35 20 5655 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/fd9434165bb6c6d0085062394f526fdf.mp3 a2e84d76a2f07eee82a6c2f7e6d476f3 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:58:01 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Aporia on the woman question in Plato's Republic Description An account of the resource Audio recording of a lecture given on November 8, 2019 by Mary Townsend as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Townsend, Mary Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-11-08 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Plato. Republic Women Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_TownsendM_Aporia_on_the_Woman_Question_in_Plato's_Republic_2019-11-08 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/be208f189dd6bf172ae37ec2768cd446.mp3 0ec9ab4d842647bb41dba1ba812738d7 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:53:20 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Joining the global great conversation : an experience in China Description An account of the resource Audio recording of a lecture given on November 1, 2019 by Martha Franks as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Franks, Martha C. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-11-01 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Education, Humanistic -- Study and teaching (Secondary) -- China. Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksM_Joining_the_Global_Great_Conversation_An_Experience_in_China_2019-11-01 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/043ca474ab123a106b9b22d5e5460e90.mp3 a54cd1121f8a3b02e180e5f9b7a597df Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 01:10:07 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Jerusalem and Athens: The Family Drama in the Bible and Greek Thought Description An account of the resource Audio recording of a lecture delivered by Ronna Burger, Professor of Philosophy at Tulane University, on November 8, 2019 as part of the Formal Lecture Series. Burger describes her lecture topic as follows: "The family is a central concern for the Bible as well as for Greek poetry and philosophy. But there seems to be an important difference in the problem on which each tradition focuses. The dominant theme of the Book of Genesis is the rivalry of brothers (or sisters)—from Cain's fratricide to the attempted fratricide by Joseph's brothers. In Greek thought, on the other hand, the dominant theme is the “Oedipal” relationship of the son to the father (and mother)—from Hesiod's Theogony to Aristophanes' Clouds, Plato's Euthyphro and the tyrannical soul in Book IX of the Republic, with its allusion to Sophocles’ Oedipus Tyrannus. What is the significance of these differing perspectives on the principal problem of the family? Does the contrast shed light on the larger issue of Jerusalem and Athens?" Burger is introduced by Tutor Jason Tipton. Creator An entity primarily responsible for making the resource Burger, Ronna, 1947- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-11-08 Rights Information about rights held in and over the resource A signed permission form has been received stating, "I hereby grant St. John's College permission to make an audio recording of my lecture, and retain copies for circulation and archival preservation at the St. John's College Greenfield Library and to make an audio recording of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Dysfunctional families in the Bible Philosophy, Ancient Poetry, Ancient Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Burger_Ronna_2019-11-08 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/44add5aaeec8dd8ed7de1fc23b2d3134.mp3 eee89888219825187b981fe45752fa07 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 01:06:58 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Philosophy Between the Lines: The Lost History of Esoteric Writing Description An account of the resource Audio recording of a lecture delivered by Arthur Melzer, Professor at Michigan State University, on November 1, 2019 as part of the Formal Lecture Series. Melzer describes his lecture topic as follows: "Is it true that most philosophers prior to 1800 concealed their true opinions behind a veneer of conventional pieties? What is the empirical evidence for this claim? What motives could have led so many philosophers to engage in such strange behavior? And why does this strike mainstream scholars today as preposterous?" Creator An entity primarily responsible for making the resource Melzer, Arthur M. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-11-01 Rights Information about rights held in and over the resource A signed permission form has been received stating, "I hereby grant St. John's College permission to make an audio recording of my lecture, and retain copies for circulation and archival preservation at the St. John's College Greenfield Library and to make an audio recording of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Secrecy Philosophy--History Censorship Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Melzer_Arthur_2019-11-01 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/42d72f0c0082cdb9f9c7f7b3c3d4c063.pdf a824b47c12c21bf762e8a8d366b05a22 PDF Text Text GADFLY Freshmen Souls • 02 On the Gadfly • 03 Civility & St. John's • 04 St. John’s College • 60 College Ave, Annapolis, MD 21401 • September 3, 2019 • Vol. XLI • Issue 01 �The Gadfly 02 The student newspaper of St. John’s College 60 College Avenue Annapolis, Maryland 21401 sjca.gadfly@gmail.com Founded in 1980, the Gadfly is the student newsmagazine distributed to over 600 students, faculty, and staff of the Annapolis campus. Opinions expressed within are the sole responsibility of the author(s). The Gadfly reserves the right to accept, reject, and edit submissions in any way necessary to publish a professional, informative, and thought-provoking newsmagazine. Submissions sent to the Gadfly should either be in Micrsoft Word or JPEG format. The deadline for submissions is the Friday prior to publication. The Gadfly meets every other Sunday at 7 PM in the BBC. We always need editors, layout designers, illustrators, and organizers. Contact us at sjca.gadfly@gmail.com for more information. Staff Athena Berreles-Luna • Co-Editor-inChief Rose Pelham • Co-Editor-in-Chief Lynn Christian • Cartoonist Contributors Kira Anderson Harrison Health Center Lynn Christian Bryan Jacobsen Rose Pelham Hope Taglich John Verdi From the Editors: Welcome back! Or, if you're a freshman, simply welcome! Don't worry, The Gadfly doesn't bite all the time. We usually just nibble a little now and then. After all, we do need a little bit of horse blood to reproduce and lay our eggs. Please enjoy these pages written by your fellow polity members––the gadflies amoung you. - The Gadfly Freshman , Freshman Souls This reprinted article first appeared in the Gadfly on September 2, 1982, and appears annually in our first issue. Without a doubt our athletic director, Mr. Krueger, is willing to stand in for the first person voice used in this article—and to answer any questions about our athletic program. -Ed. Bryce Jacobsen T A '42 he reasons, both physical and metaphysical, why everyone ought to join in our sports program are many. I list a few: 1. We have the best athletic program of any college in the country. 2. Exercise is good for the body...unless you sprain an ankle or something like that. 3. Most of us feel better, are more alert, and can get more work done if our bodies are healthy and our souls are relaxed. 4. Friendly competition is one of the really fun things in life. It is good for your soul. 5. Your circle of acquaintances will be greatly enlarged. This is good for the soul, provided you can separate the wheat from the chaff. 6. You will learn to accept, and bear with, thousands of split-second decisions from the officials, a few of which are wrong. This is very good for the soul. 7. Do you like to strive for, and achieve, specific goals? If so, consider our college blazers. They are much sought after, and the pathway is clearly laid out. Striving for goals is good for your soul. 8. It is probably true that the more pure fun occurs in the athletic program than in any other area of the college. Fun is good for your soul. 9. If you get involved in team sports, and become a “good team player,” you have realized that there are things in the universe that are more important than your own ego. This is a great good for your soul. 10. The benefits of exercise and friendly competition, learned while one is young, should be maintained for the rest of your life...i.e, they should become habitual. For virtue, as the Philosopher said, is a habit. 11. You will get to know numerous alumni, tutors, and staff members who participate in the program. This is good for your soul, or ought to be...provided that they are the proper sort of role models. 12. Our showers are the best at the college; always plenty of hot water. 13. Are you bothered by, or worried by, tobacco fumes in the air? Come to the gym. The whole building is a nicotine-free zone. 14. If you perform some sort of heroic deed on the athletic field, your name will be mentioned in our weekly column. Heroes are always acclaimed. But do not be carried away by this. Remember that “the paths of glory lead but to the grave.” 15. A high percentage of our best students are active participants in our program. 16. Those who play, stay. 17. The gym is not particularly well-equipped, as gyms go. But it has washers and dryers, and a coke machine...and I will explain to you, if you ask me, how you can get yourself in tip-top physical shape, without any equipment at all. 18. You can sit in an old-time barber’s chair in my office...you can pump yourself up and down, and adjust the slope high or low. Where else can you do that? 19. All work and no play makes Jack a dull boy. 20. It is better to light a candle than to curse the darkness. So there you have it: twenty good reasons why you should participate in our athletic program. If you are not convinced by all of this, come and talk to me. I can probably think of some more good reasons. Or better yet, talk with the upperclassmen. They will tell you all sorts of strange, interesting, and wondrous things. u �The Gadfly 03 On the Gadfly Rose Pelham A’20 Dear Polity Members, At the beginning of each academic year, we publish the first issue of the Gadfly with the following excerpt from Plato’s Apology on the cover: I was attached to the city by the god––though it seems a ridiculous thing to say––as upon a great and noble horse which was somewhat sluggish because of its size and needed to be stirred up by some kind of gadfly. It is to fulfill some such function that I believe the god has placed me in the city. I never cease to rouse each and every one of you, to persuade and reproach you all day long and everywhere I find myself in your company. (Apology 30e-31a) The tradition serves two functions. The first is to introduce the paper to anyone who is new to campus by explaining the origins of our name. The second is to renew our commitment as a publication to live up to that name insofar as we are capable. It is this latter purpose that is the more important, as it situates us within a historic tradition of critical dissent. The tradition of challenging authority and questioning the ideas taken for granted within public opinion (that is to say: of challenging δοχα) is something we should regard as essential to living freely. Today, we typically think of this as an element of freedom of the press, though, in truth, it is much more fundamental. The expression of dissent or critical social commentary is key both to social change (be it reform or revolution) and the maintenance of an equitable society. It is a principal means by which oppression and injustice may be challenged. Last, the way in which we go about it, drawing upon the example Plato gives us of Socrates, makes the Gadfly an intrinsically Johnny institution. The Gadfly is open to, and to a large extent relies upon, articles submitted to us by members of the polity––readers like you. We welcome the writing of anyone who wants to share news and new ideas with the polity: anything from student journalism and essays, to short stories and even poetry. There are relatively few restrictions on what can be published in the paper. Naturally, we will not publish hate speech, or articles that defame individuals. We also require that articles be sent to us either as Word or Google documents for ease of editing. That said, we publish the vast majority of student and tutor submissions, and generally want to facilitate the ability of polity members to share their ideas publicly. The Gadfly changes every year with the changes of the polity and the polity members who run the publication. Our readers have a substantial effect on our work as a publication, both through the feedback we receive and the articles you submit. If you want to change some aspect of the publication, the best way to do so is to work with us on it. In that respect, we can always be yours––our reader’s––the polity’s publication. Sincerely, Rose Pelham Co-Editor-in-Chief A’20 u An Appeal to Freshman Kira Anderson A’18 Freshman, St. John’s has no guidebook. We have a handbook, yes, but that exists to tell you what is or is not appropriate behavior in our polity. A guidebook, however, tells you what things are of interest and how you may reach these interesting sites. St. John’s has no guidebook, because truly everything here has a facet that will interest you. You may not yet know of your burning passion for improvisational acting or of your fiery dedication to your intramural basketball team. What interests you have now are limited to your experience outside of the college. Therefore, I am here to encourage you as a senior who knows that she has not experienced enough of the college in her three-going-on-four years; I address you as a student and as a club leader myself, as a lab assistant, and as someone who wants to see you succeed. Coming to St. John’s is a transformatory experience. You will learn many things here (and forget just as many perhaps), but I wish to impress upon you that this is a place of change. When you graduate, diploma in one hand and champagne in another, you will be a different person from the one who set foot at convocation and received a copy of Euclid. Of course, you will be older, because time waits for no one, but there will be a change of spirit as well. What you know and think you know before St. John’s are merely stepping stones. Thusly, I encourage you to take a full survey of what St. John’s has to offer, detailed to you by someone who has in fact already started panicking about a life outside of the Johnnie Bubble. I will present my solicitation in the form of a list, as pleases me and the average reader. “Listicles,” though universally despised, should be given the commendation of at least being succinct. 1. Join a club. In fact, join four or five, and when you realize you can’t balance them all with your class work and your burgeoning friend group and need for down-time away from people, drop your least favourite or least favourites. 2. Explore every building on campus (when they are open of course, not after hours.) I have personally discovered many of my favourite hiding spots by taking a Johnnie safari and trekking about for a while. Your dorm is interesting, I’m sure, but I dare you to try to find the illusive Gadfly office or determine the age of Temple Iglehart (the gym, for the uninitiated.) 3. Make friends with people who are different from yourself and your normal group of friends. This may be self-explanatory, but often the best friendships are forged in the celebration of differences. 4. Know your tutors. I did not have a meeting with a tutor of my own volition until the summer after my junior, perhaps sophomore year, and I know that now that was a mistake. I feared that tutors would think me stupid; I thought that I didn’t deserve their time. I learned, however, that I could not been more wrong. Tutors work at St. John’s because they love discussion and teaching, and every tutor that I have asked to meet with has accepted the offer with grace and kindness. 5. Know your administration and do not be afraid to speak up. This is a lesson I learned only after much experience at the college, and I fear it may be the most important that I relay to you. Know who you can report things to. Get to know who the Title IX coordinator is. Get to know the assistant dean. Meet our new President. Know your RA, and if you don’t feel comfortable with an RA, make sure to get to know one of our several (FREE) school counselors. If something occurs that you are uncomfortable with, report the incidents as soon as possible. College is a time for learning, and that cannot be done in an unsafe situation. 6. Enjoy yourself. You will hear from many upperclassman that freshman year is the easiest year at St. John’s. It is, however, the hardest. Coursework is different from anything you have experienced in highschool, and adjusting will be tough. Balancing and managing one’s life is also incredibly difficult. If someone belittles your problems with school work, saying that the year is easy, feel free to spit back some sort of biting comment like “Did you feel like being a freshman was easy when you were a freshman?” Or perhaps ask in a nice manner. My biased concept of communication is already quite evident. Along with this article worthy of a mediocre buzzfeed link, I also extend something else to you: words by which to live perhaps. This quote is now on t-shirts, and bags, and leather bracelets (which I will neither confirm nor deny that I own,) but I feel that it is nonetheless relevant. “Not all those who wander are lost,” JRR Tolkien reminds us. Freshman year is a year to wander, to discover and adventure. Ask questions and look stupid sometimes. Everyone does, even if they pretend they don’t. So, go out and wander, budding Johnnies. Try clubs and meet people and talk to leadership figures. Go and pour forth much. u �Civility and St. John's John Verdi Tutor In any community, but especially in a small one such as ours, a concern for civility, broadly conceived, ought from time to time to occupy a prominent place in thought and conversation. By civility I mean not merely a kind of public politeness (though I intend this, too), but also all the ways of twelve guidelines, principles or rules of civility, of civil discourse in particular. I think of these as beginnings to what I hope will become an extended public discussion of the forms civil behavior can take at St. John’s. (In fact, I’ll end these remarks with an opening question.) There should also be op- possible. The intrusion of authority into a discussion separates at least one member from the others, and thereby threatens the commonality of the enterprise. being a citizen in something analogous to a civis or polis. Civility at work is equivalent to good citizenship. Whether we think the College community is succeeding or failing in promoting civil behavior, it behooves us to reflect on exactly how civility and its opposite manifest themselves here. Our hope should be that with heightened awareness and dialogue about the very notion of civility, we shall find ourselves behaving in ways that further our common endeavor, that is, our own ongoing education in the liberal arts. Much of what I’m about to say will seem obvious, perhaps even platitudinous; but sometimes “the aspects of things that are most important for us are hidden because of their simplicity and familiarity.” (Wittgenstein, Philosophical Investigations, 129) portunity to investigate whether or not civility is even all that important to our living together. I for one think that it is, but the forms it takes in practice may be perceived as varying widely, and some discussion of these manifestations could be enlightening. conversation (at least usually!). In this article I would like to propose—with little comment— 3) Civil discourse shuns reliance on authority as much as 1) Civil behavior is an essential component of good citizenship, and may even be equivalent to it. Civil behavior is an acknowledgement that the concerns of others have a proper place alongside my own. 2) Civil discourse is bounded. Participants agree, either explicitly or implicitly, to limit the scope of what may be said in a discussion. This is most in evidence in our classes. 4) Formal modes of address help maintain an attitude of respect and dispassion in civil 5) Civil discourse requires that we be prepared to put ourselves and our ideas at risk. In order for this to be possible community members must feel that they and their ideas and their questions will be taken seriously at all times. 6) In a genuine discussion participants attempt to find the maximum amount of common ground with one another. It is often the case that the continuing of the conversation takes precedence over the drawing of conclusions, however tempting such closure may be. 7) Civil discourse requires both sympathetic listening and engaged speaking. The former, which might also be called civil listening, demands that we listen with our ears, not our mouths. That is, we listen to find what is �best in what someone is saying; we thereby acknowledge that the other is someone like me. Engaged speaking is carried on with broad vision and with an eye on common purposes. 8) Such civil discourse opens up spaces for disagreements to be aired without bringing the conversation to a halt. 9) It demands that when we disagree, we not become offensive, cynical or silent. 10) To discuss civilly—and perhaps to be civil in general—I must be willing to place the goals of the community above my own comfort. For a student this might mean speaking in class when he or she would rather not. For all community members it means not insisting on being left alone when by participating I can improve the community, no matter how uncomfortable participation may be for me. 11) Self-governance is a mode of civility. At St. John’s the faculty tries to govern itself in many different ways. The Dean, Assistant Dean and Associate Dean for Graduate Programs are chosen from the faculty itself, and they serve for limited terms. At faculty meetings we try to discuss all manner of College concerns, from the Program and our teaching in it, to the plans for the new dorm and the place athletics has here. We try to remain vigilant about not allowing self-governance to erode, but we often fail to achieve our ideal. 12) Civility, as it manifests itself in discussion, requires that I recognize the possibility that what someone might say has the power to change the way I see. After this long introduction, I can now ask my opening question: How goes it with civility at St. John’s? John Verdi, Tutor u �What I Did Over the Summer Hope Taglich Over the summer, I worked orexia. A'21 eyes that flash when she gets excited as an intern in the compliance reg- My eating disorder began ulation department for a regional during the summer of 2013. I was bones. She seemed blind to the fact. home healthcare company in New about to begin high school and Through my observations of her, York City. I scanned files. I called had begun internalizing the idea of I learned to associate femininity patients and asked if their medical What I Was Supposed to Be. I was with self-loathing. To be a woman equipment was delivered on time. supposed to have twiggy arms and is a sort of cardinal sin, one that I was cursed out in five different a flat stomach and a thigh gap and must be atoned for through periods languages (English, Mandarin, to make myself small and inof- of fasting, fad diets and relentless Cantonese, Spanish, Russian and fensive and palatable and hot. My exercise. And so that is what I did. French). I befriended my supervi- mother never actively projected sor, a wonderful woman named Jo- her own insecurities onto me; I high school, I swam every day for selin. She was Dominican and from never remember her calling me fat, an hour and began calorie-counting Long Island (we grew up a couple really. I grew up hearing her voice compulsively. I lost 20 pounds. towns away from one another), as her own feelings of disgust over her Over the course of the year, I lost well as a passionate Francophile. body. I remember being nine years about 10 more pounds. It was never (She gave me a copy of Shari Ben- old and joking that, at holidays enough. My memories of high stock’s Women of the Left Bank, and family gatherings, the men school are memories of standing which I devoured). I stayed with always talked about politics and the in front of the mirror, pressing my my grandparents, which was im- women always talked about losing legs and feet together to check the mensely helpful as it spared me the weight. size of my thigh gap. I remember agony of having to pay New York rent and groceries. I performed at her murmur, pinching anything she one point, I was only eating about poetry slams at the Bowery Po- could pinch. Sometimes she would 700 calories a day. etry Club. I read Don Quixote and call me over. Death in Venice. I started kickboxing. I began a brief romance with a boy named Kwai, with whom I was stuck on West 34th Street on one midsummer night, during the city’s largest power outage since 1977. I cooked for the first time in my life (grilled chicken with paprika and pesto sauce, with some steamed lentils). I also recovered from an- “Look at how fat I am.” told me. I would compliment her, “I’m so fat,” I would hear about something and great cheek- The summer before I began weighing myself every morning. At “You look sick,” my mother I would turn my face away and she would respond with, to hide my smile. “You’re sweet.” My mother very seldom says “Thank you,” when came over and devoted much someone compliments her. She of their visit to urge me to gain seems to think each compliment is weight. “You’re losing your beauty,” nothing more than an act of charity. my grandfather said. My mother is a beautiful woman. She has honey-colored skin and sense of pleasure. I felt beautiful the rich hair and almond-shaped dark more I limited myself. Self-loathing Once my grandparents That brought a rebellious �brough a sense of catharsis for me. Beauty, I felt, was a reward. I would ment. After several weeks, I realized became that one thing a day. punish myself for existing by de- I did have a problem. I didn’t know creasing inch by inch of my surface if I wanted to recover, though, and the world did not implode. I area. It was like cutting, except because I was comfortable the way I did gain weight. I don’t know how that it would enable me to fit into a was. I wasn’t happy, really, but I had much I weigh now. I recently talked size 2, and a size 2 was skinny, and learned to dissociate pleasure from to a friend who told me I looked skinny was beautiful. I felt isolated eating during that fateful summer “plumper,” and for a second, it felt at school, and it became a repose of 2013, and I was afraid of the like I could barely breathe. He was for me. It was a source of self-as- weight gain that would result from a friend I had been open about surance. Even if I was unpopular, I recovery. recovery with. was worthy and beautiful. I became more worthy and more beautiful ing and feel instantly more deter- said. with each pound I lost. As I began mined. I gradually began to stick to to stress more and more about what the meal plan that had been as- a size 2 anymore, but my body is was going on at home (which was a signed to me. I was shocked at how mine. My high school mantra was lot), I restricted more and more. much energy I had; I no longer had a quote from Kate Moss—Nothing It continued in college. I worked midday energy crashes. I had been out even when injured. I ate very, struggling with depression over the very little. I neurotically calorie- previous year, and, with each pass- counted. ing day, began to feel endorphins come creeping back. It reminded tastes as good as skinny feels. Ms. Moss’s principle is incorrect. Emilie du Chatelet, the author of one of the first modern self-help books, On Happiness, states that in order to be happy, one must be without prejudice, passionate about various arts and subjects, virtuous, healthy and loving of oneself. In fact, Chatelet states, “One of the great secrets of happiness is to moderate one’s desires and to love the things already in one’s possession.” Moreover, in Descartes’s Meditations, the essence of human self is reduced to the purely intellectual: “I think, therefore I am.” The human body exists as a vehicle for the soul; the soul is the key to the human person. It is the source of its ousia. My body, therefore, does not define me. Society might care what size jeans I wear, but Descartes does not. u Over the summer of 2019, I Reluctantly, I started treat- I didn’t wake up one morn- began my first treatment program me of an unthawing. I had crip- (three days a week after work from pling episodes of dysmorphia, but I 6 to 9, Saturday mornings from stopped weighing myself. I began to 8:30 to 1:30). This was essentially want to get better. It was a terrifying thing to want, because I could not imagine myself without my eating disorder. My skinniness had been a cornerstone of my identity since I was 14. It is terrifying, but then you just do it. And it sucks, because it’s extremely stressful to eat three full meals a day. I hadn’t had three full meals a day in six years, and my meal plan also required me to have three snacks. However, a very wise refrigerator magnet once told me to do one thing a day that scares the result of an intervention. “I don’t have a problem,” I told Alethia, my case manager at the clinic, one afternoon in May. Alethia is a trained professional; she is not supposed to cast visible judgement on patients. However, I could tell that she smelled bullshit. “Seriously, I don’t.” She asked me to describe my eating habits to her, and I told her. It was apparent, then, that according to Alethia, I definitely had a problem. me, and being meal plan-compliant I had three meals a day, “It looks good on you,” he And so it does. I am not �The Gadfly 04 Is it Time to QUIT? The Harrison Health Center is here for you. We offer 1:1 nicotine cessation services and FREE nicotine patches, gum, and lozenges. Call the Health Center at (410)626-2553 or email healthcenter@sjc.edu to take the first step toward a healthier you. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Gadfly Description An account of the resource Founded in 1980, The Gadfly is a weekly student publication distributed to over 600 students, faculty, and staff of the Annapolis campus. Click on <a title="The Gadfly" href="https://digitalarchives.sjc.edu/items/browse?collection=16&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the The Gadfly Collection</a> to view and sort all items in the collection. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Identifier An unambiguous reference to the resource within a given context thegadfly Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 4 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Gadfly, Vol. XLI, Issue 01 Description An account of the resource Volume XLI, Issue 01 of The Gadfly. Published September 3, 2019. Creator An entity primarily responsible for making the resource Berreles-Luna, Athena (Editor-in-Chief) Pelham, Rose (Editor-in-Chief) Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-03 Rights Information about rights held in and over the resource St. John's College owns the rights to this publication. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Gadfly Vol XLI Issue 01 09-03-2019 Gadfly Student publication https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/993ba75205ca43af9a3731594c500704.mp3 6a334e6495b5110ada7652a722905b5c Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:47:24 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The Pursuit of Happiness: Four French Thinkers on Our Restless Quest for Contentment Description An account of the resource Audio recording of a lecture delivered by Benjamin Storey, Associate Professor, Department of Politics and International Affairs, and Co-Director of the Tocqueville Program at Furman University, on October 18, 2019 as part of the Formal Lecture Series. The introduction to Storey's lecture was not recorded. Storey describes his lecture topic as follows: "In 1840, Alexis de Tocqueville famously observed that Americans are 'restless in the midst of their well-being'—strangely uneasy even as they enjoy unprecedented prosperity. This talk will consider the long French tradition of thinking about restlessness that lies behind that observation. It will begin by showing how the great essayist Michel de Montaigne seeks to quiet our restlessness by teaching us a new way of pursuing happiness, a way of life distinguished by the central place it accords to the modern virtues of skepticism, authenticity, and nonchalance. Turning next to Pascal and Rousseau, the paper will consider their separate and distinctive critiques of the Montaignean pursuit of happiness, which, in their view, deepens the very restlessness it is intended to palliate. The paper will conclude with a consideration of Tocqueville’s analysis of American politics in the light of his insight into the psychological unease of the American citizen, and an argument for how we might pursue happiness better by reconsidering some of the assumptions of modern skepticism from which we began." Creator An entity primarily responsible for making the resource Storey, Benjamin Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-18 Rights Information about rights held in and over the resource A signed permission form has been received stating, "I hereby grant St. John's College permission to make an audio recording of my lecture, and retain copies for circulation and archival preservation at the St. John's College Greenfield Library and to make an audio recording of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Tocqueville, Alexis de, 1805-1859 Montaigne, Michel de, 1533-1592 Rousseau, Jean-Jacques, 1712-1778 Pascal, Blaise, 1623-1662 Happiness Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Storey, Benjamin 10-18-2019 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a6bd816dd6d27c9fe8dc893b608baf1b.mp3 bfe59ea6511231a8a9630d8a1f7895b3 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:56:06 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Hegel on Reason in History Description An account of the resource Audio recording of a lecture delivered by Mark Alznauer, Associate Professor of Philosophy, Northwestern University, on September 20, 2019 as part of the Formal Lecture Series. Alznauer describes his lecture topic as follows: "Hegel says that the only presupposition that the philosopher brings to history is the simple thought of reason, that reason rules the world. In this lecture, I will compare Hegel’s philosophical histories (of the state, art, religion, and philosophy) to more empirical approaches, paying particular attention to the conceptual form of Hegel’s histories." Creator An entity primarily responsible for making the resource Alznauer, Mark Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-20 Rights Information about rights held in and over the resource A signed permission form has been received stating, "I hereby grant St. John's College permission to make an audio recording of my lecture, and retain copies for circulation and archival preservation at the St. John's College Greenfield Library and to make an audio recording of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Hegel, Georg Wilhelm Friedrich, 1770-1831 History--Philosophy Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Alznauer_Mark_2019-09-20 Alumni Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/1b87ad27305a68e8acdb3d35a12c76b2.mp3 d20ee72444dbecc7e4229e3e2f0c9676 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 01:00:51 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Beholding black holes and quickening grains of spacetime : the quest for a quantum theory of gravity Description An account of the resource Audio recording of a lecture given on October 11, 2019 by Hal Haggard as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Haggard, Hal M. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-11 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Gravity Quantum theory Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_HaggardH_Beholding_Black_Holes_and_Quickening_Grains_of_Spacetime_2019-10-11.mp3 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/4076cf9a600178cf678478821f331c65.pdf cc69968c45c5be06d9f9a13dede8202e PDF Text Text The Heptadecagon Grant Franks October 8, 2019 Review We’ve looked at the first encounters with the -1 , the development of complex arithmetic, the roots of unity, constructibility of points on the complex plane (considered algebraically) and followed the construction of the pentagon. These points will be reviewed a final time next Wednesday in the final lecture. (Remember: Wednesday, October 16, Junior Common Room, 3:15 pm.) Let’s move on to the Heptadecagon. Remember the Pentagon? We’ve seen how to approach the algebraic construction of the pentagon. The process begins with the equation: x5 - 1 = 0 One factors out the one real solution that all “roots of unity” equations share, namely (x - 1): 1 + x + x2 + x3 + x4 = 0 This equation is irreducible so long as one allows only rational numbers, Q, as solutions. (In mathjargon, it is “irreducible over the rationals.”) But if one constructs a finite quadratic field extension Q( 5 ), it can be factored into two quadratics. A second finite quadratic field extension allows it to be factored fully into four linear factors from which you can read oﬀ the solutions readily. �2 �� 5 The Heptadecagon.nb 0.31 + 0.95 ⅈ -0.81 + 0.59 ⅈ -0.81 - 0.59 ⅈ 0.31 - 0.95 ⅈ x4 + x3 + x2 + x + 1 = 0 Irreducible over Q Then adjoin 5 x2 - η2 x +1 x2 - η1 x +1 (x - ζ1 ) Irreducible over F1 Irreducible over F1 Then adjoin ζ1 Then adjoin ζ2 (x - ζ4 ) Do the Same Thing, But More O�en (x - ζ2 ) (x - ζ3 ) �5 The Heptadecagon.nb �� 3 We’ll follow the same basic plan to construct the heptadecagon. However, the procedure has a few additional complications due to the greater number of steps. The Equation For the heptadecagon, we start with the equation: x 17 - 1 = 0. Again we factor out the one real solution (x - 1) to obtain: 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = 0 This 16th degree equation has sixteen roots, all complex, that we designate: ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 , ζ11 , ζ12 , ζ13 , ζ14 , ζ15 , ζ16 Graphically, these roots appear on the complex plane as vertices of a regular 17-gon, making equal angles at the center: ζ5 ζ4 0+1i ζ3 ζ6 ζ2 ζ7 ζ ζ8 -1 + 0 i 1+0i ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 0 - 13 1i ζ ζ 14 Unsurprisingly, these occur in eight pairs of complex conjugates. (In the diagram above, complex conjugates are joined by orange dotted lines). Arrangement of the Sixteen Roots: the Eight-Periods �4 �� 5 The Heptadecagon.nb Following the general procedure seen with the pentagon, we are going to split these roots up into -two groups of eight, then four groups of four, then eight groups of two, then sixteen individuals. At each stage, we will make numbers by taking the sums of the members in each group. With the pentagon, we found that even though we didn’t know the values of any of the roots (the ζ’s), we could figure out a quadratic formula for the values of the two intermediate sums: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 because we could figure out their sum and the product, η1 + η2 and η1 ×η2 . When working on the pentagon, the way in which to subdivide the four roots presented little trouble. We had reason to believe that the complex conjugates had to stay together, so there was only one possible subdivision of the four roots into two pairs. The second division separated the two pairs roots from their conjugate mates. Now, however, we have sixteen roots and eight pairs of conjugates. For the first subdivision, there are 8 x 7 x 6 x 5 = 1,680 possible ways to separate the eight pairs into two groups of four. We don’t know a priori whether some or all, or not all or possibly only one will work. On the surface, it seems that trial and error might not work. Sorting out the roots properly is more than half the battle in doing this construction. It will require a small detour. Half the Problem is Pretty Easy First, some good news. We’re looking for a sorting of the roots that will allow us to find the sum and the product of the two groups. In that quest, the sum of the two groups will pose no problem. No matter how we divide the sixteen roots into two bunches, we will be able to get their total sum. Say we just sort out the first eight and the last eight: A = ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 B = ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 Now, when we add A + B, we get the sum of all sixteen roots. And we know that to be equal to negative one. In fact, no matter what subdivision we make, the sum of the two divisions will be negative one. Remember: �5 The Heptadecagon.nb �� 5 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = 0 which is to say: x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 + x15 + x16 = -1 Just put in ζ for x: ζ + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 So the issue that has to be addressed is η1 × η2 , the product of two eight-term sums. Modular Arithmetic Sorting out the other half of the problem will involve us in modular arithmetic. It’s not at all diﬀicult for anyone who has read clock. � ζ6 ζ ζ7 ζ2 ζ5 ζ3 ζ8 ζ4 Remember that multiplying roots of unity by themselves -- that is, raising them to powers -- will move the solution around the unit circle in the complex plane like a clock hand. (In this case the clock hand goes counterclockwise; all analogies have problems!). For example, there are five fi�h roots of unity. If I square the first one, then cube it and so forth, the result moves around the unit circle. Also, when the �6 �� 5 The Heptadecagon.nb hand has gone completely around, all later solutions are equivalent to one or another of the first five solutions. Thus, ζ 6 is equivalent to ζ 1 . The technical term for this sort of equivalence is “congruence”; we write ζ 6 ≡ (ζ 1 )Mod 5 , “zeta to the sixth is congruent with zeta one, modulo 5.” Congruence of this sort will be very useful for us, for Gauss’s solution to the sorting problem involves some very high powers of the 17th roots of unity. Primitive Roots One observation about modular arithmetic before we go on. Suppose we are working in modulo 17 -as we will be doing. Take some number, a, and raise it to successive powers. In ordinary arithmetic, it will grow continually. In modular arithmetic, it will go around the cycle of available numbers. Some numbers in doing to touch all the values available; some do not. Take 2, for instance: TableFormTablen, 2n , Mod2n , 17, {n, 1, 16}, TableHeadings → None, "n", "2n ", "(2n )mod 17 " n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2n 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16 384 32 768 65 536 (2n )mod 17 2 4 8 16 15 13 9 1 2 4 8 16 15 13 9 1 If I use “n” to designate the power to which one raises the root, notice that 2 cycles through seven values before coming to n=1 and repeating itself. On the other hand, 3 cycles through all the possible values before repeating itself �5 The Heptadecagon.nb �� 7 TableFormTablen, 3n , Mod3n , 17, {n, 1, 16}, TableHeadings → None, "n", "3n ", "(3n )mod 17 " n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 3n 3 9 27 81 243 729 2187 6561 19 683 59 049 177 147 531 441 1 594 323 4 782 969 14 348 907 43 046 721 (3n )mod 17 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1 It can be shown that, for prime numbers, there is always at least one such value. For our purposes, with the 17-gon, we only need one. The number three will work for us. (There are others; 2, 4, 8, 9, 13 and 15 don’t work; 3, 5, 6, 7, 10, 11, 12, and 14 do.) Ordering of the Sixteen Roots Gauss ordered the sixteen roots in accordance with the expression: n ζ (3 ) Since 3n modulo 17 cycles through all values from 1 to 16 before repeating, this ordering will encompass all sixteen complex roots. The ordering looks like this: n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 3n 1 3 9 27 81 243 729 2187 6561 19 683 59 049 177 147 531 441 1 594 323 4 782 969 14 348 907 43 046 721 (3n )mod 17 1 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1 ζ3 ζ ζ3 ζ9 ζ10 ζ13 ζ5 ζ15 ζ11 ζ16 ζ14 ζ8 ζ7 ζ4 ζ12 ζ2 ζ6 ζ n mod 17 �8 �� 5 The Heptadecagon.nb Notice that each term is obtained from the previous one by successive powers of three. Notice also, that if one takes every other term, one has a succession by powers of nine: 1, 9, 81, 729 … or 3, 27 = 3 x9, 243 = 3 x 81, 2187 = 3 x 729 … This will be useful in what follows. Two Eight Periods The 16-period is divided into two 8-periods by taking alternate members of the series and summing the. η1 = ζ + ζ9 + ζ13 + ζ15 + ζ16 + ζ8 + ζ4 + ζ2 η2 = ζ3 + ζ10 + ζ5 + ζ11 + ζ14 + ζ7 + ζ12 + ζ6 Notice that each period contains four pairs of complex conjugates. ζ5 ζ ζ4 ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 ζ 14 ζ 13 Complex 17th roots of unity - Two Eight-Periods Sum of the 8-periods η1 + η2 = -1, as explained above. Product of the 8-periods The product η1 η2 requires some calculation. We have the multiplication of two eight-term sums, which will yield sixty four terms: �5 The Heptadecagon.nb ζ + ζ9 + ζ13 + ζ15 + ζ16 + ζ8 + ζ4 + ζ2  ζ3 + ζ10 + ζ5 + ζ11 + ζ14 + ζ7 + ζ12 + ζ6  = … 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 exponent1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 13 13 13 13 13 13 13 13 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 8 8 8 8 8 8 8 8 4 4 4 4 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + exponent 2 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 14 7 12 6 3 10 5 11 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = sum 4 11 6 12 15 8 13 7 12 19 14 20 23 16 21 15 16 23 18 24 27 20 25 19 18 25 20 26 29 22 27 21 19 26 21 27 30 23 28 22 11 18 13 19 22 15 20 14 7 14 9 15 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ summod 17 4 11 6 12 15 8 13 7 12 2 14 3 6 16 4 15 16 6 1 7 10 3 8 2 1 8 3 9 12 5 10 4 2 9 4 10 13 6 11 5 11 1 13 2 5 15 3 14 7 14 9 15 �� 9 �10 �� 5 The Heptadecagon.nb 53 54 55 56 57 58 59 60 61 62 63 64 4 4 4 4 2 2 2 2 2 2 2 2 + + + + + + + + + + + + 14 7 12 6 3 10 5 11 14 7 12 6 = = = = = = = = = = = = 18 11 16 10 5 12 7 13 16 9 14 8 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 1 11 16 10 5 12 7 13 16 9 14 8 This is a little hard to digest. However, there are some patterns and repetitions. Looking just at the exponents of the terms: 1 + 3 = 4 ; 9 + 10 = 19 ; 13 + 5 = 18 ; 1 + 10 = 11 ; 9 + 5 = 14 ; 1 + 5 = 6 ; 15 + 11 = 26 ; 16 + 14 = 30 ; 8 + 7 = 15 ; 13 + 11 = 24 ; 15 + 14 = 29 ; 16 + 7 = 23 ; 9 + 11 = 20 ; 13 + 14 = 27 ; 15 + 7 = 22 ; 1 + 11 = 12 ; 9 + 14 = 23 ; 13 + 7 = 20 ; 1 + 14 = 15 ; 9 + 7 = 16 ; 1 + 7 = 8 ; 13 + 12 = 25 ; 15 + 6 = 21 ; 9 + 12 = 21 ; 13 + 6 = 19 ; 15 + 3 = 18 ; 1 + 12 = 13 ; 9 + 6 = 15 ; 13 + 3 = 16 ; 1 + 6 = 7 ; 13 + 10 = 23 ; 15 + 5 = 20 ; 9 + 3 = 12 ; 8 + 12 = 20 ; 4 + 6 = 10 ; 16 + 12 = 28 ; 8 + 6 = 14 ; 15 + 12 = 27 ; 16 + 6 = 22 ; 16 + 3 = 19 ; 8 + 3 = 11 ; 4 + 3 = 7 ; 2 + 3 = 5 ; 2 + 10 = 12 ; 4 + 10 = 14 ; 2 + 5 = 7 ; 8 + 10 = 18 ; 4 + 5 = 9 ; 16 + 10 = 26 ; 8 + 5 = 13 ; 15 + 10 = 25 ; 16 + 5 = 21 ; 4 + 12 = 16 ; 2 + 6 = 8 ; 2 + 11 = 13 ; 4 + 11 = 15 ; 2 + 14 = 16 ; 8 + 11 = 19 ; 4 + 14 = 18 ; 2 + 7 = 9 ; 16 + 11 = 27 ; 8 + 14 = 22 ; 4 + 7 = 11 ; 2 + 12 = 14 ; Or, with the sums reduced modulo 17: 1 + 3 = 4 ; 9 + 10 = 2 ; 13 + 5 = 1 ; 1 + 10 = 11 ; 9 + 5 = 14 ; 13 + 11 = 7 ; 1 + 5 = 6 ; 15 + 11 = 9 ; 16 + 14 = 13 ; 8 + 7 = 15 ; 4 + 12 = 16 ; 2 + 6 = 8 ; 15 + 14 = 12 ; 16 + 7 = 6 ; 9 + 11 = 3 ; 13 + 14 = 10 ; 15 + 7 = 5 ; 8 + 12 = 3 ; 4 + 6 = 10 ; 16 + 12 = 11 ; 8 + 6 = 14 ; 4 + 3 = 7 ; 2 + 3 = 5 ; 2 + 10 = 12 ; 1 + 11 = 12 ; 9 + 14 = 6 ; 13 + 7 = 3 ; 15 + 12 = 10 ; 16 + 6 = 5 ; 8 + 3 = 11 ; 4 + 10 = 14 ; 2 + 5 = 7 ; 1 + 14 = 15 ; 9 + 7 = 16 ; 13 + 12 = 8 ; 15 + 6 = 4 ; 16 + 3 = 2 ; 8 + 10 = 1 ; 4 + 5 = 9 ; 1 + 7 = 8 ; 15 + 3 = 1 ; 16 + 10 = 9 ; 8 + 5 = 13 ; 4 + 11 = 15 ; 2 + 14 = 16 ; 1 + 12 = 13 ; 9 + 6 = 15 ; 13 + 3 = 16 ; 15 + 10 = 8 ; 16 + 5 = 4 ; 8 + 11 = 2 ; 4 + 14 = 1 ; 1 + 6 = 7 ; 15 + 5 = 3 ; 16 + 11 = 10 ; 8 + 14 = 5 ; 4 + 7 = 11 ; 9 + 12 = 4 ; 13 + 6 = 2 ; 9 + 3 = 12 ; 13 + 10 = 6 ; Or, again, more graphically: 2 + 11 = 13 ; 2 + 7 = 9 ; 2 + 12 = 14 ; �5 The Heptadecagon.nb ζ4 ζ 11 ζ6 ζ 12 ζ 15 ζ8 ζ 13 ζ7 ζ2 ζ 14 ζ3 ζ6 ζ 16 ζ4 ζ 15 ζ 12 ζ ζ7 ζ 10 ζ3 ζ8 ζ2 ζ 16 ζ6 ζ9 ζ 12 ζ5 ζ 10 ζ4 ζ ζ8 ζ3 ζ 13 ζ6 ζ 11 ζ5 ζ2 ζ9 ζ4 ζ 10 ζ 15 ζ3 ζ 14 ζ 11 ζ ζ 13 ζ2 ζ5 ζ 16 ζ 10 ζ7 ζ 14 ζ9 ζ 15 ζ ζ 11 �� 11 ζ8 ζ5 ζ 12 ζ7 ζ 13 ζ 16 ζ9 ζ 14 Notice that the yellow rows have all the same members; so do the green rows. Also, that the yellow rows and the green rows have all diﬀerent members, so that a yellow row and a green row together have all sixteen elements. And all sixteen elements together equal negative one, so that the total of all sixtyfour terms is … (drum roll, please) … negative four. This is not an accident. It was carefully orchestrated by Gauss’s arrangement of the two eight-groups. In fact, the necessity that leads to this arrangement can be seen through examination and analysis of the patterns of the terms entering into the multiplication together with about a week of practice with modular multiplication. I can’t undertake that dissection in more detail here; any one interested can pursue a more complete presentation in the texts referred to in the handout. For our purposes, what is essential is that we know the sum and the product of the eight-periods, η1 and η2 . Constructing and Solving an Appropriate Quadratic We now know that the sum η1 + η2 = -1 , and the product η1 η2 = -4. We can therefore make a quadratic equation with these two numbers as its solutions. Using y as a variable, we have: y2 -  (-1) y +  ( -4) = 0 η1 + η 2 η1 η2 whose solutions will be η1 and η2 . This equation can be solved with the quadratic formula. y= -1 ± 1 + 16 2 = -1 ± 2 17 Beautiful. So I know values of the η’s, which are the sums of the 8-periods. The two values are: �12 �� 5 The Heptadecagon.nb η1, 2 = -1 ± 2 17 The approximate values for η1 and η2 are 1.56155 and -2.56155. The Four Periods Next we make four periods of four by taking every fourth root from the original series, starting with the first, second, third and fourth, respectively: ζ, ζ3 , ζ9 , ζ10 , ζ13 , ζ5 , ζ15 , ζ11 , ζ16 , ζ14 , ζ8 , ζ7 , ζ4 , ζ12 , ζ2 , ζ6 ζ13 , Period 1 = ζ, Period 2 = Period 3 = ζ3 , ζ16 , ζ5 , ζ9 , Period 4 = ζ4 , ζ14 , ζ15 , ζ10 , ζ12 ζ8 , ζ11 , ζ2 ζ7 , ζ6 Sums of the Four Periods We make the sums of each of the four-periods: μ1 = ζ + ζ4 + ζ13 + ζ16 μ2 = ζ3 + ζ5 + ζ12 + ζ14 μ3 = ζ2 + ζ8 + ζ9 + ζ15 μ4 = ζ6 + ζ7 + ζ10 + ζ11 Again, the sums present no diﬀiculy: μ1 + μ3 = η1 and μ2 + μ4 = η2 , since the the four-periods are gotten by segregating elements of the two eight-periods. Graphically, the four periods are pictured below. Notice that the 8-periods have been subdivided: the red 8-period into red and green 4-periods; the blue 8-period into blue and orange 4-periods. Once again, notice that each four-period includes two pairs of complex conjugates: �5 The Heptadecagon.nb ζ5 ζ �� ζ4 ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 12 ζ 14 ζ 13 Inner dots = 8 periods; Outer dots = 4 periods Products of the 4-Periods As for the products of the μ’s, we can work out the terms directly. First, take μ1 times μ3 : Expand[μ1 μ3] ζ3 + ζ6 + ζ9 + ζ10 + ζ12 + ζ13 + ζ15 + ζ16 + ζ18 + ζ19 + ζ21 + ζ22 + ζ24 + ζ25 + ζ28 + ζ31 Which, when simplified by re-expressing the exponents modulo 17: ζ3 + ζ6 + ζ9 + ζ10 + ζ12 + ζ13 + ζ15 + ζ16 + ζ1 + ζ2 + ζ4 + ζ5 + ζ7 + ζ8 + ζ11 + ζ14 Put in numerical order of the exponents: ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 And for μ2 times μ4 : Expand[μ2 μ4] ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 + ζ18 + ζ19 + ζ20 + ζ21 + ζ22 + ζ23 + ζ24 + ζ25 Again, reduced by re-expressing the exponents modulo 17: ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 + ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 Put in numerical order of the exponents ζ1 + ζ2 + ζ3 + ζ4 + ζ5 + ζ6 + ζ7 + ζ8 + ζ9 + ζ10 + ζ11 + ζ12 + ζ13 + ζ14 + ζ15 + ζ16 = -1 13 �14 �� 5 The Heptadecagon.nb Cool. Solving for the μ’s Once again, we know the sums and products of pairs of variables, in this case μ1 and μ3 and also μ2 and μ4 : μ1 + μ3 = η1 μ1 × μ3 = -1 μ2 + μ 4 = η 2 μ2 × μ4 = -1 With these sums-and-products, we can make two quadratic equations; we use v and w as variables: v 2 - η1 v - 1 = 0 μ1 , μ 3 = η1 ± whose solutions are μ1 and μ3 η1 2 + 4 2 w2 - η 2 w - 1 = 0 And μ2 , μ 4 = η2 ± whose solutions are μ2 and μ4 η2 2 + 4 2 Just to show where we are at this point, we can identify the values of the μ’s: μ1 = η1 + η1 2 + 4 2 = 1 2 1 2 -1 + 17  + 4 + 14 -1 + 17  μ2 = η2 + η2 2 + 4 2 = 1 2 1 2 -1 - 17  + 4 + 14 -1 - 17  μ3 = η1 - η1 2 + 4 2 = 1 2 1 2 -1 + 17  - 4 + 14 -1 + μ1 = η2 - η2 2 + 4 2 = 1 2 1 2 -1 - 17  - 4 + 14 -1 - 2 ≈ 2.04948 2 ≈ 0.344151 17  2 ≈ 0.487928 17  2 ≈ -2.9057 The Two Periods With this in hand, we look at the two-periods, obtained as before but this time taking every eighth root from the original list: β1 = ζ + ζ 16 β2 = ζ 3 + ζ 14 β3 = ζ 8 + ζ 9 �5 The Heptadecagon.nb β4 β5 β6 β7 β8 = = = = = �� ζ 7 + ζ 10 ζ 4 + ζ 13 ζ 5 + ζ 12 ζ 2 + ζ 15 ζ 6 + ζ 11 It may be worth noting that each pair of roots that make up a β is a complex conjugate pair. This is importanT, although its special importance won’t appear until the next stage. As in previous steps, these two-periods come about by separating elements of the four-periods. Their sums thus lead us back to the variables of the previous step: β1 β2 β3 β4 + + + + β5 = ζ + ζ 16 + ζ 4 + ζ 13 = μ1 β6 = ζ 3 + ζ 14 + ζ 5 + ζ 12 = μ2 β7 = ζ 8 + ζ 9 + ζ 2 + ζ 15 = μ3 β8 = ζ 7 + ζ 10 + ζ 6 + ζ 11 = μ4 We thus have sums of pairs of the β’s. It remains to figure out the products of the same pairs. Products of the 2-Periods With a little labor, we can figure out the products of the 2-periods paired in way given above. The products are given below, including the reduction of the exponents modulo 17: β1 β5 = ζ + ζ16  ζ4 + ζ13  = ζ5 + ζ14 + ζ20 + ζ29 = ζ5 + ζ14 + ζ3 + ζ12 = μ2 β2 β6 = ζ5 + ζ12  ζ3 + ζ14  = ζ8 + ζ15 + ζ19 + ζ26 = ζ8 + ζ15 + ζ2 + ζ9 = μ3 β3 β7 = ζ2 + ζ15  ζ8 + ζ9  = ζ10 + ζ11 + ζ23 + ζ24 = ζ10 + ζ11 + ζ6 + ζ7 = μ4 β4 β8 = ζ6 + ζ11  ζ7 + ζ10  = ζ13 + ζ16 + ζ18 + ζ21 = ζ13 + ζ16 + ζ1 + ζ4 = μ1 Now we have defined the products as well as the sums of the four pairs of 2-periods. Again, the way in which this multiplication works out is not an accident; it follows from Gauss’s original ordering of the roots that combinations taken by twos, by fours, and so forth will always multiply so as to produce these intermediate periods. Constructing Four Quadratic Equations With that in mind, we can construct four quadratic equations with the β’s as roots: q2 - μ1 q + μ2 = 0 whose roots are β1 and β5 15 �16 �� 5 The Heptadecagon.nb r2 - μ2 r + μ3 = 0 whose roots are β2 and β6 s2 - μ3 s + μ4 = 0 whose roots are β3 and β7 t2 - μ4 t + μ1 = 0 whose roots are β4 and β8 Their solutions can be obtained with the quadratic formula. Since we know the values of the μ’s, we can calculate the values of the β’s: β1 = μ1 + μ1 2 - 4 μ2 2 = 1.86494 β5 = μ1 - μ1 2 - 4 μ2 2 = 0.184537 β2 = μ2 + μ2 2 - 4 μ3 2 = 0.891477 β6 = μ2 - μ2 2 - 4 μ3 2 = -0.547326 β3 = μ3 + μ3 2 - 4 μ4 2 = 1.47802 β7 = μ3 - μ3 2 - 4 μ4 2 = -1.96595 β4 = μ4 + μ4 2 - 4 μ1 2 = -1.20527 β8 = μ4 - μ4 2 - 4 μ1 2 = -1.70043 The Singletons One more step remains: dividing the 2-periods into individual roots. This is in some ways the easiest step of all. Their Sums There are sixteen individual roots: ζ, ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 , ζ11 , ζ12 , ζ13 , ζ14 , ζ15 , ζ16 These, take pairwise in a particular order, constitute the β’s: β1 β2 β3 β4 β5 β6 β7 β8 = = = = = = = = ζ + ζ 16 ζ 3 + ζ 14 ζ8 + ζ9 ζ 7 + ζ 10 ζ 4 + ζ 13 ζ 5 + ζ 12 ζ 2 + ζ 15 ζ 6 + ζ 11 Here we see that we already have the sums of the sixteen ζ’s, taken pairwise. Their Products �5 The Heptadecagon.nb �� This time, the product of the roots just as simple as the sums. Since, as already noted above, each β pair constitutes a pair of complex conjugates, their product -- obtained by adding their exponents -- is always seventeen or, on the unit circle in the complex plane, +1. ζ ζ16 = ζ17 = ζ3 ζ14 = ζ17 = ζ8 ζ9 = ζ17 = ζ7 ζ10 = ζ17 = ζ4 ζ13 = ζ17 = ζ5 ζ12 = ζ17 = ζ2 ζ15 = ζ17 = ζ6 ζ11 = ζ17 = 1 1 1 1 1 1 1 1 WIth this information, we have the sum and the products of the roots (taken in this special order) we can construct eight quadratic equations whose solutions are the ζ’s. r 2 - β1 r + 1 = 0 s2 - β 2 s + 1 = 0 t2 - β3 t + 1 = 0 v 2 - β4 v + 1 = 0 w2 - β 5 w + 1 = 0 x 2 - β6 x + 1 = 0 y 2 - β7 y + 1 = 0 z2 - β8 z + 1 = 0 whose solutions are ζ and ζ16 whose solutions are ζ3 and ζ14 whose solutions are ζ8 and ζ9 whose solutions are ζ7 and ζ10 whose solutions are ζ4 and ζ13 whose solutions are ζ5 and ζ12 whose solutions are ζ2 and ζ15 whose solutions are ζ6 and ζ11 We can apply the quadratic formula to find the solutions. Since we have the values for the β’s, we can obtain values for the ζ’s: ζ and ζ16 = β1 ± β1 2 - 4 2 = 0.932472 ± 0.361242 ⅈ ζ3 and ζ14 = β2 ± β2 2 - 4 2 = 0.445738 ± 0.895163 ⅈ ζ8 and ζ9 = β3 ± β3 2 - 4 2 = -0.982973 ± 0.18375 ⅈ ζ7 and ζ10 = β4 ± β4 2 - 4 2 = -0.850217 ± 0.526432 ⅈ ζ4 and ζ13 = β5 ± β5 2 - 4 2 = 0.0922684 ± 0.995734 ⅈ ζ5 and ζ12 = β6 ± β6 2 - 4 2 = -0.273663 ± 0.961826 ⅈ ζ2 and ζ15 = β7 ± β7 2 - 4 2 = 0.739009 ± 0.673696 ⅈ ζ6 and ζ11 = β8 ± β8 2 - 4 2 = -0.602635 ± 0.798017 ⅈ Shown graphically: 17 �18 �� 5 The Heptadecagon.nb ζ4 ζ5 ζ ζ3 6 ζ2 ζ7 ζ ζ8 ζ9 ζ 16 ζ 10 ζ 15 ζ 11 ζ 14 ζ 12 ζ 13 As advertised. Full Algebraic Presentation of the Sixteen Complex Roots of Unity The stack of quadratic equations involved in calculating the heptadecagon vertices is diﬀicult to grasp when they are all assembled into a single formula. One of the roots is represented as: 1 2 1 4 1 2 17 - 17  + 1 8 -1 + 1 17  + 34+6  -4 + 1 64 -1 + 17 + 34 - 2 + 2 4 17 + 578-34 17 +  2 34 + 6 17 - 34-2 17 -8 2 17+ 17 17 + 2 578 - 34 17 - 34 - 2 17 - 8 2 17 + 17  It’s a challenge to grasp such a thing, but even a casual inspection shows that it consists exclusively of stacks of rational numbers and square roots combined with rational functions (addition, subtraction, multiplication and division). That alone is enough to guarantee its constructibility. �5 The Heptadecagon.nb �� 19 An abbreviated graphic representation of the process of constructing the heptadecagon might look like this: ζ1 … 16 = β1, 2, 3, 4, 5, 6, 7, 8 = βn 2 - 4 βn ± 2 μ1, 2, 3, 4 2 - 4 μ2, 3, 4, 1 μ1, 2, 3, 4 ± 2 μ1, 3, 2, 4 = η1,2 = η1,2 ± η1,2 2 + 4 2 17 -1 ± 2 The sixteen roots were subdivided successively as follows. ζ , ζ 3 , ζ 9 , ζ 10 , ζ 13 , ζ 5 , ζ 15 , ζ 11 , ζ 16 , ζ 14 , ζ 8 , ζ 7 , ζ 4 , ζ 12 , ζ 2 , ζ 6 The whole t The μ's The ζ's ζ ζ 16 ζ 13 , ζ 4 ζ 13 ζ 3 , ζ 5 , ζ 14 , ζ 12 ζ 9 , ζ 15 , ζ 8 , ζ 2 ζ , ζ 13 , ζ 16 , ζ 4 ζ , ζ 16 The β's ζ 3 , ζ 10 , ζ 5 , ζ 11 , ζ 14 , ζ 7 , ζ 12 , ζ 6 ζ , ζ 9 , ζ 13 , ζ 15 , ζ 16 , ζ 8 , ζ 4 , ζ 2 The η's ζ4 ζ 9, ζ 8 ζ9 ζ8 ζ 15 , ζ 2 ζ 15 ζ 3 , ζ 14 ζ2 ζ3 ζ 14 ζ 10 , ζ 11 , ζ 7 , ζ 6 ζ 5 , ζ 12 ζ5 ζ 12 ζ 10 , ζ 7 ζ 10 ζ7 ζ 11 , ζ 6 ζ 11 ζ6 The sum of the whole set was -1. The sum of the η’s required a finite quadratic field extension to include -1 ± 17 2 . Each successive subdivision required another finite quadratic field extension of what went before. But (and?) that is the sine qua non of constructibility: a point is constructible if (and only �20 �� 5 The Heptadecagon.nb if) it is defined by numbers that are either rational or the result of a succession of finite, quadratic field extensions from the rationals. Extension This is a pretty remarkable result: the 17-gon is constructible using the techniques available in Euclid’s Elements. In fact, Gauss’s result is even more remarkable than that. It has both a positive and a negative side, which I can report although we haven’t done quite enough work to demonstrate both sides fully. The Positive Side The positive side is that any figure is constructible if it the number of its sides is either, (a) a prime number equal to 2n + 1, or (b) some multiple of 2p times such a number, or (c) the sum of two of the primes in (a), or 2p times that sum. So, 20 = 1, 1 + 1 is 2, which is prime. You can’t make a 2-gon, really, but you can make polygons that are multiples of 2 times it: a 4-gon (square), and 8-gon (octagon), etc. Next, 21 = 2. 2 + 1 is 3, which is prime. You can make a 3-gon (triangle), or a 6-gon (hexagon), etc. A�er that, 22 = 4. 4 + 1 is 5, which is prime. You can make a 5-gon (pentagon). Euclid could do this. These are all the constructible prime n-gons that Euclid knew. He doesn’t say so much, but if he had known the construction of another it is hard to believe he would not have given it. A�er that, 24 = 16. 16 + 1 is 17, which is prime. You can make a 17-gon (heptadecagon). This was Gauss’s great discovery. I cannot believe that Euclid knew that the 17-gon was constructible. But there are more! Consider: 28 = 256. 256 + 1 = 257, which is prime. The 257-gon is constructible. Consider: 216 = 64 536. 64,536 + 1 = 64,537, which is prime. The 64,537-gon is constructible. In fact, if you can find another number of the form 2n + 1 which is prime, it too will be constructible. These are the so-called “Fermat numbers,” named for Pierre Fermat who conjectured that all numbers of the form 2n + 1 were prime, provided that n itself is a power of 2. Such numbers are: �5 The Heptadecagon.nb �� 21 3, 5, 17, 257, 64537, … The next one would be 4,294,967,297 … but this on turns out not to be prime. (It is 641 times 6,700,417). As of last year, only the first eleven such numbers have been fully tested. The last one, 11 22 + 1, has 617 digits; it has two prime factors. The next one has 1,234 digits; whether it is prime or not is still undetermined. Las Vegas is not giving odds. In fact, now it is conjectured that apart from the first five, no other Fermat numbers are prime, although as far as I know that guess hasn’t been proven or disproven. The Negative Side The negative claim is that only these polygons are constructible. Gauss did not demonstrate that his construction method was the only one possible. The negative claim was demonstrated later by Pierre Wantzel in 1837. The seven-gon, which Euclid just skips, cannot be constructed. Neither can the 9-gon, the 11-gon or the 13-gon. Euclid constructs the (non-prime) 15-gon by the combination of the triangle and the pentagon. His leap from the hexagon to the 15-gon is completely unexplained in the Elements, and to my knowledge few students remark on it. (They should.) So What? What follows from this exercise? That is the subject of next week’s talk. Further Reading For further explication of Gauss’s sorting of the roots and modular congruences, you may wish to consult Hadlock, Field Theory and Its Classical Problems (Mathematical Association of America, 1978) � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 21 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The heptadecagon Description An account of the resource Transcript of a lecture given on October 8, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-08 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Polygons Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_The_Heptadecagon_2019-10-08 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/c243d0e09b752feef01b9ac5cc9990b8.pdf 40ff74d84f2a2b7d84d208491d3d62f3 PDF Text Text The Pentagon and the Heptagon Recapitulation Two weeks ago, we saw how Rafael Bombelli began to suspect that imaginary numbers might be meaningful as he worked on the cubic equation x 3 - 15 x - 4 = 0 Using the formula Cardano stole from Tartaglia, got x= 3 2 + 11 -1 + 3 2 - 11 -1 which he was then able to solve by intuiting that 2 + 11 -1 = 2 + 3 -1  . The second lecture described Caspar Wessel’s graphic presentation of the arithmetic of complex numbers. On the complex number plane -(i) complex numbers can be expressed in polar coordinates by giving a distance (modulus) and an angle (argument); (ii) multiplication of complex numbers amounts to (a) multiplication of their distances (moduli) and (b) adding their angles (arguments); and (iii) the solutions of equations of the form x n - 1 = 0, known as the “Roots of Unity,” appear graphically as the vertices of an equilateral n-gon in the unit circle on the complex plane. Last week, we encountered the idea of “Constructible Numbers.” We showed that Euclid’s postulates allowed construction of lengths that correspond to the field of rational numbers, a collection of numbers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to irrational numbers). However, Euclid’s postulates do not permit construction of all incommensurable magnitudes. We can only construct those that correspond to numbers that can be found in towers of finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the countable infinity of the algebraic numbers. 3 �2 bers that is closed under the operations of addition, subtraction, multiplication and division. In addition, Euclid’s postulates allow the construction of incommensurable magnitudes (which correspond to irrational numbers). However, Euclid’s postulates do not permit construction of all incommensurable 4 The Pentagon and the Heptagon.nb magnitudes. We can only construct those that correspond to numbers that can be found in towers of finite quadratic field extensions, that is, field extensions that have a degree of 2n over the rational numbers. Plenty of numbers are not included. For example, 2 is not constructible; neither are the non-algebraic (transcendental) numbers, which form an uncountable infinity far greater than the countable infinity of the algebraic numbers. 3 We now turn to an application of what we have seen so far: construction of a regular pentagon in a given circle. The Lesser-Know Impossibility Problem Ancient geometry knew several classical problems that seemed impossible; the three most famous were trisecting the angle, doubling the cube and squaring the circle. These were daunting challenges. No one had found how to accomplish any of them, but the ancients did not know whether they were really impossible or only difficulties awaiting clever solutions Only in the late 18th and 19th centuries did we learn that these three problems really are impossible, at least with the tools of Euclidean geometry. In addition to these three, lesser-known but equally interesting problem is that of the heptagon, the regular seven-sided polygon. In book IV of the Elements, Euclid shows how to construct in a given circle an equilateral triangle (IV.2), a square (IV. 6), a pentagon (IV. 11) and a hexagon (IV. 15), regular figures with three, four, five and six sides. He then shows how to construct a regular 15-gon (IV. 16). Then he stops. The reader might be expected to wonder, why? Why jump from 6 to 15? Euclid, in his customary laconic way, says nothing. Some of the figures he skips over were easily constructible. The octagon is easily made by bisecting the angles of the square. The 10-gon can be gotten similarly from the pentagon and the 12-gon, from the hexagon. But the orderliness of Euclid’s sequence really falls apart with the 7-gon. With what we learned last week, it is easy to see that the 7-gon is impossible to construct. It is obtained from the polynomial: x 7 - 1 = ( x - 1)  x 6 + x 5 + x 4 + x 3 + x 2 + x 1 + 1 = 0 That sixth-degree polynomial is irreducible and, since its degree over the rationals is not a power of two, we can see right away that these complex roots are not constructible. Did Euclid know that the 7-gon was impossible? He probably suspected it. He surely knew that he couldn’t do it, which is not quite the same thing. Beyond the Impossible: the Unsuspected Possible In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what is mor, he realized at the same time that there are other polygons that can be constructed. Looking only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what Euclid knew. To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic �4 The Pentagon and the Heptagon.nb 3 In 1796, at the age of 19, Carl Friedrich Gauss realized the impossibility of constructing the 7-gon; what is mor, he realized at the same time that there are other polygons that can be constructed. Looking only at those with a prime number of sides, in his book Disquisitiones Arithmeticae, he not only showed that the 17-gon is constructible, he showed how to do it. This is remarkable advance beyond what Euclid knew. To help us get to Gauss’s result, it will be helpful to begin with a slightly simpler project: the algebraic construction of the pentagon. Euclid’s Construction of the Pentagon Of course, Euclid knew how to construct a regular pentagon in a given circle. To begin, let’s review how Euclid’s construction works. First, a Special Triangle He begins with construction of a very special triangle, one that is isosceles and whose base angles are both twice as big as its vertex angle. θ 2θ 2θ A little reflection shows why this triangle might be important to the construction of a regular pentagon: the three angles of the triangle total up to 180°, of course, but they also add up to five times the vertex angle. This triangle creates one angle that is one-fifth of 180°, and two that are one-fifth of 360°. If this triangle can be made, it will be the key to constructing the pentagon. However, constructing this triangle is no simple matter. To make it, Euclid recalls that back in book II, proposition 11, he had shown how to cut a line at a point so that the square on one portion of the line is equal to the rectangle contained by the whole line and the remaining portion of the line. �4 4 The Pentagon and the Heptagon.nb Digression This kind of division is known as one into "mean and extreme ratio," sometimes also referred to as the "Golden Ratio." It has many cool features, including connection to Fibonacci numbers and logarithmic spirals, but we haven' t time to get into all these things right now. If we take the whole AB to be “1” and the distance AC to be “x”, then finding this ratio can be understood as analogous to solving the equation: x 2 = (1 - x ) x2 + x - 1 = 0 or whose solutions are: 1± 1 - 4 (-1) 2 = 1± 5 2 You may note that these values are not rational, since they contains the square root of five. They are, of course, constructible, which we know because (a) we are dealing with a quadratic extension of the rationals and (b) because Euclid in fact constructs one of them. (No surprise there.) �1± 1 - 4 (-1) 2 = 1± 5 4 The Pentagon and the Heptagon.nb 2 5 You may note that these values are not rational, since they contains the square root of five. They are, of course, constructible, which we know because (a) we are dealing with a quadratic extension of the rationals and (b) because Euclid in fact constructs one of them. (No surprise there.) Returning to the Construction Euclid takes a line divided in this way and, using one end as a center, draws a circle with the whole line as a radius: A C B He then makes a chord in the circle equal to the larger segment of the divided line: A C B D He completes the triangle ABD, and joins CD: �6 4 The Pentagon and the Heptagon.nb A C B D Finally, he draws a circle that goes through points A, C and D: A C B D Thanks to a proposition from earlier in Book III, he knows that when from a point outside a circle (like point B) a line cuts a circle (as line BCA), and another line is draw to the circumference of the circle (as line BD), and when the rectangle on AB, AC is equal to the square on BD, then the line (BD) is tangent to the circle (ACD). With that established, another proposition of Book III allows him to say that the angle CDB (angle 1) is equal to the angle CAD (angle 2): �4 The Pentagon and the Heptagon.nb A C 2 B 4 5 3 1 D Add angle CDA to both. Thus angles 2 + 3 are equal to angles 1 + 3. But because AB = AD (in the circle around A), angles 1 + 3 are equal to angle 5 . So: angle 5 = angles 1 + angle 3 = angle 2 +angle 3 and because of exterior angles in triangle CBD angle 4 = angle 2 + angle 3 Therefore, triangle ABD is isosceles and line DB = line DC. And line DB = line AC. Therefore, angle 3 = angle 2 = angle 1. This, then, is the isosceles triangle with its base angles equal to twice the vertex angle. The Pentagon With the isosceles triangle having the base angles equal to the vertex angle now available, the rest is easy. 7 �8 4 The Pentagon and the Heptagon.nb Simply bisect the arcs standing on the longer sides, which are each twice the arc on the shorter side. Now you have five equal sides and your pentagon is complete. Join the vertices and you have not only a pentagon, but a pentangle (a regular five-pointed star). �4 The Pentagon and the Heptagon.nb 9 This construction is completely rigorous and very clever. However, it offers no clues at all about how to pursue construction of other such prime-sided polygons, such as the 7-gon, the 11-gon, the 13-gon, etc. Preliminary: the Pentagon The algebraic construction of the pentagon amounts to finding the roots of the fifth degree cyclotomic polynomial. That is, we begin with the equation: x5 = 1 or x 5 - 1 = 0. The number 1 is evidently a solution to this equation. It is, in fact, the only rational solution. Therefore, the equation can be factored by removing the factor (x - 1): x 5 - 1 = ( x - 1)  x 4 + x 3 + x 2 + x + 1 = 0 The Fifth Order Cyclotomic Polynomial The second expression,  x 4 + x 3 + x 2 + x + 1, is irreducible “over the rationals”; that is, it can’t be simplified by showing it to be the product of factors of lower degree among the rationals. We can be completely sure that this expression is irreducible because we know that the four roots of the polynomial x 4 + x 3 + x 2 + x + 1 = 0 are complex with imaginary components. They are the four non-real fifth roots of unity. �10 4 The Pentagon and the Heptagon.nb ζ1 ζ2 ζ3 ζ4 But being irreducible over the rationals doesn’t mean that this thing can’t be factored in an extended field. In fact, it has been shown that every polynomial of nth degree can be factored into n linear factors in the full complex number field. Our challenge is to find which factors need to be appended to the rationals in order to factor or “split” this fourth degree polynomial. Complex Conjugates We haven’t discussed complex conjugates, but this diagram presents the idea nicely. Notice that the complex roots of this polynomial appear as two pairs of complex numbers, symmetrically arranged above and below the real number axis. Root ζ1 is paired this way with root ζ4 and root ζ2 is paired with root ζ3 . Being so arranged, these roots are written in this form: a+bi and a - bi The expressions are the same except for the positive and negative signs attached to the imaginary portions. Complex conjugates have this handy feature: when a pair of complex conjugates are added, their sum is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a real number. This feature is handy because we are often looking for roots of polynomials whose coefficients are rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots: (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = … But if you multiply this trio out, you will have some imaginary coefficients. �is a real number. Also, when a pair of complex conjugates are multiplied together, their product is a real number. 4 The Pentagon and the Heptagon.nb 11 This feature is handy because we are often looking for roots of polynomials whose coefficients are rational (or, in any case, do not involve imaginaries). Of course, you can construct an arbitrary polynomial with a random selection of complex roots: (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = … But if you multiply this trio out, you will have some imaginary coefficients. (x - (2 + 7 i)) (x - (9 - 3 i)) (x - (-15 + 4 i)) = (813 + 699 ⅈ) - (142 - 41 ⅈ) x + (4 - 8 ⅈ) x2 + x3 In fact, the only way to eliminate the imaginary components from the expanded polynomial is if the coefficients occur in pairs of complex conjugates. That way, when the conjugates are multiplied, the imaginary components disappear. Return to the Problem To solve our fourth-degree cyclotomic polynomial: 1 + x + x2 + x3 + x4 = 0 We will proceed in the usual, brash algebraic way: we will pretend that we already have the solutions. Then we’ll work to discover what they are. The Fundamental Theorem of Algebra tells us that this fourth degree equation has four solutions, which we will designate (as in the picture) ζ1 , ζ2 , ζ3 and ζ4 . Roots ζ1 and ζ4 are one pair of complex conjugates; ζ2 and ζ3 are another pair. Two-Stage Solution Take the sums of ζ1 , ζ4 and of ζ2 , ζ3 , like this: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 When added together, η1 and η2 sum up to -1 (because all the fifth roots of unity together sum to zero, and η1 and η2 include all the roots except (+1 + 0 i): η1 + η2 = ζ 1 + ζ 4 + ζ 2 + ζ 3 = -1 Also, the product of η1 and η2 works out like this: ζ 1 + ζ 4  ζ 2 + ζ 3  = ζ 3 + ζ 4 + ζ 6 + ζ 7 Restate this result with the exponents taken Modulo 5, because, on the unit circle in the complex plane, ζ 5 = ζ 0 = 1. Thus, we have ζ6 = ζ5 ζ1 = ζ1 ζ7 = ζ5 ζ2 = ζ2 Substitute: �12 Restate thisand result with the exponents 4 The Pentagon the Heptagon.nb 5 taken Modulo 5, because, on the unit circle in the complex 0 plane, ζ = ζ = 1. Thus, we have ζ6 = ζ5 ζ1 = ζ1 ζ7 = ζ5 ζ2 = ζ2 Substitute: ζ3 + ζ4 + ζ6 + ζ7 = ζ3 + ζ4 + ζ1 + ζ2 = -1 Presto! We have the sum of the four non-real roots of the equations x 5 - 1 = 0. We know that these sum to -1. Building a Quadratic Equation for η1, η2 Great! We have two terms, η1 and η2 . We don’t know what they are, but we do know that their sum is -1 and their product is also -1. Does that sound like a familiar situation? When we know that when we know the sum and product of two terms, we can construct a quadratic equation that has these terms as roots. In this case, we have: x2 + x - 1 = 0 whose roots are given by the quadratic formula: η1 and η2 = -1 ± 1+4 2 = 1 2 -1 + 5  and 1 2 -1 - 5 . (The approximate values of these are 0.61803 and -1.61803.) Behold! Now It Factors! Remember that we said that the expression x 4 + x 3 + x 2 + x + 1 = 0 could not be factored over the rationals? Now it can be factored in an extended field when we append 12 -1 + append 5  -- or even if we just 5 -- to the rationals. We have: ( x - ζ1 ) ( x - ζ4 ) = x 2 - ζ1 x - ζ4 x + ζ1 ζ4 ( x - ζ1 ) ( x - ζ4 ) = x 2 - (ζ1 + ζ4 ) x + ζ1 ζ4 = x 2 - (ζ1 + ζ4 ) x + 1 = η1 This expression, x 2 - (ζ1 + ζ4 ) x + ζ1 ζ4 , has coefficients that are in the extended field. The = η1 coefficient of x is the sum of the two roots ζ1 + ζ4 ; we don’t know them individually yet, but we know that they sum to η1 , which is in the extended field. The constant term is ζ1 ζ4 ; we know right away that the product of these two is 1 (product of their moduli, sum of their arguments). �4 The Pentagon and the Heptagon.nb 13 The Four Singletons Now look at the four roots individually: ζ1 , ζ2 , ζ3 , ζ4 We know how they sum in pairs: η1 = ζ 1 + ζ 4 η2 = ζ 2 + ζ 3 We also know the products of the same pairs : ζ1 ζ4 = ζ5 = 1 ζ2 ζ3 = ζ5 = 1 So we can make two more quadratic equations: w2 - η1 w + 1 = 0 whose roots are ζ1 and ζ4 , which are solved as w = η1 ± y 2 - η2 y + 1 = 0 whose roots are ζ2 and ζ3 which are solved as y = η2 ± η1 2 - 4 2 η2 2 - 4 2 We now have enough information to solve for the four roots: ζ 1 ζ 4 ζ 2 ζ 3 = = = = η1 + η1 2 - 4 2 η1 - η1 2 - 4 2 η2 + η2 2 - 4 2 η2 - η2 2 - 4 2 1 = 2 2 2 -1+ 5  - 1 -1- 5  + 1 2 -1- 5  - = -0.809017 + 0.587785 ⅈ 2  2 -1- 5  - 4 2 = 0.309017 - 0.951057 ⅈ 2  2 -1- 5  - 4 1 = 0.309017 + 0.951057 ⅈ 2  2 -1+ 5  - 4 2 1 = 2 2 1 = 1  2 -1+ 5  - 4 2 1 = -1+ 5  + = -0.809017 - 0.587785 ⅈ You can see that these solutions contain radicals of radicals. These expressions are not in the first extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that it includes these four solutions. �14 4 The Pentagon and the Heptagon.nb You can see that these solutions contain radicals of radicals. These expressions are not in the first extended field, but we can extend that field again (in a finite quadratic algebraic field extension) so that it includes these four solutions. These can be plotted on the complex plane: 0.31 + 0.95 ⅈ -0.81 + 0.59 ⅈ -0.81 - 0.59 ⅈ 0.31 - 0.95 ⅈ Voila. More Important Than the Answer To summarize and review. More important that getting the answer or than drawing the pentagon is to notice how the field extensions were built. Beginning with the rationals, which are all constructible, we first got the values for η1 and η2 , which were the sums of ζ 1 + ζ 4 and ζ 2 + ζ 3 respectively, the two pairs of complex conjugates. These values were 1 2 -1 ± 5 , and thus required that we move into an extended field: Q ⟶ Q(η1, 2 ) This is a quadratic extension and is thus constructible. Then, getting the four roots themselves required another field extension. The four roots are η1,2 ± η1,2 2 - 4 2 , and each will require one more quadratic field extension. Q ⟶ Q(η1, 2 ) ⟶ Q(η1, 2 , η1,2 ± η1,2 2 - 4 2 ) Sequences of quadratic field extensions are constructible. Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots. 1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs �4 The Pentagon and the Heptagon.nb 15 Look again at what is happening here. At the outset, we knew that we had a fourth degree equation with all complex roots. 1 + x + x2 + x3 + x4 = (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) By segregating out the pairs of complex conjugates, we separated the factors on the right into two pairs : 1 + x + x2 + x3 + x4 = {(x - ζ1 ) (x - ζ4 )} × {(x - ζ2 ) (x - ζ3 )} 1 + x + x2 + x3 + x4 = x2 - (ζ4 + ζ1 ) x + ζ1 ζ4  × x2 - (ζ2 + ζ3 ) x + ζ2 ζ3  1 + x + x2 + x3 + x4 = x2 - η1 x + 1 × x2 - η2 x + 1 Is this interesting? Yes! If we confine ourselves to rational numbers, then our original equation could not be factored. If we admit η1 and η2 , it could be factored into two factors. If we admit all the complex numbers -- really, we needed go so far; a finite field extension adding η1,2 ± η1,2 2 - 4 2 to the mix would be enough -- then it factors into four factors: In Q 1 + x + x2 + x3 + x4 In Q(η1, 2 ) "" factors to x2 - η1 x + 1 × x2 - η2 x + 1 In Q(η1, 2 , η1,2 ± η1,2 2 - 4 2 ) “” is irreducible factors to (1 - ζ1 ) (1 - ζ2 ) (1 - ζ3 ) (1 - ζ4 ) The procedure we have followed does exactly what is required for specifying constructible figures: it has made a sequence of finite field extensions, starting with the rationals, Q, and proceeding by quadratic field extensions until the polynomial with our desired points as roots is completely factored. This stepwise factorization works for the pentagon because at each step it was possible to subdivide the roots into two groups, each of which could be shown to be a quadratic expression of the preceding group. That is not always possible. Conclusion We have seen here an application of the technique of algebraic decomposition. The equation we are trying to solve is broken into simpler and simpler parts as the field in which we operate is expanded step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors. Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which polygons are constructible and which are not. We will see this method play out on a larger stage next week with the construction of the hep- �16 We have seen here an application of the technique of algebraic decomposition. The equation we are trying to solve is broken into simpler and simpler parts as the field in which we operate is expanded 4 The Pentagon and the Heptagon.nb step-by-step until we arrive at a final field, the “splitting field,” in which the polynomial can be completely decomposed into linear factors. Unlike Euclid’s way of working, this methodical procedure provides a framework for evaluating which polygons are constructible and which are not. We will see this method play out on a larger stage next week with the construction of the heptadecagon. Thank you. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 16 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The pentagon and the heptagon Description An account of the resource Transcript of a lecture given on October 1, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-10-01 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Polygons Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_The Pentagon_and_the_Heptagon https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/475e77346471eb8b8d17111a59ad06cb.pdf e7795fa0d668289e15620ac4c85c0693 PDF Text Text Constructible Numbers Grant Franks June 3, 2019, revised September 21, 2019 Introduction Recapitulation Two weeks ago, we saw that Rafael Bombelli confronted the possibility that the square root of negative one might have some positive significance when he solved the cubic equation x 3 - 15 x - 4 = 0 and, using the formula Cardano stole from Tartaglia, got x= 3 2 + 11 -1 + 3 2 - 11 -1 which he was then able to solve by intuiting that 2 + 11 -1 = 2 + 3 -1  . Last week, we followed Caspar Wessel in developing a graphic understanding of the arithmetic of complex numbers. In particular, we saw that (i) complex numbers can be expressed in polar coordinates by giving a distance (modulus) and an angle (argument); (ii) multiplication of complex numbers amounts to (a) multiplication of their distances (moduli) and (b) adding their angles (arguments); and (iii) the solutions of equations of the form x n - 1 = 0, known as the “Roots of Unity,” appear graphically as the vertices of an equilateral n-gon in the unit circle on the complex plane. So far, so good. Today’s Agenda �2 �� 3 Constructible Numbers.nb Today, we’re going in diﬀerent direction. Today, we will talk about constructing numbers and the emptiness of the Euclidean plane. The Geometrical Plane The Awkward Question “Are there holes in the geometric plane?” This is a question that rarely gets asked when doing geometry. There is no reason to raise such a question until some awkward questions get asked, and there is no way to answer the question without being able to step back from intuitively given geometrical space by thinking algebraically. Construction In Euclid’s geometry, one begins with five postulates, three of which authorize constructions: Postulate One: to draw a line between any point and any other point; Postulate Two: to continue a line indefinitely; and Postulate Three: to draw a circle with any center and radius. The references to “any point,” “any center” and “any radius” give the impression that the Euclidean plane contains all possible points. But that isn’t quite true or, to be more precises, there are many points in a plane to which Euclidean geometry provides no access. For instance, suppose you wanted to “square the circle.” This is a classical problem that amounts to drawing a rectangle with an area equal to that of a given circle. Archimedes shows that such a rectangle would have a height equal to the radius of the circle and base equal to half its circumference. If you can draw a circle with any center and radius, could you draw one whose radius is equal to the semicircumference of the circle you are trying to square? If you could do that just by saying it, your problem would be solved. That would not satisfy a mathematician. She or he would want to know how to get that radius length, that is, how to construct it. Beginning with the one length that is given -- the radius of the circle to be squared -- and using only permissible manipulations, how can we construct the desired straight line with a length equal to the semi-circumference of the given circle? The Meno Problem This problem sounds a little like the geometrical problem of the Meno: Socrates asks the slave boy, “If you are given this square with these sides, can you show the side of a square with double the area?” The slave boy is stumped. Some modern students think they have a better answer than the slave boy and say, “That’s easy! It’s the square root of two!” They don’t realize that they are not giving an �3 Constructible Numbers.nb �� 3 answer at all. The phrase “square root of two” is just a slightly shortened version of “the magnitude which, when multiplied by itself, gives two.” So, to Socrates’ question, “What is the magnitude which, when multiplied by itself, yields two?” they have answered, “The magnitude which, when multiplied by itself, gives two.” True, no doubt, as any tautology is true, but it doesn’t really advance our knowledge of, well, anything. Socrates provides what is really needed: a geometrical construction for finding the square root of two by looking at the diagonal of a square with sides equal to one. (Really, he gives the construction for 2 2 because his original square had sides equal to two, but that is minor detail.) For the squaring of the circle, we want a construction for a straight line whose length is equal to the semidiameter of the given circle. Alas, not all that humankind desires does it obtain! There is no Euclidean construction for that length. The story that leads to that result culminates in the late nineteenth century. It involves Ferdinand Lindemann’s demonstration of that π is not just an irrational magnitude -- that had been known for over a century (Johann Lambert 1761) -- but that it is a particular kind of irrational magnitude. Getting to that result, if anyone here is interested in it, would require work that goes beyond what this series of talks will cover. If it is any consolation, however, the ideas we will cover tonight are necessary preliminaries to that work. The Delian Problem Once upon a time, a long time ago, a great plague aﬀlicted the city of Delos. The citizens consulted Apollo’s oracle at Delphi who told them that the god was dissatisfied with the altar of his temple. The altar was made in the shape of a cube, and the oracle said that the god wanted an altar twice as big. The citizens, eager to be rid of the plague, got a great piece of marble and built an altar twice as long, twice as deep and twice as tall as the one that was there. The plague continued. The priest of the oracle corrected the people saying the god wanted an altar with twice the volume of the present one. The newly built altar had eight times the volume, and was not what was wanted. The people were understandably annoyed, but everyone in the ancient world knew that gods love to mess with people by issuing weirdly misleading oracular pronouncements. Gods are cruel, that’s all there is to it. Geometers realized immediately that what was needed was, in eﬀect, an altar whose side was 2 times bigger than the present one. When they set out to design it, however, they found that the god had been even crueler than expected. They couldn’t figure out how to construct the altar. Finding a construction for 2 was easy, but finding one for the 2 was surprisingly diﬀicult. 3 3 To get to the bottom of the problem that they faced, we have to sketch out a new kind of algebraic operation, finite field extensions. Ordinarily this would be a semester-long study, but since this is St. John’s College, I will try to compress it into about fi�een minutes. �4 �� 3 Constructible Numbers.nb Rational Operations What is a Field? To begin, we need to define “a field.” For our purposes, a “field” is a collection of objects -- we’re going to be talking about numbers -- that are closed under the operations of addition (and its inverse, subtraction) and multiplication (and its inverse, division). Consider first, then, the whole numbers: 1, 2, 3, …. These are closed under addition, that is, if you add any two whole numbers you get a whole number. But they are not closed under subtraction. Although you can subtract 5 from 7 to get 2, you cannot subtract 7 from 5. So set aside the whole numbers and take up the integers: … -3, -2, -1, 0, +1, +2, +3, …. Now you have closure under addition and subtraction. You also have closure under multiplication but not division. 10 divided by 5 is 2, but 10 divided by 3 is not among the integers. So set aside the integers and take up the rational numbers: … -1 -1 -2 , 5 , 11 , 3 …0… 1 , 2, 5 10 7 3 …. That is, all the numbers made up by ratios of integers with one another (forbidding division by zero). Now we have it: this is a field. It is closed under addition and subtraction, it is closed under multiplication and division. For convenience’ sake, we will give the rational numbers a symbolic name, Q. (Why Q and not R? Because R is reserved for the real numbers. Alas.) Euclid’s Operations Allow Us to Form a Field The operations of Euclid’s geometry allow us to construct lengths on a line that correspond to the field of rational numbers. If we begin with a given length that we will call the “unit,” we can with straightedge and compass easily make a double length, a triple length, etc. If we define “negative” to be mean motion in one direction from an arbitrary starting point and “positive” to mean going in the opposite direction, we can construct lengths corresponding to all integers. By an easy construction, we can also divide our given unit length into equal parts corresponding to any whole number. Thus we can make lengths corresponding to any positive proper fraction; by multiplying these we can make any proper or improper fraction, and by directing them toward the negative side of our arbitrary zero point, we can identify places corresponding to any rational number. The lengths from zero to these points can be added, subtracted, multiplied and divided at will and the result will always be another rational length. We have a field. If we erect two such lines at right angles to one another, we can locate and label any point on a plane �3 Constructible Numbers.nb �� 5 that corresponds to (a, b), where a and b are rational numbers. As my grandfather used to say, “Now we’re cookin’ with gas!” Other Lengths All points with rational coordinates is a lot of points, but we know that there are other lengths that can be found in Euclidean geometry. There is, for example, 2 , which is the diagonal of the square with sides of unit length. In fact, we can construct lengths equal to the square roots of any lengths we can find through other means. D A C B If you want to find a length equal to r , draw line AB in length equal to r + 1. Here, let Ac = r and let CB = 1. Erect a semi-circle on line AB and a perpendicular at C meeting the semicircle at D. Join AD and DB. Triangle ADC is similar to triangle DCB and to the combined triangle ADB. Therefore: AC : CD :: CD : CB AC × CB = CD2 But AC = r and CB = 1; therefore: r × 1 = 2 = CD2 r = CD. Combinations A little examination will show that these are all the operations that are available to us. We have addition, subtraction, multiplication, division -- these are suﬀicient to find any rational lengths. In addition to this, we can take the square root of any length that we can find. Not only that: we can do so as many times as we please. So, can construct 2 , or 5 , or 17 3 , or of any rational length. And that’s not �6 �� 3 Constructible Numbers.nb all! We can construct 2 . Or 2+ 2 . Or 17 3 + 2+ 7 3 … or any sequence or combina- tion of the rational operations and repeated extraction of square roots. With these techniques in hand, someone might easily jump to the conclusion that these Euclidean operations can construct any length whatsoever. That’s probably what I would have said if anyone had asked me back when I was a Johnnie Freshman more years ago than I care to think about. But I would have been wrong. Jumping to Conclusions When I learned about the Pythagorean theorem and irrational numbers (or their equivalents, incommensurable lengths), I didn’t take time to think about these new numbers as carefully as, in retrospect, I should have. Looking back, I think my understanding ran something like this: “We had the whole numbers, but they weren’t enough to do subtraction so we added the negative numbers and got the integers. But the integers weren’t enough to do division, so we added the fractions and got the rational numbers. But even the rationals weren’t enough to account for all the lengths we could find in geometry -- the 2 is irrational! (Hey! I was just as surprised by this as the Greeks were!) -- so we added the irrational numbers to the rational numbers and now we have all the real numbers, which is all that there are!” That understanding didn’t get challenged for decades until I began working on a preceptorial on Galois Theory and Professor Charles Hadlock, author of Field Theory and Its Classical Problems, introduced me to finite field extensions. It was here that I learned the humbling lesson that not all irrational numbers are the same. Some numbers are more irrational than others, and lumping them all together blurred together distinctions that are best kept separate. Baby Steps Let’s start over. Go back to when we had just the rational lengths and could find any point with rational coordinates. Now, we read the Meno and find out about 2 . Instead of pretending that we are now able to generate all possible irrational lengths, look carefully at what we have. We can make any rational length, and we can make the square root of two. If we continue now to use just with our rational operations (+, -, x, ÷) on the two lengths we have at hand, 1 and 2 , we can make any number that looks like this: a+b 2 where a and b are rational numbers. Notice something important about these numbers: we can add them, subtract them, multiply them, and divide them any way we please and we always get other �3 Constructible Numbers.nb �� 7 numbers of this same kind. So, if we have: 3 + 5 2  + -1 + 7 2  = 2 + 12 3 + 5 2  × -1 + 7 2  = -3 + 21 2. or 2 -5 2 2 + 35  2  = 67 + 16 2 Division is a bit more complicated, but it works as well. The upshot is this: the numbers a + b 2 form a field of their own. This new field is called an extension field. Because we added a finite number of elements to form it (in this case, just one), it is called a finite field extension. And because the element we added was a solution of a polynomial with elements of the original field as coeﬀicients -- in this case, x 2 - 2 = 0 -- it is called a finite algebraic field extension. And because it was made by adding an element that is the square-root of a member of the original field, it is called a quadratic finite algebraic field extension. Let’s call it F1 and write F = Q( 2 ) to signify that F was formed by appending 2 to Q and making all the numbers of the form a + b 2 where a and b are elements of Q. F is big. It’s bigger than Q, the rational numbers. It includes Q as a subset, so we write: F1 ⊃ Q. But F does not include all the numbers (lengths) that we can construct because we can adjoin other elements if we wish. We can even take the square root of some squirrely element of F1 that already has a square root of two, say: 3+7 2 We can append this element to F1 and form the numbers: c+d 3+7 2 where c and d are elements of F. This is a quadratic finite algebraic field extension of F1 . Let’s call it F2 and write F2 = F1 ( 3+7 2 ) to signify that F2 was formed by appending ing all the numbers of the form c + d 3+7 3+7 2 to F1 and mak- 2 where c and d are elements of F1 . F2 is big. It’s bigger than F1 and much bigger than Q. It includes F1 as a subset, so we write: F2 ⊃ F1 ⊃ Q. Do you see where this is going? We can continue this process as long as we wish. �8 �� 3 Constructible Numbers.nb … F5 ⊃ F 4 ⊃ F 3 ⊃ F 2 ⊃ F 1 ⊃ Q When I put them all together, I have a tower of finite quadratic field extensions. Every number that corresponds to every possible constructible length is somewhere in that tower. Altogether, they are called the constructible numbers. The set of constructible numbers is very big. But it’s not everything. There are Non-Constructible Numbers There are, as it turns out, non-constructible numbers, as can be shown in several ways. For instance, 2 (the real cube root of two) is not a constructible number. It does not belong to any tower of quadratic field extensions over the rationals. 3 For, proceeding in the time honored way of reductio ad absurdum, suppose that 2 were constructible. Also remember that, since in the real numbers y = x 3 is strictly increasing, there is only one real cube root of two. Now, if 2 were constructible, it would belong to some quadratic field extension of a field that was itself part of a tower of quadratic field extensions leading back to the rationals. 3 3 2 ∈ Fn 3 That means that include 2 . 3 where 2 =a+b Fn ⊃ Fn-1 ⊃ Fn-2 ⊃ Fn-3 ⊃ … F1 ⊃ Q c , where a, b and c are all parts of Fn-1 , but where Fn-1 does not itself 3 Cube both sides of this equation. 3  2  = a + b 3 3 c  = a3 + 3 a 2 b 2 = (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c ) c + 3 a b 2 c + b3 c c c The number 2 is a part of Fn - 1 ; we know this because it is a member of Q. Therefore, it has no component multiplied by c , which means that ( 3 a2 b + b3 c ) = 0. Next consider (a3 + 3 a b2 c ) - ( 3 a2 b + b3 c ) c (notice the minus sign). Since 3 a2 b + b3 c = 0, this 3 has the same value as (a3 + 3 a b2 c ) + ( 3 a2 b + b3 c ) c . But it unpacks into a - b c  . So we have two cube roots of 2: a+b c a-b c But there is only one value ; therefore b = 0. That means that a + b c is really just a, and 3 2 is in �3 Constructible Numbers.nb �� 9 Fn - 1 . By the same reasoning, it is a member of Fn - 2 and Fn - 3 and so on until we discover it is a member of Q, that is, that it is rational. But 3 2 is NOT rational. (Those who doubt this can see Appendix 1.) Thus, 2 is NOT in ANY tower of quadratic field extensions beginning with the rationals. It is not constructible. 3 The Delian Problem is Thus Solved At this point, the architects and engineers at Delos should despair: if 2 is not constructible, then they cannot double the size of the altar of Apollo with Euclidean mathematics. The gods are cruel, but they are are mathematically well-informed. 3 There are Lots of Non-Constructible Numbers. The demonstration that the 2 is not constructible is all well and good, but it is rather ad hoc. It doesn’t immediately produce any broad conclusions about constructible vs. non-constructible numbers. 3 A slightly more detailed investigation of field theory allows broader conclusions. It is possible to characterize the size or “degree” of one algebraic field extension over another. Compare, for example, the quadratic extension that results from adjoining 2 to the rationals with what would be called the “cubic” extension that occurs when you adjoin 2 . In the first case, we can express any number in the extended field by an expression that looks like this: 3 2. a+b These numbers form a field. You can add, subtract, multiply and divide to your heart’s content and never leave the field. If you try this with 2 , however, a problem arises. Form a number like: 3 a+b 3 2. You can add and subtract alright, but as soon as you start multiplying, you’ll find yourself running into: 3 2 × 3 2 = 3 4 The cube root of four is not the same as the cube root of two. It can’t be expressed by combinations of rational numbers and the cube root of two. It is outside of the (purported) field. This problem did not arise with 2 because 2 × 2 =2 �10 �� 3 Constructible Numbers.nb which is within the field defined by a + b 2 . With the cube root, it is not enough to add one term; you need to add two. To get a field that includes 2 , you need numbers of this form: 3 a+b 3 2 +c 3 4. A little experimentation will persuade you that these numbers do form a field. Notice that when you formed a extended field with 2 , your new numbers had two terms, a + b 2 . With 2 of two, your new numbers have three terms. The quadratic extension is of “degree two,” while the cubic extension is of “degree three.” Field extensions can be compounded: an extension of degree two followed by an extension of degree three will yield an extended field of degree six over the original field. Field extensions can get remarkably complex, but for our purposes it will be enough to focus on relatively simple extensions of relatively small degrees. 3 3 The degree of an extension is measured by complexity of the minimal polynomial needed to produce the new elements whose addition to the original field leads to the extension. Determining whether a polynomial is “minimal” poses some problems, but this approach can produce sweeping knowledge about whole classes of extensions. So, for instance, every quadratic field extension over the one before it, so that a tower of quadratic field extensions -- that is, the collection leading to any constructible numbers -- will have powers 2, 4, 8, 16 … 2n over the rationals. At the same time, it can be shown that for equations of the form: xn - 2 = 0 the number n gives the degree of the extension resulting from appending one of the solutions of the equation to a field. This result allows is to know that Solutions of Appended to Q are of degree And thus are generally 2 2 Constructible 2 3 Not Constructible - 2 4 2 4 Constructible - 2 5 2 5 Not Constructible x6 - 2 6 2 6 Not Constructible x7 7 2 7 Not Constructible x2 I.e. - 2 2 x3 - 2 3 x4 x5 - 2 Also, it can be show that the solutions of the equations for the nth roots of unity, a�er factoring out (x 1), are minimal when n is prime. So Roots of structible Roots of structible x3 - 1 = 0 appended to Q are of degree 2 over the rationals, so con- x5 - 1 = 0 appended to Q are of degree 4 over the rationals, so con- �3 Constructible Numbers.nb Roots of structible Roots of structible Roots of structible Roots of structible Roots of structible etc. �� x7 - 1 = 0 appended to Q are of degree 6 over the rationals, so NOT con- x 11 - 1 = 0 appended to Q are of degree 10 over the rationals, so NOT con- x 13 - 1 = 0 appended to Q are of degree 12 over the rationals, so NOT con- x 17 - 1 = 0 appended to Q are of degree 16 over the rationals, so con- x 19 - 1 = 0 appended to Q are of degree 18 over the rationals, so NOT con- 11 Many -- indeed, most -- of these create extensions whose degrees are not powers of two over the rationals. Thus they create field extensions filled with numbers that are not constructible. Literally infinite fields of non-constructible numbers emerge. Hierarchy of Irrationals Viewing all the irrationals as an undiﬀerentiated mob is a mistake. We can distinguish between those irrationals that are constructible and those that are not. The constructible numbers are built by successive quadratic field extensions starting from the rationals. Non-Constructibles Constructibles Rationals The distinction between the constructible and non-constructible numbers is interesting enough, but situation is even stranger than that. As we have seen, the Constructibles are made from towers of field extensions all of degree two. We �12 �� 3 Constructible Numbers.nb have just seen that many polynomials have solutions which, when appended to a field, give an extension of degree other than two; the equation x 3 - 2 = 0, for instance, has solutions of degree 3, which takes us away from the powers-of-two towers of constructible numbers. Suppose we toss away that restriction. Suppose we consider all algebraic field extensions of any degree. What if we allow ourselves to build towers in which each step can be of any degree -- that is, to append to a field the solutions of a polynomial of any degree. In this way, we could make towers of fields that include the solutions of any finite polynomial equations. That would include all the constructible numbers and much, much more. This immense collection is known as the algebraic numbers. It includes the rational numbers and the constructible numbers and much, much more. It does not, however, include everything. There are numbers that are not included among the algebraic numbers. You know a few: π is not an algebraic number. Neither is e, the base of the natural logarithm system. Leibniz and later Euler called these non-algebraic numbers “transcendental numbers,” a wonderfully mystical “woo-woo” name that stuck and is in common use today. Transcendentals Algebraics Constructibles Rationals There are LOTS of Transcendental Numbers When I name π and e as transcendental numbers, you may be misled into thinking that there are only a few such numbers and that each of them is a precious rarity, much treasured by mathematicians like these two specimens. Au contraire! Far from being scarce, the transcendental numbers not only surpass all other numbers in �3 Constructible Numbers.nb �� 13 quantity, they do so by an infinite amount. Of course, to characterize one infinity as greater or lesser than another is a controversial project first pioneered by nineteenth century mathematician Georg Cantor. According to Cantor, the smallest sort of infinity is like that of the natural numbers which can be ordered in such a way that one can count oﬀ the members of an infinite set sequentially and be sure eventually to encounter every member. The natural numbers are obviously countable in this way, as are the integers if we number them like this: 0 +1 -1 +2 -2 +3 -3 … 1st 2nd 3rd 4th 5th 6th 7th … It takes a little more work to see that the rationals can be placed in countable order (they can), and a bit more still to figure out that the algebraic numbers can also be ordered and counted. But they can. Cantor designates this infinity by the symbol ℵ0 . The complete collection of real numbers -- and also the collection of all complex number a + b i where a and b can be any real number -- cannot be so ordered. These numbers, according to Canto, form a higher degree of infinity, the infinity of the continuum, ℵ1 which is widely take to be equivalent to 2 ℵ0 , a quantity distinctly diﬀerent, and distinctly bigger than ℵ0 -- insofar as “bigger” is a concept applicable to infinities. From this perspective, the relation of the algebraic numbers to the entire set of complex numbers is pretty much what Ptolemy would call “the ratio of a point to a line.” The whole realm of all our manipulations, geometric and algebraic, occur in a vanishingly small subset of the totality of real numbers. Yet though we speak of infinities, do not imagine that these transcendentals are far away. They are not far away in heaven, so that you have to ask, “Who will ascend into heaven to get them?” Nor are they beyond the sea, so that you have to ask, “Who will cross the sea to get them?” No, they are very near to you always on every side, crowding about with incredible density. And the net of constructible numbers seems now to spread across the Euclidean plane like ever-thinning gossamer network of barely perceptible points, each separated from the next by gulfs teeming full with inaccessible points. While we all contemplate the miserable smallness of all our endeavors, let us take a brief break and then there will be time for questions. Appendix 1: Irrationality of 2 3 This result follows from Euclid, Book X, proposition 9. It can also be shown from arithmetic principles as follows: Suppose that 3 2 is rational. Then it can be expressed as a fraction numbers. We may assume also that the fraction a b a b where a and b are finite whole is expressed in lowest terms, so that a and b have no �14 �� 3 Constructible Numbers.nb factors in common. We have: 3 a b 2 = 2= a3 b3 2 b3 = a 3 That means that a3 is even, which means that a is even; thus a3 is divisible by 8. Let it be 8 c3 . 2 b3 = 8 c 3 b3 = 4 c 3 That means that b3 is divisible by 4, which means that b is even (and that b3 is in fact divisible by 64). But we began with the hypothesis ab was a fraction where a and b have no factors in common. Therefore, 3 2 cannot be expressed as a fraction ab . � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 14 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Constructible numbers Description An account of the resource Transcript of a lecture given on September 24, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-24 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Number theory Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_Constructible_Numbers_2019-09-24 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/37f22ac1e1a3ae9f9901d4932a844796.mp3 d80fed73e0576147a7d961997391c542 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:56:13 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource "Tel us som moral thing" : moral certainty and the art of story in Chaucer's Canterbury Tales Description An account of the resource Audio recording of a lecture given on September 27, 2019 by Maggie Evans McGuinness as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource McGuinness, Meagan E. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-27 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Chaucer, Geoffrey, -1400. Canterbury tales Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_McGuinnessM_Tel_us_som_moral_thing_Moral_Certainty_and_the_Art_of_Story_in_Chaucers_Canterbury_Tales_2019-09-27 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/5aae95441dc8d0a3ffd5ae6d11895a18.mp3 5c7c854ebc80de9c75297a8d0edde237 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 01:23:59 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource The United States, China, and Thucydides' many, many traps Description An account of the resource Audio recording of a lecture given on September 20, 2019 by Karl Walling as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Walling, Karl-Friedrich, 1957- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-20 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Thucydides. History of the Peloponnesian War Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_WallingK_The_United_States,_China,_and_Thucydides_Many,_Many_Traps_2019-09-20 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/a6aeec4018399afe1e0c4484b56ccabd.mp3 d1a5fe93723d18c3f6bacb707f2e3f31 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. 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. During the Fall and Spring semesters, lectures are given on Friday nights. Items include audio and video recordings and typescripts. For more information, and a schedule of upcoming lectures, please visit the <a href="http://www.sjc.edu/programs-and-events/annapolis/formal-lecture-series/" target="_blank" rel="noreferrer noopener">St. John's College website</a>. Click on <a title="Formal Lecture Series" href="https://digitalarchives.sjc.edu/items/browse?collection=5&sort_field=Dublin+Core%2CDate&sort_dir=d">Items in the St. John's College Formal Lecture Series—Annapolis Collection</a> to view and sort all items in the collection. A growing number of lecture recordings are also available on the St. John's College (Annapolis) Lectures podcast. Visit <a href="https://podcasts.apple.com/us/podcast/st-johns-college-annapolis-lectures/id1695157772">Apple Podcasts</a> or <a href="https://open.spotify.com/show/6GDsIRqC8SWZ28AY72BsYM?si=f2ecfa9e247a456f" title="Spotify">Spotify</a> to listen and subscribe. Contributor An entity responsible for making contributions to the resource St. John's College Greenfield Library Title A name given to the resource St. John's College Formal Lecture Series—Annapolis Identifier An unambiguous reference to the resource within a given context formallectureseriesannapolis Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 50:33 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Philosophic Anthropology and the Emergence of Liberal Constitutionalism Description An account of the resource Audio recording of a lecture delivered by Michael Zuckert, Emeritus Professor of Political Science at the University of Notre Dame, on September 13, 2019 as part of the Formal Lecture Series. Zuckert describes his lecture topic as follows: "Constitution Day is intended to honor and cogitate on the Constitution. My talk focuses not so much on the constitution itself but on constitutionalism. To a large degree, I maintain, constitutionalism is a product of developments in political philosophy, especially in the realm of philosophic anthropology. In my lecture I explore the underpinnings of the theory of human nature that supported constitutionalism of the sort the American Constitution embodies." Creator An entity primarily responsible for making the resource Zuckert, Michael P., 1942- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Annapolis, MD Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-13 Rights Information about rights held in and over the resource A signed permission form has been received stating, "I hereby grant St. John's College permission to make an audio recording of my lecture, and retain copies for circulation and archival preservation at the St. John's College Greenfield Library and to make an audio recording of my lecture available online." Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context Zuckert_Michael_2019-09-13 Subject The topic of the resource Hobbes, Thomas, 1588-1679 Locke, John, 1632-1704 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ce789f61b6997180ca1dc76624bae159.pdf 96136207810f761155e2fa715e4dd4fa PDF Text Text Trigonometric Interpretation of Complex Numbers Grant Franks June 3, 2019, revised September 16, 2019 Dedication Caspar Wessel (1745 - 1818) Let us pause for a moment to remember and give thanks for Caspar Wessel, Norwegian mathematician and cartographer, who conceived the idea that complex numbers might be usefully portrayed on a map. Introduction We ended the last talk with Rafael Bombelli staring at Cardano’s Formula as applied to the cubic equation x 3 - 15 x - 4 = 0 The formula gives for a solution: x= 3 2 + 11 -1 + 3 2 - 11 -1 which at first glance appears to be nonsensical since it consists of two terms both containing the cuberoot of expressions involving the square-root of negative one, which mathematicians in other contexts �2 �� 2 Trignometry and Complex Numbers.nb agreed meant that no solution was possible. But Bombelli knew that there was a solution to this equation. In fact, he knew what at least one of the solutions was: the integer + 4 solves the equation. 43 - (15) (4) - 4 = 64 - 60 - 4 = 0 Bombelli figured out by some combination of guesswork, deduction and just inspired staring that there are expressions which, when cubed, give 2 + 11 -1 and 2 - 11 -1 . They are 2 + -1 and 2 + -1 , respectively. Hard as it is to find them, it is easy to confirm that they work. All you need do is to multiply them by themselves three times, keeping in mind the one rule we know about -1 , namely, that when multiplied by itself it gives -1. -1  = 4 + 4 2 + -1  2 + 2 + -1  = 2 + 2 - -1  2 - 2 - -1  = 2 - 3 -1 - 1 = 3 + 4 2 -1  2 + -1  = 3 + 4 -1 -1  2 + -1  = 2 + 11 -1 -1  = 2 - 11 -1 And -1  = 4 - 4 3 2 -1  2 - Knowing the cube roots of 2 ± 11 x= 3 2 + 11 -1 -1 - 1 = 3 - 4 + 3 -1  = 3 - 4 -1 -1  2 - -1 allowed Bombelli to solve the particular problem facing him: 2 - 11 -1 = 2 + -1 + 2 - -1 = 4. However knowing the answer to this problem doesn’t show us how to deal more generally with other numbers involving -1 . The answer to that problem is the subject of tonight’s talk. Arithmetic of Complex Numbers Their Real and Imaginary Parts of a Complex Number The most evident problem with -1 is that it doesn’t stand in a relation of “more” or “less” with regard to other numbers we have come across. That feature more than any other makes -1 seem especially weird. When one takes the step from the whole numbers to fractions (that is, to positive rational numbers), things like “one-half” or “five and a quarter” could be related as greater or less than whole numbers we were already familiar with. Later, for all its undefinable strangeness, an irrational like 2 at least sat snugly between rational numbers, greater than some and less than others. (That, in fact, is how Dedekind defined irrational numbers, namely, by identifying which rationals each was �2 Trignometry and Complex Numbers.nb �� 3 greater than and which it was less than.) Even the very strange negative numbers aren’t as peculiar as -1 . If your idea of a number is that it should respond to counting something or measuring something, negative numbers are nonsense because there is less than nothing there to count or to measure. But if you can get over that problem, at least negative numbers still stand in greater-and-lesser relations to one another. Not so -1 . It is neither greater than nor less than any real number. That much is pretty clear: the square of every real number is positive, or at least “non-negative.” The -1 is not anywhere on the real number line that stretches from enormous negatives to enormous positives. So, if we are to imagine it at all, we have to picture it being “somewhere else.” Caspar Wessel set the imaginary numbers apart from the reals on an axis of their own at right angles to the real number line. He thus established a complex number plane. One axis represents the real numbers, the other the numbers that include -1 . A real number and an imaginary number together form a two-part entity called a “complex number.” Each point on the complex number plane represents a single complex number. A complex number can look like: 2+ -1 or -3 + 9 -1 or 0 +5 -1 or -4 + 0 -1 . The first and second examples have both a real and imaginary part. The third has only an imaginary part; the real part is zero. The last example has only a real part; the imaginary part has a zero coeﬀicient. At the risk of seeming overly pedantic, I want to note here that “having an imaginary part with a zero coeﬀicient” is not quite the same thing as “being a real number.” The complex number “-4 + 0 -1 ” is not quite the same thing as the real number “-4.” The reason for this hyper-technicality and squeamishness about nomenclature is not at all clear at this point and it won’t become clear until the last lecture. It’s not unusual to overlook this distinction and, for now, doing so won’t cause any problems. It is common, even convenient, to skip over the zero terms and to write “-4 + 0 i” as just “-4” I mention this not-yet-developed distinction only so that, when it comes back again in the final lecture, I can say “As I have already said …”, and you will all nod sagely in agreement. The real and imaginary parts of a complex number stand in diﬀerent orders and, when they are added or subtracted, they act independently of one other. In modern parlance, one might say that a complex number can be represented as a vector on a plane with a real axis in one direction (generally, le�-right) and an imaginary axis orthogonal to it (up-down). That’s not how Caspar Wessel spoke because the term of a “vector” wasn’t introduced until the middle of the 19th century, decades a�er Wessel died. But the fundamental idea is there: a complex number is a two-part object whose parts add independently of one another. That idea had been around for years, at least since Isaac Newton had analysed motions into components towards and parallel to the sides of a parallelogram. �4 �� 2 Trignometry and Complex Numbers.nb 5i 4i 3+4i 3i 2i -3 + 2 i i -5 -4 -3 -2 -1 0 1 2 3 4 5 -i -4 - i -2 i -3 i -4 i 2-4i -5 i Here, then, is the representation of four complex numbers on a complex number plane: 3 + 4 i, 2 - 4 i, -4 - i and -3 + 2 i. So far, this is just a picture. Its value appears as we see how it is used. Addition and Subtraction of Complex Numbers In addition, the real and imaginary parts act separately. So, if one adds -3 + 2 i to 4+2i one gets (-3 + 4) + (2 + 2) i The real parts add ordinarily, and the imaginary parts do too, thanks to the (formerly implicit, now explicit) understanding that “distribution of multiplication over addition” works for the number i as it does for other numbers, so that we have: 2 i + 2 i = (2 + 2) i = 4 i. This procedure is just what one would do with components of a vector or of a decomposed Newtonian force or velocity. Graphically, as Wessel proposes envisioning complex numbers, the result looks like this: �2 Trignometry and Complex Numbers.nb �� 5i 4i 1 + 4i 4+2i 3i 2i -3 + 2 i i -4 -3 -2 -1 0 1 2 3 4 -i Multiplication by a real number (or real part of a complex number) is like ordinary multiplication Multiplying a complex number by a real number amounts to multiplying each of the real and complex parts of the complex number as you would expect. For multiplication by positive integers, the result is just like repeated addition of the vector representing the complex number. Multiplication by negative numbers is like repeated subtraction. 5 �6 �� 2 Trignometry and Complex Numbers.nb 5i 4i Multiplication of 3 + i by 3 + 0 i 3i 2i i -2 -1 3+i 0 1 2 3 4 5 6 7 8 9 10 -i -2 i So far, so good. The graphic representation hasn’t yet shown us anything novel about complex numbers or given us new, but there is more and better yet to come. The Crux of the Problem: Imaginary Multiplication Next we have to deal with complex numbers times other complex numbers. This is where things get interesting. It’s not immediately clear what that means graphically, but we do have an algebraic understanding. The one thing we know for sure about -1 is that when you multiply it by itself, it gives -1. Let’s go back to the example we have already seen: Bombelli’s discovery that 2 + i is the cube root of 2 + 11 i. As we showed already, we can multiply 2 + i times itself: (2 + i) (2 + i) = (2 ⨯ 2 )+ (2 ⨯ i )+ (2 i ⨯ 2) + ( i ⨯ i) = 4 + 2 i + 2 i - 1 = 3 + 4 i. �2 Trignometry and Complex Numbers.nb 5i (2 + i)(2 + i) = 3 + i 4i 3i 2i i -2 -1 2+ i 0 1 2 3 4 -i -2 i So far, this is not too revealing. Multiply the product by 2 + i again: 5 �� 7 �8 �� 2 Trignometry and Complex Numbers.nb 12 i (2 + i)3 = 2 + 11 i 11 i 10 i 9i 8i 7i 6i 5i 4i (2 + i)(2 + i) = 3 + i 3i 2i i -2 -1 2+ i 0 1 2 3 4 5 -i -2 i What sense does that make? The meaning appears more easily with Polar Coordinates So far, we’ve been writing complex numbers like points on a plane using Cartesian coordinates. For some purposes, it is a LOT easier to understand what is going on if you use polar coordinates. (Trust me.) To start, consider a circle with a radius of one centered on the origin. This is the “unit circle in the complex plane.” �2 Trignometry and Complex Numbers.nb �� 1.51 = A 1.11841 = θ A (cos θ + i sin θ) Now if you choose any angle θ, the point (Cosine(θ) + i Sine (θ)) will necessarily fall on the unit circle. As the angle θ goes through the complete cycle from 0 to 2 π -- we measure angles in radians, which is easier for all sorts of reasons once you get used to it; if you are thinking in degrees, say “0° to 360°” -the point (Cosine(θ) + i Sine (θ)) goes around the circle. If the angle continues to grow, the point spins endlessly around the unit circle. If you want a point, that is to say “a complex number,” inside or outside the unit circle, multiply the result by some constant A. If A is greater than one, the corresponding point (complex number) will be outside the unit circle; if it is between zero and one, the point (complex number) will be inside the circle. Any point on the complex plane can be designated with a pair of numbers A (for length) and θ (for angle). In complex-number-speak, the angle of the complex number expressed in polar coordinate form is called the “argument”; the length is called the “modulus” of the number. 9 �10 �� 2 Trignometry and Complex Numbers.nb Multiplication of Two Arbitrary Complex Numbers Try multiplication again with two arbitrary complex numbers, this time expressed in polar form. Let the two numbers be: A B (cos(θ) cos(ϕ) - sin(θ) sin(ϕ) + i (cos(θ) sin(ϕ) + cos(ϕ) sin(θ)) “Okay,” you say. “How has this helped me?” The answer to that would be clear if you had been careful about memorizing trigonometric identities, in particular, the identities for the sine and cosine of the sum of two angles. On the oﬀ chance that you don’t have those identities burned into the forefront of your minds, let me show you what you need to “remember” or, as Socrates might say, “recollect.” Digression: Trigonometric Identities for Sine and Cosine of the Sum of Two Angles. Consider a portion of a unit circle with center at O. From center, draw a line OA at any (acute) angle; call the angle ϕ. Drop a perpendicular AF to the horizontal diameter of the circle. The right triangle formed as lengths that represent cos ϕ (horizontal OF) and sin ϕ (vertical AF). Now draw a line OB, creating another angle, θ, on top of the first one. Drop a perpendicular BC to OA, the hypotenuse of the first triangle. The segments OC and BC represent cos θ and sin θ, respectively. Drop perpendicular CH to the original diameter OA. Also, drop a perpendicular from BD at the top of angle θ down onto the original diameter. The segments thus created, OD and BD, represent cos (ϕ + θ) and sin (ϕ + θ) respectively. Note draw a horizontal CE from C to the line BD. In triangle BEC notice that angle EBC is equal to ϕ. Since segment BC is equal to sin ϕ, we conclude that BE = sin θ cos ϕ and that EC = sin θ sin ϕ. Meanwhile, since OC = cos ϕ, we conclude that CH = cos θ sin ϕ and that OH = cos θ cos ϕ. Examination will show that: BD = sin (θ + ϕ) = BE+ EC = sin θ cos ϕ + cos θ sin ϕ ; and CD = cos (θ + ϕ) = OH - DH = OH - EC = cos θ cos ϕ - sin θ sin ϕ. �2 Trignometry and Complex Numbers.nb �� 11 �� ϕ �� θ �� -�� B Sin θ Sin θ Cos ϕ ϕ A G θ C Sin ϕ sθ Co Sin θ Sin ϕ Cos θ Sin ϕ E ϕ O D Cos θ Cos ϕ Cos ϕ H F Now look back at the product that we just obtained in multiplying two complex numbers. A B (cos (θ) cos (ϕ) - sin (θ) sin (ϕ) + i (cos (θ) sin (ϕ) + cos (ϕ) sin (θ)) cos (θ + ϕ ) sin(θ + ϕ ) The collection of trigonometric terms associated with the real portion of the expression is cos (θ + ϕ). The collection of trigonometric terms associated with the imaginary portion of the expression is sin (θ + ϕ). The numbers associated with the lengths (modulus) are multiplied; the angles (arguments) are added. The significance of the imaginary multiplication is now visible: In multiplying two complex numbers, whether written as A (cos(θ) + i sin(θ)) and B (cos(ϕ) + i sin(ϕ)) or as a + b i and c + d i, graphically speaking what happens is that one �12 �� 2 Trignometry and Complex Numbers.nb (i) multiplies the distances of each number from the origin of the plane (the moduli), and (ii) add the angles (arguments) made between the positive real axis and the line from the origin to the point representing the number. In short, again: in complex multiplication, distances from the center (moduli) multiply; angles from the center add. All sorts of neat things follow from this observation. Raising Complex Numbers to Powers Causes Them to Spin! If you raise a complex number to a (real) power, the argument (angle) of the result will grow continually as the distance from the center grows (if it begins outside the unit circle) or shrinks (if it begins inside the unit circle). Raising complex numbers to real powers therefore causes the results to trace spirals in the complex plane. Here is the exponentiation of a complex number represented by a point a little bit outside the unit circle: �� {Modulus =, 1.0435} If we reduce the modulus (the “length”) so that the point falls inside the unit circle, the spiral will go inwards because increasing powers of a length (modulus) less than one will shrink. �2 Trignometry and Complex Numbers.nb �� 13 Between these two cases is the balanced point, where the modulus is one and the point lies on the unit circle. Then, increasing powers of the complex numbers will result in a representative point that spins forever around the circumference of the unit circle. The investigation of complex numbers is a vast field. Thick textbooks are devoted to “functions of a complex variable.” The Mandelbrot set, which lies at the beginning of complexity studies, exists in the complex field. (It is defined as the set of complex numbers c that do not diverge when the function fc (z) = z2 + c is iterated from z = 0.) All this would be subject matter for an immense study. However, for the present , I want only to point to two results that are relevant to the particular path that these talks are taking toward their goal, constructing the heptadecagon. Taking Integral Roots First, now that we understand how complex numbers are multiplied and raised to powers, we can easily find how to find integral roots of any complex number and thereby develop a general solution to the problem that faced Rafael Bombelli. His great triumph, recall, was finding the cube root of one complex number, 2 + 11 i, which he did by a combination of great genius, immense labor and fabulous luck. (Almost any other complex number would have been much harder for him to deal with.) However now we can see how easily to take the cube root of any complex number. Remember, to cube a complex number, you cube the real number that is its modulus and triple the angle (argument). So, �14 �� 2 Trignometry and Complex Numbers.nb to take the cube root of a number, all you need do is to (i) take the cube root of the length (the “modulus”) and (ii) and divide the angle (the “argument”) by three. The Cube Root of 2 + 11 i The particular problem that Bombelli faced was finding the cube root of 2 + 11 i. To take its cube root the new way, first calculate its modulus (length) and argument (angle). The length of the vector from the origin to (2 + 11i) we can get with the Pythagorean Theorem: length (modulus) = 22 + 112 = 4 + 121 = 125 If we allow ourselves some trigonometry, the angle is easy enough, too:  = 1.39094 radians (79.7 degrees). angle (argument) = ArcTan 11 2 To take the cube root, take the cube root of the length (modulus). In this case, we are assisted because 125 = 53 : 3 125 = 3 125 = Take the angle and divide by three: 5. 1.39094 3 = 0.463648 radians. So we get: 5 (Cos(0.463648) + i Sin (0.463648) ) = (2.236) (0.894427 + i 0.447214) =2+i Just the result that Bombelli arrived at by genius, sweat and divine guesswork. Here, for comparison, are the values Bombelli worked on plotted atop the graph of the spiral z = (2 + i)n �2 Trignometry and Complex Numbers.nb �� 15 12 i (2 + i)3 = 2 + 11 i 11 i 10 i 9i 8i 7i 6i (2 + i)n 5i 4i (2 + i)(2 + i) = 3 + i 3i 2i i -2 -1 2+ i 0 1 2 3 4 5 -i -2 i The Roots of Unity When the Modulus Equals One We have seen that when complex numbers whose representative points lie outside the unit circle spiral outward when squared, cubed, or generally raised to powers greater than one. Those that lie inside the unit circle spiral inward. Those that lie on the unit circle -- those with a modulus that is exactly equal to one -- spin around the unit circle with out moving inward or outward. These are very interesting, very handy numbers. Because the cosine of a given angle and the sine of the same angle can form the sides of a right triangle whose hypotenuse is equal to one, we can write these complex numbers with modulus one in the form: z = cos θ + i sin θ �16 �� 2 Trignometry and Complex Numbers.nb We have seen that multiplying two complex numbers adds their angles (arguments) and multiplies their lengths (moduli). In the case of these numbers, the modulus is one, so multiplying it any number of times leaves it unchanged. For these numbers, multiplying means just adding the angles. So, if we take a number and multiply it by itself, we get: z2 = (cos θ + i sin θ) (cos θ + i sin θ) = (cos 2 θ + i sin 2 θ) If we do it again, we get: z3 = (cos θ + i sin θ) (cos θ + i sin θ) (cos θ + i sin θ) = (cos 3 θ + i sin 3 θ) And in general, zn = (cos θ + i sin θ)n = (cos n θ + i sin n θ). If two diﬀerent modulus one numbers are multiplied, we get: z1 z2 = (cos θ + i sin θ) (cos ϕ + i sin ϕ) = cos (θ + ϕ) + i sin (θ + ϕ) �2 Trignometry and Complex Numbers.nb �� θ ϕ Cos ϕ + i Sin ϕ Cos θ + i Sin θ Cos (θ + ϕ) + i Sin (θ + ϕ) In this operation, multiplication of the complex numbers is tightly bound up with addition of the angles. Such tight linkage of multiplication and addition is characteristic of exponentiation and logarithms, and in fact it is a very short step from what we have seen here to a formula expounded by Leonhard Euler in his work Introduction to the Analysis of the Infinite that identifies the two: ei θ = cos θ + i sin θ. (A few years ago I gave a whole lecture on this identity; I’ll see about having it available on the library web-site alongside this one.) For now, we will be especially interested in a subset of these numbers that bear the intriguing and evocative name, the “Roots of Unity.” Roots of Unity The “Roots of Unity” sounds like a New Age metaphysical treatise or the name of a theologically inclined folk-rock ensemble, but in our present context it means something rather diﬀerent and more precise. It refers to numbers that, when raised to integral powers come to the result 1. Numbers like: 17 �18 �� 2 Trignometry and Complex Numbers.nb 2 1, 3 1, 4 1, 5 1 … etc. To put the matter slightly diﬀerently, we are talking about numbers that are the solutions of equations like: x2 x3 x4 x5 - 1 1 1 1 = = = = 0 0 0 0 or in general, xn - 1 = 0 Based on what I learned in high school, these equations are not hard to solve. For x 2 - 1 = 0, I know that there are two solutions, + 1 and -1. For x 3 - 1 = 0, there is only one solution, +1, because (-1)3 = -1. That pattern continues down the line, with even numbered powers having two solutions and odd numbered powers having only one. That understanding works so long as one considers only the real numbers. But in the complex number field the answer is more complete, more interesting and in some ways more satisfying. Take x 3 - 1 = 0 for example. We are looking here for a number which, when cubed, is equal to one, that is, the cubed root of one. Easy! One, when cubed, is equal to one. That’s fine, but it’s not the full story. Consider the number on the unit circle whose angle is 120°: when squared it is still on the unit circle and its angle is 120° × 2 = 240°; when cubed, it is still on the unit circle and its angle is 120° × 2 = 360° = 0°. That number is +1 + 0 i. Thus, the complex number at 120° on the unit circle is also a cubed root of one! So, for that matter, is the number on the unit circle at 240°: squared, its angle is 480° = 120°; cubed, its angle is 360° = 0°. There are, in fact, three cube roots of one, and the points that represent them form an equilateral triangle in the unit circle. �2 Trignometry and Complex Numbers.nb �� 2i The Cube Roots of Unity i 1 - , + i , 2 -2 3  2 0 -1 1 3 - , + i ,  2 2 {1, + i , 0} 1 2 -i -2 i The Algebraic Approach The graphical approach to the cube root of unity is simple: take the 360° of the circle and divide them by three. One can also take a strictly algebraic approach which is a little more intricate but which reaches the same result. Begin with the equation: x 3 - 1 = 0. As you noticed at first, the integer 1 (or, better, the complex number, 1 + 0 i) is solution. Therefore, we expect that this polynomial will be divisible by the linear factor (x - 1), as indeed it is: x 3 - 1 = (x - 1 ) (x 2 + x + 1) = 0. The new factor, (x 2 + x + 1), can easily be broken down into two linear factors by applying the quadratic formula to the equation x 2 + x + 1 = 0 : x= -1 ± 1 - 4 (1) 2 = -1 2 ±i 3 2 . So, the complete breakdown of the equation x 3 - 1 = 0 into linear factors is: (x - 1), x -  -12 + i 3 2 , x -  -12 - i 3 2  19 �20 �� 2 Trignometry and Complex Numbers.nb You can verify this result by multiplying any of the solutions -- 1,  -12 3 2 + i  or  -12 3 2 - i  -- by itself three times and seeing that you get the result 1 + 0 i. More Roots of Unity It should not surprise you to learn that the equation x 4 - 1 = 0 gives four fourth roots of unity: +1, -1, +i and- i. And the equation x 5 - 1 = 0 gives five fi�h roots of unity, like this: 2i 2i i -2 -1 i 0 1 2 -2 -i -2 i -1 0 1 2 -i -2 i And so forth. Generally speaking, there are always n nth roots of unity. This tidy fact is a special case of a more general result proved by Gauss and called the “Fundamental Theorem of Algebra” which states that in the complex number field a polynomial equation of the nth degree always has n solutions. That is a wonderful result, but we don’t need its full generality for our task-at-hand. Look again at the polynomials that define the n roots of unity. We can see from the graphic representations that the number 1 + 0 i is a solution of each of these “roots of unity” equations. Consequently, we can divide any of them by the factor (x - 1), just as we did with the cube-root of unity equation: x 4 - 1 = (x - 1) (x 3 + x 2 + x + 1) = 0 x 5 - 1 = (x - 1) (x 4 + x 3 + x 2 + x + 1) = 0 and generally: x n - 1 = (x - 1) (x n - 1 + x n - 2 + … + x 2 + x + 1) = 0 The increasingly lengthy remainder terms are of special interest to us. The solutions to the corresponding polynomial equations �2 Trignometry and Complex Numbers.nb �� 21 x3 + x2 + x + 1 = 0 x4 + x3 + x2 + x + 1 = 0 and generally: xn - 1 + xn - 2 + … + x2 + x + 1 = 0 are precisely what we need in order to find the vertices of regular polygons inside a unit circle. For fairly evident reasons, these equations are called collectively the cyclotomic (that is, “circle-cutting”) polynomials. They will be the subject, not of the next talk, but the one a�er that. Oh, By the Way … One More Thing to Note About Roots of Unity! Before closing, I want to note one more feature about the roots of unity that will show up in a later talk. It is this: for any whole number n, the sum of the nth roots of unity comes to zero. This can be seen pretty easily by looking at the case of the four fourth roots of unity: 2i i -2 -1 0 1 2 -i -2 i The four roots are +1, +i, -1 and -i. It is evident (isn’t it?) that when these four are added together, the sum is zero. A�er all, the pair + 1 and -1 add to zero, as do the pair +i and -i. �22 �� 2 Trignometry and Complex Numbers.nb Only a little less evident is what happens with the three third roots of unity: 2i i -2 -1 0 1 2 -i -2 i The two red vectors are parallel and equal to the blue vectors to the two complex third roots of unity. Placing the three vectors end-to-end in the usual way for vector addition gives a closed triangle, beginning and ending at (0, 0). Similarly for all nth roots of unity: their sum always comes out to zero. As a quick corollary, if one takes all nth the roots of unity for any n, the whole collection excluding the number +1 sum up to -1. This follows easily from the fact that all the roots of unity sum to zero; if one excludes +1, the rest must sum to -1 so that all of them together come to zero. These facts will be used repeatedly in what follows. If you don’t remember them, I’ll remind you of them when they come up again. Conclusion So these are the fundamentals of the arithmetic of complex numbers. The next talk will concern itself with another topic altogether, the algebraic diﬀerence between points that can be constructed and those that can’t. In the fourth lecture, these two topic will come together to demonstrate how algebra can decide whether a construction is possible or not; we’ll look at two classical problems -- the trisection of an angle and the doubling of the cube -- and then at a new problem: the construction of the seven-gon. The fourth Tuesday lecture will bring all that has been said to bear on Gauss’s surprise, the construction of the seventeen-gon. � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 22 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Trigonometric interpretation of complex numbers Description An account of the resource Transcript of a lecture given on September 17, 2019 by Grant Franks. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-17 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Numbers, Complex Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_Trigonometric_Interpretation_of_Complex_Numbers_2019-09-17 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/5a4db880c5c90ae2c98431316292db7a.mp3 ec8f26175bdac2498ba0edca6967bbd7 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:58:46 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Exploring Aristotle's Poetics through the eyes of a modern novelist Description An account of the resource Audio recording of a lecture given on September 13, 2019 by William Kowalski as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Kowalski, William, 1970- Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-13 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Aristotle. Poetics Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_KowalskiW_Exploring_Aristotle's_Poetics_through_the_Eyes_of_a_Modern_Novelist_2019-09-13.mp3 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/cbdf472cef32aef794932253cabe71bd.mp3 18d6b9eaedeceff4756a2c619c2e50c6 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 01:12:49 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Reconciling subjectivity and substance : Hegel's critique of pure personhood Description An account of the resource Audio recording of a lecture given on September 2, 2019 by Jonathan Hand as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Hand, Jonathan Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-06 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Hegel, Georg Wilhelm Friedrich, 1770-1831 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_HandJ_Reconciling_Subjectivity_and_Substance_2019-09-06mp3 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/19d3ad04ce55bbde7c5555119c349471.pdf 02cc71f518cf3202042cdf38d8f9eed9 PDF Text 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 Plotx3 - 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) � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 18 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Meet your new best friend : the square root of negative one Description An account of the resource Transcript of a lecture given on September 11, 2019 by Grant Franks as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Franks, Grant H. Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-11 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Algebra Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_FranksG_Meet_Your_New_Best_Friend_The_Square_Root_of_Negative_One_2019-09-11 https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/83a1685f9a71b0f08cd3ef6eb83ce714.pdf 896bd63aa4109a9079d7a93815931b15 PDF Text 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. J. Hand Hegel Person Lecture Given September 6 2019 Page 4 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 5 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 6 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 7 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 8 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 9 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 10 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 11 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 12 of 38 �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: J. Hand Hegel Person Lecture Given September 6 2019 Page 13 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 14 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 15 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 16 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 17 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 18 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 19 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 20 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 21 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 22 of 38 �“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. J. Hand Hegel Person Lecture Given September 6 2019 Page 23 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 24 of 38 �“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 J. Hand Hegel Person Lecture Given September 6 2019 Page 25 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 26 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 27 of 38 �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): J. Hand Hegel Person Lecture Given September 6 2019 Page 28 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 29 of 38 �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. J. Hand Hegel Person Lecture Given September 6 2019 Page 30 of 38 �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 J. Hand Hegel Person Lecture Given September 6 2019 Page 31 of 38 �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 � Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Transcripts—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Description An account of the resource Items in this collection are part of a series of lectures given every year at St. John's College at the Santa Fe, NM campus. Text A resource consisting primarily of words for reading. Examples include books, letters, dissertations, poems, newspapers, articles, archives of mailing lists. Note that facsimiles or images of texts are still of the genre Text. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data pdf Page numeration Number of pages in the original item. 35 pages Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource Reconciling subjectivity and substance : Hegel's critique of pure personhood Description An account of the resource Transcript of a lecture given on September 6, 2019 by Jonathan Hand as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Hand, Jonathan Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-09-06 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource text Format The file format, physical medium, or dimensions of the resource pdf Subject The topic of the resource Hegel, Georg Wilhelm Friedrich, 1770-1831 Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_HandJ_Reconciling_Subjectivity_and_Substance_Hegel's_Critique_of_Pure_Personhood_2019-09-06 Friday night lecture https://s3.us-east-1.amazonaws.com/sjcdigitalarchives/original/ea0224501ba2d0795847c24af1193597.mp3 d091ce51dc23419988871a3502539734 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource St. John's College Lecture Recordings—Santa Fe Contributor An entity responsible for making contributions to the resource St. John's College Meem Library Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Sound A resource primarily intended to be heard. Examples include a music playback file format, an audio compact disc, and recorded speech or sounds. Original Format The type of object, such as painting, sculpture, paper, photo, and additional data wav Duration Length of time involved (seconds, minutes, hours, days, class periods, etc.) 00:59:20 Dublin Core The Dublin Core metadata element set is common to all Omeka records, including items, files, and collections. For more information see, http://dublincore.org/documents/dces/. Title A name given to the resource "Something before her time delivered" : wonder, grace, and time's creation in The winter's tale Description An account of the resource Audio recording of a lecture given on August 30, 2019 by J. Walter Sterling as part of the Dean's Lecture and Concert Series. Creator An entity primarily responsible for making the resource Sterling, J. Walter Publisher An entity responsible for making the resource available St. John's College Coverage The spatial or temporal topic of the resource, the spatial applicability of the resource, or the jurisdiction under which the resource is relevant Santa Fe, NM Date A point or period of time associated with an event in the lifecycle of the resource 2019-08-30 Rights Information about rights held in and over the resource Meem Library has been given permission to make this item available online. Type The nature or genre of the resource sound Format The file format, physical medium, or dimensions of the resource mp3 Subject The topic of the resource Shakespeare, William, 1564-1616. Winter's tale Language A language of the resource English Identifier An unambiguous reference to the resource within a given context SF_SterlingW_Something_Before_Her_Time_Delivered_2019-08-30 Deans Friday night lecture