of Socrates, however—as well as slavery and the status of women-- gave my friend’s
Hellenophilia a marked ambivalence. When pressed on these issues, she said that despite
showing up the fragmentation, alienation, and aimlessness of the modern world, the polis
was not a world to which we either could or even should want to return. We are all
“individuals” now.
J. Hand Hegel Person Lecture
Given September 6 2019
Page 1 of 38
�Of course, this ambivalence about “modernity” is hardly unique to my friend: it
has been a fundamental aspect of European civilization at least since 1754, when in his
Discourse on Inequality Rousseau distinguished the ancient “citizen” from the modern
“bourgeois,” to the detriment of the latter. Rousseau gave powerful voice to this
dissatisfaction, but he certainly did not “cause” it: it is an understandable human reaction,
one probably at the root of the feelings—ranging from ambivalence to hostility—that
non-Western countries have about “Westernization,” however understood. Dissatisfaction
with or at least ambivalence about modernity is part of modernity. The attempt to
resolve that ambivalence, and reconcile us to modernity, lies at the center of Hegel’s
thought.
This project of reconciliation is most obvious in the presentation of the modern
“ethical world” in his Elements of the Philosophy of Right, first published by Hegel in
1821 on the basis of his Berlin lectures, and expanded by editors after his death to include
additions culled from student lecture notes clarifying various points. However, a
1
concern about modernity as a problem, via a contrast with ancient Greece, runs through
all of Hegel’s work. As a recent biography by Terry Pinkard points out, even before
Hegel reached the age of 20, ancient Greece represented for him, as it did for many
young Germans at the time, a world of lost wholeness.
2
Hegel’s word for the quality the Greek world had that is seemingly lacking in his,
and our, time: substantial. A full understanding of what Hegel means by substance is
well beyond the limits of this lecture: Hegel, in using the Latinate term substanz (as
opposed to a term with German roots), puts himself in relation to a tradition starting with
Citations from the Philosophy of Right are styled “PR” and are by section number in Elements of
the Philosophy of Right, translated by H. Nisbet and edited by Allen Wood (Cambridge, UK: Cambridge
U.P., 1991). For the Editorial Notes, I use page numbers.
Terry Pinkard, Hegel: A Biography (Cambridge UK: Cambridge U.P., 2000), p. 32.
1
2
J. Hand Hegel Person Lecture
Given September 6 2019
Page 2 of 38
�Aristotle’s reflections on οὐσια and passing through Spinoza and Kant. However, we
need not be adepts in the history of ontology to understand what Hegel is driving at with
the term in a political or human context. For example, Hegel says in PR §40 that a view
of right that divides the world into “things” and “persons” having “rights” to those
things—the classic “Lockean” view, if you like—“jumbles together rights which
presuppose substantial relations, such as family and state, with those that refer to abstract
personality.” My relations to my sister, or to my country, are more concrete and weightier
(I hope) than my relation to my contractor when I hire him to fix my roof, that lengthy
contract notwithstanding. Indeed that contract is required because there is nothing else
grounding, standing under, sub stance, my relation to my contractor. Our relation, Hegel
calls one of “abstract” personhood, because any two parties could make such a contract,
whomever they happened to be.
Helpful in understanding what Hegel means by “substance” and “substantial” are
his remarks at the beginning of the chapter on Spirit, chapter VI, of the Phenomenology
of Spirit. By “Spirit,” Hegel refers to his claim that all consciousness, all mind, is social
and historical “all the way down”: there is never “mind” without its being Greek mind,
French mind, etc. The first 5 chapters of the Phenomenology are an extended reductio
argument where all other views of consciousness are shown to fail on their own terms.
Hegel begins chapter VI with a section on the Greek world, which he calls “True Spirit.”
In the introduction to the chapter, just prior to that section, he says (¶439) :
3
Citations to the Phenomenology (PhG) are by paragraph number in the translation of A.V. Miller
(Oxford: Oxford U.P., 1977). I have occasionally revised these using the bilingual version of the PhG
published on-line by Terry Pinkard in 2010, downloaded October 2014 from
http://terrypinkard.weebly.com/phenomenology-of-spirit-page.html. Since Pinkard has subsequently
published the English part of the translation with Cambridge U.P., the site has been taken down, but as of
3
J. Hand Hegel Person Lecture
Given September 6 2019
Page 3 of 38
�Spirit, being the substance and the universal, self-identical, and abiding essence,
is the unmoved solid ground and starting-point for the action of all, and it is their
purpose and goal, the in-itself of every self-consciousness expressed in thought.
This substance is equally the universal work produced by the action of all and
each as their unity and identity, for it is the being-for-self, the self, action.”
(Hegel’s italics).
Athens makes Athenians, and Athenians make Athens. For Hegel, the “substantiality” of
the Greek ethical world was something the Greeks experienced immediately, i.e.
without reflection or the giving of reasons. Thus, Hegel claims in the Philosophy of
Right, §147, that to speak of the Greeks as having “faith” or “believing in” their gods
understates their immediate identity with their ethical world:
Faith and trust arise with the emergence of reflection, and they presuppose
representations and distinctions. For example, to believe in pagan religion and to
be a pagan are two different things.
The Greeks did not “believe in” paganism; they were pagans. A nice example of this
immediacy is the character Anytus in Plato’s Meno. When Socrates asks him to whom
one should go to become virtuous, Anytus responds irritably that any Athenian gentleman
would be better (92e) than those who make a living claiming to teach such things, i.e.
sophists. Anytus admits (92b) to never having met a sophist, but for Anytus, virtue is
what virtuous people do. That Anytus is one of Socrates’s accusers in the Apology only
underscores the fact that the well-brought up Athenian as such has no patience with those
who ask what virtue is.
For Hegel, “substance” and “immediacy” are hardly peculiar to the Greek world.
It’s the reverse: Greece is important because it is there that spirit experiences a decisive
crisis or division, a crisis evident in the fate of Socrates, who as “free infinite
personality,” as Hegel describes him, has no place in the Greek ethical world. Rather, for
November 2018, the bilingual version was still available at
https://www.marxists.org/reference/archive/hegel/works/ph/pinkard-translation-of-phenomenology.pdf.
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�Hegel, all ethical worlds prior to Rome—what one might call “traditional societies”—
show this immediate, unreflective substantiality. In his Lectures on the Philosophy of
World History, Hegel infamously begins with “the oriental world” (by which he means
China, India, and the Persian Empire) which “has as its inherent and distinctive principle
the Substantial [the Prescriptive] in morality.”
4
In the case of China, Hegel claims the
ethical world rests on the pillars of the Emperor’s absolute authority and the spirit of
family piety. The latter of course appears in many other places—it is what is defended
5
by Antigone in Sophocles’ play of that name.
For Hegel, what was undeveloped in all pre-modern societies, and what modern
society has in spades, is “subjectivity.”
Hegel traces the origins of modern subjectivity
ultimately to Christian inwardness—to one’s personal relation to God—an inwardness he
calls in the Phenomenology the “unhappy consciousness” and which emerged in the
alienated subject of the Roman empire. However, it is only in modern times that
subjectivity comes into its own, as a secular and political principle. In an “Addition”
(p.13) to the preface to PR, Hegel describes his (and I think, our) situation:
The human being does not stop short at the existent, but claims to have within
himself the measure of what is right; he may be subjected to the necessity and
power of external authority, but never in the same way as to natural necessity, for
his inner self always tells him how things ought to be, and he finds within himself
the confirmation or repudiation of what is accepted as valid.
This claim is not made everywhere and always, but it might make it seem that modern
subjectivity is a mere “mentality,” the way we think now. Far from it. To be real, a part
of the modern European spirit, subjectivity has to be embodied, made objective, as it has
now become in the modern “state,” a political form which gives an unprecedented
4
5
Cited from the translation of J. Sibree (New York: Dover, 1956; first published 1899), p. 111.
Op. cit, pp. 120-1.
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�autonomy to the economy, religion, and the arts: what we call “society.”
Forerunners,
such as Socrates, were mere anomalies.
Hegel is no simple celebrant of modern subjectivity—quite the contrary. Much
like a Socrates swatting off many and varied forms of sophist pests, Hegel spends much
of his argument, in PR, the Phenomenology, and other places cataloguing and combatting
all the various pathological or at least incomplete forms of subjectivity that are sprouting
in Europe like mushrooms after the rain. Thus, in the Preface to PR, Hegel gets in his
crosshairs one “Herr Fries,” “a leader of this superficial brand of so called philosophers,”
who claims that “truth consists in what wells up from each individual’s heart, emotion,
and enthusiasm in relation to ethical subjects, particularly to the state, government, and
the constitution.” Fries and his young enthusiasts, however, are just one flavor in the
contemporary subjectivity smorgasbord. In the penultimate section in the “Morality”
division of PR, §140, Hegel goes through a series of modern variants of subjectivity to
show just where taking truth to be solely “in oneself” finally leads:
It is not the thing which is excellent, it is I who am master of both law and thing; I
merely play with them as my own caprice, and in this ironic consciousness in
which I let the highest of things perish, I merely enjoy myself. In this shape,
subjectivity is not only empty of all ethical content in the way of rights, duties,
and laws and is accordingly evil (evil, in fact of an inherently wholly universal
kind); in addition, its form is that of subjective emptiness, in that it knows itself as
this emptiness of all content and, in this knowledge knows itself as the absolute.
(PR §140, p. 182)
This concern about what we would call “nihilism,” and what Hegel calls “evil” and
“absolute sophistry” (§140, pp. 182-3), are definitive of Hegel’s entire philosophic
effort: to bring together substance and subject, or as we might say, subjectivity and
objectivity. As Hegel famously says in the Preface to the Phenomenology (¶ 17): “In my
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�view, which can be justified only by the exposition of the system itself, everything turns
on grasping and expressing the True, not only as Substance, but equally as Subject.”
How, then, in his political philosophy, does Hegel attempt a reconciliation
between substance and subjectivity? To flesh this out, I shall focus on Hegel’s treatment
of the concept of personhood, Persönlichkeit, a notion whose Latin etymology points to
its roots in Roman law, but which only comes into its own in modern times. First, I shall
compare Hegel’s discussion of personhood in the Phenomenology of Spirit with that in
the first section of the Philosophy of Right, “Abstract Right.” While initially the latter
appears far more positive than the former, the differences are traceable to different
manners of treatment, rather than different evaluations. Secondly, I shall look at the role
personhood plays in Hegel’s presentation of the concrete forms of modern society in the
third, and longest, section of the Philosophy of Right, “Ethical Life.” Hegel confines the
applicability of personhood to only one sphere of our common or ethical world, namely
“civil society,” by which Hegel primarily means the economy, the world of property and
contracts. For Hegel, “civil society” is, crucially, bounded (below and above, as it were)
by two other domains of our common existence, the family and the state, which are
constituted by principles qualitatively different from, and inherently more “substantial”
than, personhood. I shall conclude with a few thoughts about how successful Hegel’s
project is, and what its relevance might be to our world.
The origin and the ground of Personhood—from the
Phenomenology to the Philosophy of Right
In the Phenomenology, Personhood first makes its appearance in a sub-section
entitled “Legal Status,” Rechtszustand, which comes at the end of the first major section
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�of the Spirit chapter (chapter VI, part A), “True Spirit, the Ethical World” (Die Wahre
Geist, die Sittlichkeit). This first section is largely devoted to ancient Greece because, as
I have said, the polis is Hegel’s model for what Spirit is, namely a social world whose
parts form a whole, and to which the individual has a substantial relation. At the same
time, as the motion of World History shows, Spirit is not static. “Legal Status” (VI.A. c.)
marks the transition from the world of the Greek polis to that of Roman law, from
citizens to subjects administered by a bureaucracy. For Hegel, this transition is not
simply due to the contingent facts of Rome’s military conquest of Greece, or of the scale
of Empire as opposed to that of the Greek city. Rather, the Greek world’s passing is due
to, as it were, natural causes. There is an internal tension in the Greek spirit, a tension
whose playing out leads to its evisceration—a crisis to which, in Hegel’s treatment, the
world of Roman Law is somehow, if unintentionally, responsive, the next phenomenon
in the Phenomenology.
As our seniors know, the center of Hegel’s treatment of the Ethical World in the
Phenomenology is his discussion of Sophocles’ Antigone. I only have time for a very
superficial sketch here. Hegel begins (¶446) by noting that “The simple substance of
Spirit, as consciousness, is divided,” namely between human law and divine law. On the
one hand, there is law as made in the public realm by citizens or rulers, as something
explicit, mediated, and subject to change. On the other hand, there is divine law, which
centers on the family and burial rites, and is something known implicitly and
immediately as unchanging. In the Greek world the human law is a masculine principle,
the divine law feminine. The divine and the human law exist harmoniously until
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�Antigone acts: she buries, against Creon’s proclamation, her brother Polyneices. To
Creon’s question “And so you dared to transgress these laws,?” she says
Yes, for it was not Zeus who proclaimed these things to me,
Nor was it She, Justice, who dwells with the gods below,
Who defined these laws for human beings;
Nor did I think that such strength was in your
Proclamations, you being mortal, as to be able to
Prevail over the unwritten and steadfast lawful conventions of the gods!
For not as something contemporary or of yesterday, but as everlasting
Do these live, and no one knows from where they appeared.
(Antigone lines 449-457)
6
For Hegel, the play is a tragedy because neither Creon or Antigone are “right.” In a way,
both are, because human and divine law are integral parts of the Greek Spirit, in fact, in a
certain sense, of Spirit generally. In another way, both perspectives are wrong because
incomplete: Hegel notes (¶472) that the play’s dénouement shows
The movement of the ethical powers against each other and of the individualities
calling them into life and action have attained their true end only in so far as both
sides suffer the same destruction.
Creon asserts the superiority of the city over the family, only to lose his son and heir;
Antigone asserts the family’s superiority only to die, as her name suggests, childless.
Sophocles’ play thus represents a moment of growing self-awareness within the Greek
world that there is a problem with its fundamental principles.
In the last paragraph before “Legal Status,” ¶476, Hegel speaks of “this ruin of
the ethical substance,” and concludes with a sentence that suggests why Rome is the
“next” phenomenon after the Greek Spirit and its crisis:
The substance emerges as a formal universality in them, no longer dwelling in
them as a living Spirit; instead, their simple unadulterated individuality has been
shattered into a plurality of multiple points.
Cited from the translation of Peter Ahrensdorf and Thomas Pangle in The Theban Plays (Ithaca,
NY: Cornell, 2014).
6
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�The key phrase is “formal universality.” Are we most fundamentally citizens, or
family members? The formal universality of law sidesteps this question altogether: as
“persons,” we are all subjects of the law, no matter how we “identify” as individuals.
The Latin etymology of “Person,” which derives from the word for an actor’s mask, is
revealing: in the world of legal status, what matters is not the substance of who we are,
what is under the mask, but our “role,” our rights and obligations under a system of rules.
Hegel’s judgment of this new Roman world is harsh. He begins his discussion of
“Legal Status” this way (¶477):
The universal unity into which the living immediate unity of individuality and
substance withdraws is the soulless (geistlose) community which has ceased to be
the substance—itself unconscious—of individuals, and in which they now have
the value of selves and substances, each possessing a separate being-for-self. The
universal is split into the atoms of absolutely multiple individuals; this lifeless
(gestorbene, having died) Spirit is an equality, in which all count the same, i.e. as
persons (Personen).
This world is “soulless” and “lifeless” because its rules seem artificial and alien
impositions, a taste of which we all get around April 15, courtesy of the IRS. (Hegel may
have more than Rome in his sights here.) It’s this soulless world that produced, as a
response, Christian interiority, i.e. the notion that we all have souls, a “depth” that
transcends any civic or family identity. Hegel explicitly links the discussion of “Legal
Status” to his earlier discussion, in the chapter on Self-Consciousness (IV), of the
movement from Stoicism to Skepticism to the Unhappy Consciousness. The world of
Roman law is profoundly unsatisfying; legal personhood (¶478) “is an abstract
universality because its content is the rigid unyielding self, not the self that is dissolved in
substance.” At the same time, this world is, in its way, an “advance” over Greece (¶479):
Personhood (Persönlichkeit), then, has stepped out of the life of the ethical
substance. It is the independence of consciousness, an independence which has
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�actual validity (die wirklich geltende Selbstständigkeit). The non-actual thought of
it which came from renouncing the actual world appeared earlier as the Stoical
self-consciousness.
It’s one thing to be independent or free in thought, as the Stoic asserts; with Roman law
come the beginnings of an actual or embodied independence, one we will come to know
as rights. As we shall see, however, for Hegel our status as “bearers of rights” is hardly
the full story of what we are.
One aspect of Hegel’s discussion in the Phenomenology which reappears in the
Philosophy of Right is that personhood is fundamentally an “economic” understanding of
the self. It is worth quoting ¶480 of the PhG at some length:
For what counts as absolute, essential being is self-consciousness as the sheer
empty unit of the person…This empty unit of the person is, therefore, in its reality
a contingent existence, and essentially a process that comes to no lasting result.
Like Skepticism, the formalism of legal right is thus by its very nature without a
peculiar content of its own; it finds before it a manifold existence in the form of
‘possession’ and, as Skepticism did, stamps it with the same abstract universality,
whereby it is called ‘property.’…The actual content or the specific character of
what is mine—whether it be an external possession, or also the inner riches or
poverty of spirit and character—is not contained in this empty form, and does not
concern it.
Persons, qua persons, assert “This is mine” about an indifferent, purely “personal” “this.”
Your possessions may show the worst possible taste, but they are still yours. The law, by
protecting property, makes the assertion of “mine”—unlike the illusory because selfcontradictory claim of the Skeptic that all truth is relative, just “my truth”-- “recognized
and actual” (¶480). From Hegel’s description, one is tempted to call personhood an
essentially Hobbesian view of the self, both because of the undetermined nature of the
good that bare “persons” pursue, and because its pursuit being “without lasting result”
recalls the “restless search for power after power ending only in death” that Hobbes says
characterizes the natural state of selves trying to preserve themselves, namely war. As
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�we shall see, in the Philosophy of Right, Hegel calls “civil society” the aspect of
difference, i.e. of opposition, in the modern Ethical world-- the jostling, competing selves
of the economy are the Hobbesian aspect of society, the domesticated version of his state
of war.
In the Philosophy of Right, Hegel treats personhood and the larger question of
“Right” in a different mode: not phenomenologically and historically, but conceptually or
“logically.” The first sentence of the “Introduction” reads “The subject-matter of the
philosophical science of right is the Idea of right—the concept of right and its
actualization.” (§1). Through the book’s three main sections—“Abstract Right,”
“Morality,” and “Ethical Life” (Sittlichkeit, from Sitte, customs or habits), the
development of the concept moves from the most abstract or immediate notion of right to
the most concrete and reflective, right as embodied in an actual social world. [Another
way of seeing the book’s three main divisions: a movement from Locke to Kant to
Aristotle]. Following in the path of the distinction between nature and freedom opened
up by Rousseau, and deepened by Kant, “right” for Hegel is not found in nature but is the
result of freedom, of being asserted and developed by human beings. For Hegel,
however, freedom is not just an “ideal,” the will of the Kantian moral agent who acts out
of pure principle or duty rather than natural or physical motives such as desire or fear.
Rather, freedom is something actual, embodied in the various historical worlds produced
by Spirit. Hegel says (§4):
The basis of right is the realm of spirit in general and its precise location and point
of departure is the will; the will is free, so that freedom constitutes its substance
and destiny, and the system of right is the realm of actualized freedom, the world
of spirit produced from within itself as a second nature.
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�In this account, personhood comes at the very beginning, “right’s” first word, as it
were. In the first sentence of the last section of the Introduction (§33), Hegel says:
In accordance with the stages in the development of the Idea of the will which is
free in and for itself, the will is [to begin with] immediate; its concept is therefore
abstract, as that of personality, and its existence is an immediate external thing;
the sphere of abstract or formal right.
What this means, I hope, will become clearer from what follows. Hegel begins the next
section, “Abstract Right” (§34), with the following doozy:
The will which is free in and for itself, as it is in its abstract concept, is in the
determinate condition of immediacy. Accordingly, in contrast with reality, it is its
own negative actuality, whose reference to itself is purely abstract.
Why “negative,” you ask? Take the barest assertion of the self, of the will: “I want an
apple.” By not just wanting an apple, as any other animal might, but saying “I” want it, I
distinguish the “I” from my desires. I am not my desire for the apple, or any other desire
I might have. Freedom begins with an act of selfhood that is a negation or distinction,
namely the distinction between the self and its empirical contents. This bare self, as
“abstract” or contentless, is the same for you as it is for me (and in asserting it, I know it
to be so): it is universal. Thus Hegel can say (§35):
The universality of this will which is free for itself is formal universality, i.e. the
will’s self-consciousness (but otherwise contentless) and simple reference to itself
in its individuality; to this extent, the subject is a person. It is inherent in
personality that, as this person, I am completely determined in all respects (in my
inner arbitrary will, drive, and desire, as well as in my relation to my immediate
external existence), and that I am finite, yet totally pure self-reference, and thus
know myself in my finitude as infinite, universal, and free…Personality begins
only at that point where the subject has...a consciousness of itself as a completely
abstract ‘I’ in which all concrete limitation and validity are negated and
invalidated.
In the “Addition” to §35, the point is made more simply:
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�The person is essentially different from the subject, for the subject is only the
possibility of personality, since any living thing whatever is a subject. A person is
therefore a subject which is aware of this subjectivity.
With this self-awareness, by saying “I,” human beings, as persons, even surpass nature.
The addition continues:
As this person, I know myself as free in myself, and I can abstract from
everything…And yet as this person, I am something wholly determinate.
Personality is thus at the same time the sublime and the wholly ordinary; it
contains this unity of the infinite and the utterly finite, of the determinate
boundary and the completely unbounded. The supreme achievement of the person
is to support this contradiction, which nothing in the natural realm contains or
could endure.
To sum up: in §35, Hegel explicates personhood as subjectivity’s reflection,
turning back upon, itself. What does this reflexivity have to do with “right”? What
Hegel says in §36 seems a leap:
Personality contains in general the capacity for right and constitutes the concept
and the (itself abstact) basis of abstract and formal right. The commandment of
right is therefore: be a person and respect others as persons.
It is hard to see how we get to a notion of right, to a commandment, unless we start from
a notion of personhood as legal status, yet this is precisely what Hegel has not done here.
7
A simple thought experiment might help. When I go into my yard, my neighbor’s dog
barks. One might say, he’s defending his yard. The critical point, though, is that he is not
saying it. If he could, he’d be a person. And, as soon as a hypothetical talking dog said
“This is my yard,” in effect saying “I”, he would also have to admit that I am also an “I.”
In other words, in asserting “mine” he has to admit “yours,” as in “That is your yard,”
and stop barking. (Of course, if he could talk, he might dispute exactly where the
See Friedrike Schick, “The concept of the person in Hegel’s Philosophy of Right,” Rev. Fac. Direito
UFMG, Belo Horizonte, n. 66, pp. 177 - 200, jan./jun. 2015. Found online (October 2018) at
https://www.direito.ufmg.br/revista/index.php/revista/article/download/1685/1601.
7
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�property line is, but that is beside the point). To my great frustration, I cannot explain this
to him. Dogs—despite what some people imagine—are not people.
In other words, in the reflexive act of distinguishing myself from my desire—“I
want an apple”—I am asserting an ownership of that desire: it’s my desire. From there,
it’s a very short step to say—it’s my apple—especially if I worked to get it, or as Locke
would say, mixed my labor with it. For Hegel, it’s only really my apple when I sell it—
i.e. when someone else recognizes it as mine by paying for it. The “commandment” of
§36—“be a person and respect others as persons” sounds like a Kantian moral
imperative, but it isn’t. The form of “right” that goes with personhood is purely formal: it
isa bare assertion of rights which contains no positive content or duty beyond respecting
the rights of others. It’s my apple—whether I eat it, or smash it in the street, is up to
me—my right to it leaves me free for various possibilities, free from you. Thus Hegel
says (§38):
With reference to concrete action and to moral and ethical relations, abstract right
is only a possibility as compared with rest of their content, and the determination
of right is therefore only a permission or warrant. For the same reason of its
abstractness, the necessity of this right is limited to the negative—not to violate
personality and what ensues from personality.
The “negative” character of abstract right makes its subject matter clear: property. Don’t
touch! Property is the way that persons—the most elementary form of selfhood—make
their selfhood objective or real. Hegel makes this clear (§40):
Right is primarily that immediate existence freedom gives itself in an immediate
way, (a) as possession, which is property…(b)A person, in distinguishing himself
from himself, relates himself to another person, and indeed it is only as owners of
property that the two have existence for each other. Their identity in themselves
acquires existence through the transference of the property of the one to other by
common will and with due respect to the rights of both—that is, by contract.
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�When you hear the phrase “It’s ok, they are two consenting adults,” you know you are on
the terrain of personhood.
Thus, while starting from a different point, and proceeding in a different manner
than he did in the Phenomenology, Hegel, in his presentation of personhood in the
Philosophy of Right, arrives at the same place: self-atoms asserting their rights. At the
same time, the practice of slavery shows that the Roman version of personhood is, from
the larger perspective of the concept, defective (§40): “But as for what is called the right
of persons in Roman law, it regards a human being as a person only if he enjoys a certain
status…hence in Roman law even personality itself, as opposed to slavery, is merely an
estate [Stand] or condition [Zustand].” Personhood is, as such, universal. Moreover, by
making only the male head of the family a “person,” and by making his authority over
them a matter of personal rights or ownership, Roman law also distorted the family:
…the content of the right of the so-called right of persons in Roman law is
concerned with family relationships….The right of persons in Roman law is
therefore not the right of the person as such, but no more than the right of the
particular person; it will later be shown that the substantial basis of family
relationships is rather the surrender of personality. (§40)
These things noted, it is still true that there is a profound connection between the
discussion of personhood in the Phenomenology and in the Philosophy of Right. In the
former, personhood appears as a soulless and abstract world, with the death of the Greek
Ethical life; in the latter, personhood, if in a more perfect form, is still in its inherent
abstraction insufficiently substantial to constitute the whole of the modern form of
Sittlichkeit. It is to that which we now turn.
The boundaries of personhood in modern Sittlichkeit
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�For Hegel, right—to be actual, rather than a mere form or ideal—must be
embodied in institutions and practices: “Ethical substance…is the actual spirit of a family
and a people” (PR §156). Furthermore, conceptually, or when thought through,
Sittlichkeit has as three different aspects or, in Hegel’s terminology, “moments”:
A. immediate or natural ethical spirit—the family. This substantiality passes over
into loss of unity, division, and the point of view of relativity, and is thus
B. civil society, i.e. an association of members as self-sufficient individuals in
what is therefore a formal universality, occasioned by their needs and by the legal
constitution as a means of security for persons and property, and by an external
order for their particular and common interests. This external state
C. withdraws and comes to a focus in the end and actuality of the substantial
universal and of the public life which is dedicated to this—i.e. in the constitution
of the state. (PR §157).
The important point here is that, in the ethical world, “civil society” is bounded—both
“below” and “above” as it were—by two other realms which have different principles,
and are inherently more substantial. “Personhood” doesn’t fit neatly into any of the
three, but it seems closest to the “formal universality” of civil society that protects private
property. (This is complicated because Hegel also includes in “civil society” what we
would call the “welfare state,” the government’s moderation of the effects of unfettered
“capitalism” or the pursuit of self-interest). The affinity between the universal notion of
the “person” and economics explains a curious remark of Hegel’s: he says that it is only
in the context of the “system of needs,” the economy, that he will refer to “human
beings.” (PR §190). The implication, I think, is that the family and the state are more
“concrete” realms than civil society. This is particularly true of the family, which is not
made up of mere “human beings” but of husbands and wives, mothers and fathers,
parents and children, brothers and sisters. It is true of the state is well, which is composed
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�in one way of the various social classes (farmers, craft and tradesmen, bureaucrats) and in
another way of citizens and rulers.
For Hegel, ethical life “begins,” not historically but conceptually, with the family,
because its connections seem immediate or “just there,” facts. Immediacy is inherently an
aspect of consciousness, as it always, in the first instance, confronts an “object,”
something “just there.” Immediacy is thus also, as Hegel’s treatment shows, an inherent
aspect or “moment” of ethicality. Hegel says in the Phenomenology (¶460) that the
community’s human law “possesses” “in the divine law its power and authentication:”
by “divine” Hegel means here what Antigone means, law as not made by somebody but
“just there,” a fact, immediate. We have the sense that it is wrong to break a contract,
more so a law, but for Hegel these senses of “wrongness” rest on a more primordial sense
of right and wrong, one that is immediate and unreflective, even unconscious, that lives
in the family. Upon reflection, this immediacy may prove illusory, the family’s form
depending on civic customs such as the incest taboo, spousal monogamy, etc. (This
dependence is what Antigone cannot see). However, one might say that the family’s
“ethicality” is one that is experienced as immediacy, and this is connected, paradoxically,
to the fact that the family emerges from something that for Hegel is not, strictly speaking,
“ethical” at all, feeling (PR §158):
The family, as the immediate substantiality of spirit, has its determination in the
spirit’s feeling of its own unity, which is love. Thus, the disposition [appropriate
to the family] is to have self-consciousness of one’s individuality within this unity
as essentiality which has being in and for oneself, so that one is present in it not as
an independent person but as a member.
In the “Addition” to §158, Hegel usefully explains:
Love means in general the consciousness of my unity with another, so that I am
not isolated on my own, but gain my self-consciousness only through the
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�renunciation of my independent existence and through knowing myself as the
unity of myself with another and of the other with me… Love is therefore the
most intense contradiction; the understanding cannot resolve it…[it] is both the
production and the resolution of this contradiction. As its resolution, it is ethical
unity.
Now, this all sounds very exalted—what could be more “ethical” than knowing
myself as the unity of myself with another? Indeed, such knowledge is the very task of
Spirit, the implicit telos of consciousness itself in its desire to make the “object” its own
by grasping it, com-prehending. Yet, Hegel also says here that “love is a feeling, that is,
ethical life in its natural form, ” which means that strictly speaking, love can only be an
image or prefiguration of the ethical. As he makes clear both here and in the
Phenomenology, in the family love is inherently bound up with nature, such as the
parents’ sexual desires for each other, and their affection for their children. Only
Spirit—freedom made actual—is ethical, not nature (PhG ¶451):
However, although the Family is immediately determined as an ethical being, it is
within itself an ethical entity only in so far as it is not the natural relation of its
members, or in so far as their connection is the immediate connection of separate
actual individuals…it is only as a spiritual entity that it is ethical.
In his indispensable commentary on the Phenomenology, Peter Kalkavage gives a
simple example of this dictum: “As a natural father, I am fond of my children. As their
ethical father, I must see that they are properly cared for and educated.”
8
Now, this
might make it seem that Hegel is taking a Kantian position, i.e. that the duty to one’s
children is ethical only in so far as it is motivated purely by the idea of duty as opposed to
feeling. Yet—and this is a long story—Hegel’s entire practical philosophy is an extended
critique of this Kantian view. Put simply, that critique has two related strands: the test of
the categorical imperative is too “formal” to generate any specific content, and all action,
8
The Logic of Desire (Philadelphia: Paul Dry Books, 2007), p. 242.
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�for Hegel, has an element of passion (see especially PhG ¶622). Implicitly referring to
Kant, in PR §124 Hegel says “it is an empty assertion of the abstract understanding…to
take the view that, in volition, objective and subjective ends are mutually exclusive.” In
this same section, as well as in the Phenomenology (which Hegel cites here in PR), and
in his Lectures on the Philosophy of World History, Hegel is critical of those who,
having the “morality of valets,” (as in the saying, “no man is a hero to his valet de
chambre”) would deprecate the actions of great men as unheroic because impure,
containing an element of passion or interest.
9
Like Aristotle, Hegel maintains that feelings can be shaped or habituated by
custom or habits (Sitte), but not simply transcended. As Robert Pippin has pointed out,
unlike for Kant, for Hegel spirit or freedom is not a different order of causality than
nature, but rather a mode of self-relation, via a purposive taking up of, a reflective stance
towards, nature. Pace Kant, for Hegel practical reason is inherently impure, which is
10
why in developing the concept of Right, the Philosophy of Right moves from “Morality”
to “Ethical Life.” For Hegel, feeling is as it were only the “matter” of action or our
connections with each other; ethicality or spirit gives this matter form or principle. This
PhG ¶665 with Lectures on the Philosophy of World History, pp. 31-2.
Robert Pippin, “Naturalness and Mindedness,” European Journal of Philosophy 7:2 (1999), pp.
194-212. See esp. p. 207: “Kant’s dualism may not be metaphysical but it is strict; the realms of nature and
spirit are either/or, never both/and, while for Hegel, spirit is, as we shall see, a kind of achievement which
some natural beings are capable of, and so there can be a continuity between natural and spiritual
dimensions.” Pippin points out (pp. 198-9) that, for Hegel, a proto-form of Spirit is present in other
animals: “In the Encyclopaedia passages that describe the ‘transition’ [between nature and spirit], Hegel’s
position is that some sentient creatures do not merely embody their natures, in the way a stone or planet or
an insect might be said simply to be what it is. Some come to be in some sort of relation to their
immediately felt or experienced dispositions, sensations and inclinations…Such creatures do not, say, just
register threatening stimuli; they experience what is taken to be a threat, take up the threat ‘in a way’,
fearfully, feelingly. Feelings thus in Hegel’s language are said to be ‘modes of negativity’ – or nonidentity: a mode of self-relation within an experience, not merely (although certainly also) being in a
state…Soul [as in Aristotle, the principle of all animal life] is said by Hegel to be what it is in its
‘sublation’ of (cancelling the independence of while yet preserving) nature, not in ‘being’ other than
nature.”
9
10
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�formation of nature through custom and habit, Hegel calls culture, Bildung, from Bild,
image or form, similar to the Greek eidos (see PR §20 with §151, 187).
11
Locutions such as “ethical immediacy” or “ethical life in its natural form” might
seem merely self-contradictory, but they are not, or rather they are more than that: they
are paradoxical. As Rousseau answered those who accused him of self-contradiction, the
contraction is “not in me, but in things.” [cite?] For Hegel, the ‘understanding’ (Verstand)
may balk, but ‘reason’ (Vernunft) or “speculation” can grasp unity within contradiction,
identity within difference, paradox.
The paradoxical role of “nature” in ethical immediacy—as both origin and
something surpassed—is evident in Hegel’s discussion of marriage. He begins thus (PR
§161):
Marriage, as the immediate ethical relationship, contains first the moment of
natural vitality; and since it is a substantial relationship, this involves life in its
totality, namely as the actuality of the species [Gattung] and its process.
The translator, H. Nisbet, notes here “In this context of marriage and the family, the word
Gattung (genus, species) carries with it strong overtones of the closely related word
From Hegel’s discussion of Antigone in the Phenomenology, however, one might have the
impression that his view shades ultimately into Kant’s. Since “the ethical is intrinsically universal,” (¶451),
the only ethicality that belongs solely to the family is the duty of burial, a duty towards all of its members.
Burial is a “spiritual” act because it asserts and preserves an individuality by protecting the body from the
ravages of nature, i.e. from becoming mere indifferent food for birds and dogs, “at the mercy of every
lower irrational individuality.” (¶452). Through this act, the individual becomes, in the family’s memory,
a daimon, a protective spirit. Hegel says “this last duty thus constitutes the perfect divine law…every other
relationship…which does not remain simply one of love but is ethical belongs to human law…” (¶453).
For example, the duty to provide for one’s wife and children is enforced by human, civic, ordinance. For
her opposing a spiritual act, burial, to nature, and because, allegedly, siblings “do not desire each other,”
Antigone is for Hegel the paradigm of the family’s ethicality. However, despite her love for Polyneices not
being tinged with “desire,” she is not a Kantian moral agent: she does not act from the dictates of “pure
practical reason.” On the contrary: “the feminine, in the form of the sister, has the highest intuitive
awareness of what is ethical.” (¶457; Hegel’s italics). Moreover, she doesn’t risk death by insisting that all
of the unburied, other women’s brothers, be buried—she asserts no duty to them. She buries her brother,
whose individual loss is “irreparable” (¶473; Hegel cites here Antigone ln. 910). She is very much in the
realm of immediacy, of feeling, of particularity—although one might also say that with her, and in the
divine law, that realm—the family—surpasses itself (the etymology of her name—“against generation”—
suggests as much). Indeed Spirit is self-surpassing, the self-surpassing of nature by culture’s formative
power.
11
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�Begattung (mating, copulation).” Simply put: but for the “facts of life,” i.e. the fact that
human beings reproduce sexually, they never would have come up with the ethical form
that is marriage. That’s not the end of the story, however. §161 continues:
But, secondly, in self-consciousness, the union of the natural sexes, which was
merely inward (or had being in itself) and whose existence was for this reason
merely external, is transformed into a spiritual union, into self-conscious love.
This looks like, but is not, romanticism. Hegel merely argues that, once humans no longer
just have desires, but become selves aware of having desires because aware of other
selves, the nature and object of desire changes. Human lovers look into each other’s
eyes; unlike animals, they desire to be desired. Hegel here follows Rousseau: the
acquisition of self-consciousness means that the road back to simple animality is forever
blocked. Viewed from the perspective of the self, desire as purely animal is “merely
external.”
What Hegel calls the “immediate concept” (§160) of the family’s ethicality,
marriage, seems to emerge from a space somewhere between desire and contract (§162):
The subjective origin of marriage may lie to a greater extent in the particular
inclination of the two persons who enter this relationship, or in the foresight and
initiative of parents, etc. But its objective origin is the free consent of the persons
concerned, and in particular their consent to constitute a single person and to give
up their individual personalities within this union. In this respect, their union is a
self-limitation, but since they attain their substantial self-consciousness within it,
it is in fact their liberation.
Three points bear fleshing out. First, the contrast here between subjective and objective
recalls that between subject and substance; Hegel says here that as an “objective
determination,” marriage is an “ethical duty, ” but that aspect coexists with its subjective
aspect. More pointedly, Hegel makes it clear that “ethical” or spiritual does not mean
separate from the sexual; those who understand these as separate are limited by a
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�“monastic attitude,” asserting “what is falsely called Platonic love” “which mistakenly
views the moment of natural life [Lebendigkeit, lit. livingness] as “utterly negative.”
(§163). Marriage is the result of this combination of subjective and objective, a (§163)
“consciousness of this union as a substantial end, and hence in love, trust, and the sharing
of the whole of individual existence.”
Secondly, the contrast between the subjective and objective side of marriage is,
as it were, recapitulated within the subjective or “external” side, where, it seems, Hegel
tries to find a middle. At one “extreme,” that of a low “level of development [Bildung] of
reflective thought,” marriages were arranged. But, “At the other extreme, it is the mutual
inclination of the two persons.” In the “Addition” to §162, this contrast of “extremes” is
sharpened:
Among those peoples who hold the female sex in little respect, the parents arrange
marriages arbitrarily, without consulting the individuals concerned; the latter
accept this arrangement, since the particularity of feeling [Empfindung] makes no
claims for itself as yet…In modern times, on the other hand, the subjective origin
[of marriage], the “state of being in love” is regarded as the only important factor.
Here, it is imagined that each must wait until his hour has struck, and that one can
only give one’s love only to a specific individual.
Hegel’s tone here shows his typical ambivalence about “modern times,” as does this
remark in the body of §162, which highlights Hegel’s concern about the problematic
status of Sittlichkeit in modernity:
But in those modern dramas and other artistic presentations in which love
between the sexes is the basic interest, we encounter a pervasive element of
frostiness which is brought into the heat of passion such works portray by the total
contingency associated with it. For the whole interest is represented as resting
solely upon these particular individuals. This may well be of infinite importance
for them, but it is of no such importance in itself.
What does Hegel mean by “contingency”? Harry might never have met Sally.
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�While not an advocate of arranged marriage, Hegel seems worried that the
pendulum has swung too far in the other direction. This concern about the contemporary
primacy given to love and feeling shows up in his remarks about divorce in the Addition
to §163:
Since marriage contains the moment of feeling [Empfindung], it is not absolute
but unstable, and it has within it the possibility of dissolution. But all legislations
must make such dissolution as difficult as possible and uphold the right of ethics
against caprice.
For Hegel, divorce should be hard; to dissolve a union that is Sittlich, ethical, the fact that
“we are no longer in love” should not suffice (and until recently, it didn’t). The
“objective” side of marriage is also the reason it is traditionally held in a church, but this
aspect is far older than Christianity (§163):
…viewed in a shape appropriate to representational thought, this spirit has been
venerated as the Penates etc.; and in general it is in this spirit that the religious
character of marriage and the family, i.e. piety, is embodied. [The translator’s
endnote, p. 438, informs us that “In Roman religion, the penates were the spirits
of the cupboard (penus); together with the lares (spirits of the hearth), they were
worshiped as guardians of the house.”]
We are now mostly unsure about why marriage is held in a church, and often use words
such as “commitment” when speaking its basis, both signs of the advance of
Persönlichkeit.
Third, the phrase “free consent of the persons involved” makes it sound like
Hegel views marriage is a contract—and it is, but of a most peculiar kind, one that
surpasses itself because it is substantial (§162):
For the precise nature of marriage is to begin from the point of view of contract—
i.e. that of individual personality as a self-sufficient unit—in order to supersede it
[ihn aufzuheben]. That identification of personalities whereby the family is a
single person and its members are its accidents...is the ethical spirit.
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�“Accidents,” since Aristotle, are the qualities which inhere in substances; Hegel cites here
the definition of “substance” in his Encyclopedia. Criticizing a more purely contractual
understanding of marriage, Hegel calls Kant out by name (§161, Addition):
Formerly…[marriage] was considered only in its physical aspect or natural
character. It was accordingly regarded only as a sexual relationship, and its other
determinations remained completely inaccessible. But it is equally crude to
interpret marriage merely as a civil contract, a notion [Vorstellung] which is still
to be found even in Kant…and is thus debased to a contract entitling the parties
concerned to use another.
By “crude,” Hegel refers to Kant’s notorious view that marriage is a contract giving the
parties the use and enjoyment of each other’s sexual organs. Sentimentalism, however, is
not an adequate response. Hegel goes on:
A third and equally unacceptable notion is that which equates marriage with love;
for love, as a feeling, is open in all respects to contingency, and this is a shape
which the ethical may not assume.
For Hegel, one key (and now controversial) aspect of marriage that makes it
substantial—more than a contract between a party of the first part, and a party of the
second part—is that it is, as a whole rather than a mere aggregate, internally
differentiated. Marriage is between two people with essentially different, and
complementary, characteristics: namely a man and a woman. To be fully actual, their
substantial unity depends, for Hegel, on a third thing: the issue of a child (§173,
addition). However, for Hegel, what we now call “gender” is far from being a simple
natural fact: it is part of spirit. As we saw in the section on “True Spirit” in the
Phenomenology:
“simple substance…equally exhibits in its own self the nature of consciousness,
that is, to create distinctions within itself…It thus splits up into distinct ethical
substances, into a human and divine law. Similarly, the self-consciousness
confronting the substance assigns to itself according to its essence, wesen, (Miller
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�has “nature”), one of these powers-- teilt sich nach seinem Wesen der einen dieser
Mächte zu. (¶445).
What does this last obscure sentence mean? What is this assigning (zuteilen), and what is
its basis, “according to its essence”? The subsequent discussion makes some of this a
little more clear: the assignment is gender. When the young man leaves the family, the
realm of ethical immediacy, for the public or political world (¶459):
He passes from the divine law, in whose sphere he had lived, over to the human
law. But the sister becomes, or the wife remains, the head of the household and
the guardian of the divine law. In this way, the two sexes overcome their [merely]
natural being and appear in their ethical significance, as diverse beings who share
between them the two distinctions belonging to the ethical substance. These two
universal beings of the ethical world have, therefore, their specific individuality in
naturally distinct self-consciousnesses…the ethical Spirit is the immediate unity
of the substance with self-consciousness—an immediacy which appears,
therefore, both from the side of reality, and of difference, as the existence of a
natural difference.
Contemporary commentators tend to seize on the “appears” (erscheinen) in that
last sentence, and argue that what Hegel means by “nature” and “natural difference” are
simply what the individuals in a particular historical-spiritual world take to be such—it is
their mistaken view that their relations are based on some immediate available truth. This
view, while having some truth, and consistent with contemporary sensibilities, sidesteps
too neatly a very vexed question: how Hegel understands the status of nature.
12
Erscheinen, appearance, is not scheinen, mere deceptive seeming; the phenomena of the
Phenomenology are appearances (from φαινω, appear), one sided to be sure, of the truth.
Admittedly, gender, appearing immediately in spirit as an essential or simple natural
difference, oversimplifies: “This moment loses both the indeterminateness which it still
had there, and the contingent diversity of aptitudes and capacities.” (¶459). Still, retail is
See the critical remarks about Robert Pippin’s claim (op. cit., p. 204) that in Hegel there is “no
missing ontology” [of nature] in Raoni Padui, From the Transcendental to the Ontological: Hegel,
Heidegger, and the Legacy of Transcendental Idealism (Ann Arbor, MI: UMI, 2012), pp. 224-5.
12
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�one thing, wholesale another; if society makes men and women more different than they
are by nature, where there is a “contingent diversity of aptitudes and capacities,” for
Hegel they are still by nature, on the whole, different. While “nature” appears
differently in different historical-spiritual worlds, these appearances are all appearances
of something. Times change, but nature persists as the “matter” somehow conditioning
the forms Spirit takes in the practical world.
13
Thus, for Hegel, it seems that nature or sex has as it were a continuing
gravitational pull on the differences that Spirit posits within itself, namely gender. In
speaking of gender as “assigned,” Hegel is the ancestor of contemporary feminism; in
maintaining the mysterious persistence of nature, he is its public enemy number one. The
persistence of nature is implicit in how closely the conceptual treatment of the family in
This claim, about the persistence of nature in conditioning spirit, admittedly, seems to contradict
the central claim of Hegel’s Science of Logic, that the Concept, as self-grounding, gets its content entirely
from itself. See the discussion in Padui, op. cit., pp. 224-37, especially pp. 234: “The coherence of this
account relies on understanding the development of nature into the realm of freedom as nature’s own selfovercoming, and it is this aspect that is most difficult to fully comprehend, let alone accept. The selfdetermination present within the sphere of Geist can only be absolute if it can somehow ‘shed’ its
conditionality vis-à-vis its natural conditions, as Hegel himself saw extremely clearly.” The position I am
attributing to Hegel here is closer to what Padui identifies as Schelling’s critique of Hegel, wherein
Schelling insists, contra Hegel, on nature’s “pre-categorical” or “noumenal” reality (Padui, p. 237), as a
reality prior to Spirit. In the phrase quoted above, “the indeterminateness which it still had there” in PhG
¶459 refers, I would surmise, to nature in this pre-categorical sense.
However, it seems to me, that by referring to an “absolute” “self-determination within the sphere
of Geist,” Padui mistakes the relation between the Logic and the Phenomenology, between the eternal
movement of the Concept depicted in the former, and the necessarily time-bound nature of Spirit depicted
in the latter. Even though, as the Phenomenology attempts to show, that it is only in the “last” stage of
Spirit, that of Hegel’s own time, that philosophy can become sophia (see PhG, Preface, esp. ¶‘s 5, 19, 37),
and the philosopher achieve “absolute knowing” or “science,” still: absolute knowing is a thinking of
thinking which is also a thinking of being and is identical to what the Logic poetically and somewhat
misleadingly (if taken literally) calls “God as he is in his eternal essence, before the creation of nature and
of a finite spirit” –this thinking is not, strictly speaking, a human thinking or a form of Geist. See the
discussion in pp. 423-51 in Kalkavage, Logic of Desire, cited supra, as well as Jean Hyppolite, Genesis and
Structure of Hegel’s Phenomenology, translated by Samuel Cherniak and John Heckman (Evanston Illinois:
Northwestern U.P., 1974), pp. 581ff, esp. the quote from the Logic on p. 582. Thus, the “ethical” is not
part of the Concept’s self-motion: God’s thinking, the Logic, is beyond Good and Evil. As not embodied, it
is not conditioned by nature or the basis of an actual social world. Hegel thus preserves the PlatonicAristotelian priority of theory to practice. However, there is still a problem: if nature is the “other” of the
Concept, the divine Logos, then, in order to be truly “other,” nature would have to be not totally
comprehended by the Logos: contrary to Hegel’s explicit teaching: the real is not “absolutely” rational.
13
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�the Philosophy of Right tracks the more “historical” discussion of Antigone in the
Phenomenology. In the former, Hegel cites the latter explicitly (§166). The difference
between the husband and wife in the Philosophy of Right is not identical with the tragic
antithesis between Creon and Antigone. But: the modern difference is not totally different
from ancient difference: plus ça change, plus c’est la même chose. Here is how Hegel
sets out what are now called the “roles” of man and woman in the family (§166):
Man therefore has his actual substantial life in the state, in science, etc., and
otherwise in work and struggle with external world and with himself, so that it is
only through his division that he fights his way to self-sufficient unity with
himself. In the family, he has a peaceful intuition of this unity, and an emotive
and subjective ethical life. Woman, however, has her substantial vocation in the
family, and her ethical disposition consists in this [family] piety.
This description seems now both like ancient history and only yesterday; how much of
this, too, will pass is the question of the hour. Hegel in effect places the entire burden of
there even being a sphere of ethical immediacy distinct from civil society entirely upon
women. Without Penelope’s fidelity, Odysseus would have no home to which to return.
If the family provides a kind of “floor” underlying the impersonal relations of
“persons” that characterize civil society, the state—the third division of “Ethical Life—is
the ceiling, Sittlichkeit’s architechtonic principle and guiding end. What Hegel means
by the state is a huge topic; I limit myself to a few brief remarks. First, contra Locke, for
Hegel the state should not be understood as a contract made between individuals in a
“state of nature” for the securing of their rights against each other (§77; cf. §187, 194).
A “state of nature” does not and cannot exist because the individual is “already by nature
a citizen” (§77, addition). Put more precisely, “individuality” as a reality depends upon a
certain kind of political order, namely a modern one, in which state and society are
distinct (§182, addition):
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�Civil society is the [stage of] difference which intervenes between the family and
the state, even if its full development occurs later than that of the state; for as
difference it presupposes the state, which it must have before it as a self-sufficient
entity in order to subsist itself. Besides, the creation of civil society belongs to the
modern world, which for the first time allows all determinations of the Idea to
attain their rights. If the state is represented as a unity of different persons, as a
unity which is merely a community [of interests], this applies only to the
determination of civil society.
Secondly, the state’s higher unity, which encompasses the moments of immediate
unity (the family) and difference (civil society, the “economy”) is not, for Hegel, based
upon feeling, love, or any “family-like” notion such as common descent or ethnicity. The
state, for Hegel, has a rational basis. To be affirmed, it must be understood, which is why
Hegel thinks that most people will find themselves not in it, but in the family, their
profession, and/or in religion (see §§201, 255, 270). The constitution and the laws are the
self-conscious articulation and ground of what we in fact always are, members of a
whole, or spirit.
Thirdly, the state for Hegel is necessarily one of many states. On the question of
universality, there is a fundamental divide between theory and practice (§209):
It is part of education, of thinking as consciousness of the individual in the form
of universality, that I am apprehended as a universal person, in which [respect]
all are identical. A human being counts as such because he is a human being, not
because he is a Jew, Catholic, Protestant, German, Italian, etc. This
consciousness, which is the aim of thought, is of infinite importance, and it is
inadequate only if it adopts a fixed position—for example, as cosmopolitanism, in
opposition to the concrete life of the state.
Hegel rejects cosmopolitanism, the view that “I am a citizen of the world,” because the
state, as Right made actual, concrete, or embodied, contains logically an element of
particularity. But, Hegel goes further than that, maintaining that “war should not be
14
Strictly speaking, ‘individuality’. See §259, addition: “The state as actual is essentially an
individual state, and beyond that a particular state. Individuality should be distinguished from particularity;
14
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�regarded as an absolute evil and as a purely external contingency” (§324). This limited
praise of war forms part of his critique of classical liberal social contract theory (§324):
It is a grave miscalculation if the state, when it requires this sacrifice, is simply
equated with civil society, and if its ultimate end is seen merely as the security of
the life and property of individuals. For this security cannot be achieved by the
sacrifice of what is supposed to be secured.
In his remarks on war in the Philosophy of Right, Hegel develops a thought that he
had stated in the Phenomenology (¶475):
For the community is a nation, is itself an individuality, and essentially is only
such for itself by other individualities being for it, by excluding them from itself
and knowing itself to be independent of them. The negative side of the
community, suppressing the isolation of individuals within it, but spontaneously
active in an outward direction, finds its weapons in individuality. War is the Spirit
and the form in which the essential moment of the ethical substance, the absolute
freedom of the ethical self from every existential form, is present in its actual and
authentic existence.
War is certainly an assertion of a state’s “individuality,” and reigns in the atomizing
tendencies of civil society, but what does Hegel think is “ethical” about that? It turns out
that for Hegel what gives war an “ethical” aspect is the sacrifice of life and property
which war requires. These goods, as “finite” and “transient” are contingent, subject to
necessity, to nature. “But,” Hegel claims (§324), “in the ethical essence (i.e. the state)
nature is deprived of this power, and necessity is elevated to a work of freedom, to
something ethical in character.”
The analysis of war in PR §324 parallels his analysis of burial in the
Phenomenology, and culminates with a dig at Kant’s essay “Perpetual Peace”:
War is that condition in which the vanity of temporal things and temporal
goods—which tends at other times to be merely a pious phrase—takes on serious
significance, and it is accordingly the moment in which the ideality of the
particular attains its right and becomes actuality. The higher significance of war is
it is a moment within the very Idea of the State, whereas particularity belongs to history.” There is no
logical derivation of a particular national character, such as Frenchness.
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�that, through its agency (as I have put it on another occasion), ‘the ethical health
of nations is preserved in their indifference towards the permanence of finite
determinacies, just as the movement of the winds preserves the sea from that
stagnation which a lasting, not to say perpetual, peace would produce among
nations.’
In the “addition,” Kant is mentioned explicitly:
Kant proposed a league of sovereigns to settle disputes between states…But the
state is an individual, and negation is an essential moment of individuality. Thus,
even if a number of states join together as a family, this league, in its
individuality, must generate opposition, and create an enemy.
Here, Hegel also argues that in a condition of peace, the “particularities” of civil society
become “rigid and ossified.” War is in effect part of the “life cycle” of the body politic,
occasionally necessary for it to be a body at all: “the unity of the body is essential to the
health, and if its parts grow internally hard, the result is death.”
Conclusion: Where are we now?
In the animated montage of the opening credits of the TV series Madmen, an
office and its furnishings dissolve, and a man, in silhouette, falls between the concrete
canyons of Manhattan skyscrapers. This sense of groundlessness, of lack of substance, is
the theme of the show. Not only is the identity of the main protagonist, Donald Draper,
based on a lie; in almost all of the marriages depicted, infidelity and divorce are rampant,
as self-atoms fly off on their centrifugal paths to go bump in the night. The show is set in
the advertising world of the 1960’s, but its popularity suggests that it hits a contemporary
nerve. This current sense of a lack of substance is hardly confined to the United States.
One sees it, for example, in the works of the French novelist Michel Houellebeq, whose
“elementary particles” are, likewise, descendants of the same atomicity that Hegel found
in ancient Rome. Given how far we are from the vision of Sittlichkeit Hegel lays out in
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�the Philosophy of Right, it might very well seem that his project of reconciling
subjectivity and substance is a colossal failure.
It might, however, be a very instructive failure. In the past, Hegel was often
vilified by superficial commentators as a source of both extreme left and right, of
Marxism and fascism. Contemporary scholars now see Hegel as a man of the center, a
friendly corrector of liberalism, not its enemy. Yet, we seem to living in a time where, as
Yeats put it in “The Second Coming,” the “center cannot hold.” At the moment, the
politics of the United States and other western countries seems to be caught, once again,
in a gyre of ever-widening partisanship. A re-consideration of Hegel’s thought might
prove useful in identifying just who the two main parties are. In section 125 of the Gay
Science, Nietzsche announces, through the mad man, the death of God. What we in the
West are seeing today—and perhaps have been seeing ever since World War I-- is the
death of Hegel, where the two sides of the political world divide up the body between
them.
For those left of center, what is most congenial in Hegel are the regulatory and
welfare aspects of his state over civil society (PR §§240-245). Poverty, Hegel notes, is a
particularly intractable problem in modern society, the unfortunate result of the workings
of the economy, of individuals pursuing their interests; therefore, “public conditions
should be regarded as all the more perfect” (§242) to the degree that the alleviation of
poverty is not left to individual charity but becomes a matter of state concern. Where
today’s left parts company with Hegel are issues of family, gender, and marriage on the
one hand, and nationalism and war on the other. What unites these positions is a
rejection of the confines in which Hegel had placed “personhood.” Thus, changing a
J. Hand Hegel Person Lecture
Given September 6 2019
Page 32 of 38
�definition several thousand years old, marriage is no longer exclusively between a man
and a woman, but between two persons. Similarly, in recent protests against
immigration restrictions and enforcement, one hears the rubric “no human being is
illegal.” All are persons. Taken to its logical conclusion, such an idea would require the
elimination of the nation-state—a goal which, in its way, the E.U. has been pursing.
In politics, as in physics, Newton’s third law of motion holds: for every activist,
there is an equal and opposite reactionary. The contemporary right looks, unsurprisingly,
like a photographic negative of the left. Civil society, or the economy, is where many on
the right insist on the rights of abstract personhood (including those of fictitious or legal
persons, namely corporations): they are impatient with taxation, redistribution, and
regulation. As far as questions of the family, marriage, and gender, the situation with
respect to “personhood” is the opposite. True, few on the right would go as far as Hegel
does and relegate women to the private sphere. At the same time, these citizens—about
one half of the country-- are not exactly happy warriors in the battle against the enemy
named by their liberal brethren: “gender stereotypes,” along with its evil siblings
“hetero-normativity,” and “patriarchy.” In a mostly inchoate way, they believe or rather
feel that not all the old understandings and definitions of man and woman should be
junked; it’s as if, resisting the tide, they whisper to each other “Vive la différance!”
Expel nature with a pitchfork, they wink, it always returns.
To this persistent muttering and foot dragging, those on the other side of the fence
shout back that this allegedly essential dichotomy of man and woman is “socially
constructed,” insubstantial: it is an “ideology,” a smoke screen blown to mask and
hence uphold a “hierarchy.” All essentialism is, essentially, bad. On the issue of gender,
J. Hand Hegel Person Lecture
Given September 6 2019
Page 33 of 38
�the left would thus appear to be the party of fluidity, of Heraclitus’ panta rhe, everything
flows. Yet, even here, Parmenides—substance—makes an appearance. Those
undergoing “gender reassignment”—for whom liberals are supposed to be sympathetic-often elect onerous medical procedures to give themselves the body that corresponds,
whatever that might mean, to their gender “identity.” Gender in this case is not fluid or
“insubstantial” at all but a cold hard fact, one that is extremely personal—a fact that
demands, in some cases, corrective medical action. The antinomy between left and right
on the question of gender—fluidity versus fixity-- is thus contained, in different form,
within the left.
As Hegel often argues, each opposing partial view, as an abstraction, contains,
but does not resolve, both opposites. To the old feminist rallying cry “biology is not
destiny,” there appears to be an addendum: the destiny can be fulfilled only in so far as
the biology can be changed. Persons should be free to choose their gender, people say,
but this claim leaves the basis of freedom unclear: are we free because as persons we are
indeterminate, capable of being whatever we choose, or are we free because, as fully
determined, we have to express what we are? For Hegel, as we have seen, “personhood”
as such mysteriously contains this duality. “Person” is at once the most universal and
public, and particular and private, of attributes. We assert our rights saying “I am a
person too!” and refuse to answer questions saying “that’s personal.” As persons, we
should be complete, wholes, like individually wrapped Kraft singles. It’s been nearly
half a century since some feminist proclaimed that “A woman without a man is like a fish
without a bicycle.” Yet, when Harry finally comes together with Sally, there’s hardly a
dry eye in the house.
J. Hand Hegel Person Lecture
Given September 6 2019
Page 34 of 38
�However, it is on the questions of the state, and of citizenship, where the
difference between left and right on personhood really becomes clear and distinct. In
accord with Hegel’s critique of cosmopolitanism, those on the right are unapologetic in
maintaining that America should pursue its national self-interest, even as neo and paleo
cons go at each other’s throats in arguing about how to do so. In some cases, true, such
“nationalism” slides into an ethnic or racial nationalism, going beyond the rational or
strictly political confines in which Hegel had placed the state’s self-assertion. The recent
fracas about a “crisis at the border” makes the continental divide manifest. Is the crisis
primarily a “humanitarian” crisis—too many people without adequate food and medical
attention—or is the fundamental problem that too many people, including a small
percentage but significant number of really bad people, are entering the country without
permission? You can choose your media outlet, because the media no longer mediates.
In the cacophony of the parties—of the parts—one wonders, where is the whole?
Is there a whole, a unum e pluribus? Can a center be found? Or are we headed back to
Rome, to a substance-less “legality” attempting to contain, by sheer force, the warring
factions and isolated atoms? Hegel, if not answering these questions, helps us think
about them. His questions—What is a modern society? What is a modern society?—are
still ours.
J. Hand Hegel Person Lecture
Given September 6 2019
Page 35 of 38
�
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
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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.
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pdf
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35 pages
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Reconciling subjectivity and substance : Hegel's critique of pure personhood
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Transcript of a lecture given on September 6, 2019 by Jonathan Hand as part of the Dean's Lecture and Concert Series.
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Hand, Jonathan
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St. John's College
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Santa Fe, NM
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2019-09-06
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text
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pdf
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Hegel, Georg Wilhelm Friedrich, 1770-1831
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English
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SF_HandJ_Reconciling_Subjectivity_and_Substance_Hegel's_Critique_of_Pure_Personhood_2019-09-06
Friday night lecture
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
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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
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paper
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11 pages
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Returning to Lucretius
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Transcript of a lecture given on April 5, 2019 by Thomas Nail as part of the Dean's Lecture and Concert Series.
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Nail, Thomas
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St. John's College
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Santa Fe, NM
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2019-04-05
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text
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pdf
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Lucretius Carus, Titus. De rerum natura
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English
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Nail, LecN2
Friday night lecture
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Dublin Core
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
Coverage
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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.
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paper
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35 pages
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On teachers and students : Heidegger in America
Description
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Transcript of a lecture given on September 21, 2018 by Michael Gillespie as part of the Dean's Lecture and Concert Series.
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Gillespie, Michael Allen
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St. John's College
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Santa Fe, NM
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2018-09-21
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text
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pdf
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Heidegger, Martin, 1889-1976
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English
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Gillespie, Lec. G555
Friday night lecture
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Dublin Core
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
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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.
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paper
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28 pages
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The (Plato's) Cave, and the cave beneath the cave, in Hegel's Phenomenology of spirit
Description
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Transcript of a lecture given on October 10, 2014 by Jonathan Hand as part of the Dean's Lecture and Concert Series.
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Hand, Jonathan
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St. John's College
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Santa Fe, NM
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2014-10-10
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text
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pdf
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Hegel, Georg Wilhelm Friedrich, 1770-1831. Phänomenologie des Geistes
Philosophy, Ancient
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English
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24004304
Friday night lecture
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Dublin Core
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St. John's College Lecture Transcripts—Santa Fe
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St. John's College Meem Library
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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
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pdf
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31 pages
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Recognizing Odysseus
Description
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Transcript of a lecture given on February 1, 2019 by Margaret Kirby as part of the Dean's Lecture and Concert Series.
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Kirby, Margaret Anne, 1956-
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St. John's College
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Santa Fe, NM
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2019-02-01
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text
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pdf
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Homer. Odyssey
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English
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Kirby, M. Recognizing Odysseus
Friday night lecture
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