On Self-delusion and Unimaginable Beauty: a Mathematician's Reveries from the Margins

J.D. Phillips

LaFollette Lecture

Wabash College

19 October 2007

[Intro music: "A Little Less Conversation ", played loudly]

Section one: Twenty-eight.

Thank you, all of you, but especially the LaFollette family for your generosity and your loyalty to this College (and after today, I'm hopeful we can add: for your patience and forbearance with a faculty that occasionally can be a challenge). It's an honor for me to be here today to give this, the 28^th Lafollette lecture. And serendipitous, too! You see, 28 is an especially seductive number, and I, well, I’m a professional mathematician. Pray; attend! The proper divisors of 28—that is, the counting numbers less than it, and that divide into it evenly, without remainder—are 1, 2, 4, 7, and 14. Now, just to pass the time, let’s add them up: 1 plus 2 is 3, plus 4 more is 7, plus 7 is 14, plus 14 more is, brace yourself, 28, the number we started with. Yeah.

Now, you may be unimpressed by this, if you have some sort of grim personality disorder, I suppose. But just try to find another counting number with this property. Go ahead; pick a number and give it a try. You may, for instance, try 10. Well, let’s see; its proper divisors are 1, 2, and 5, and 1 + 2 + 5 = 8, which is less than 10, so 10 doesn’t have the sexy property that it’s equal to the sum of its proper divisors. In fact, the sum in this case is too small; vulgar little number. Okay, fine, let’s try another number. How about 12? Its proper divisors, as even the div. 2 faculty can check, are 1, 2, 3, 4, and 6. But alas, the sum of these numbers—1 + 2 + 3 + 4 + 6 —is 16. So 12 doesn’t work either. In this case, the sum is too large; corpulent little beast.

I should point out that mathematicians gave names to these properties many centuries ago. And unencumbered as they were by the linguistic gloom of political correctness, they called numbers for which the sum is too small (numbers like 10), deficient. Naughty boys! With a bit more decorum, although just a bit, they called numbers for which the sum is too large (numbers like 12) abundant. That leaves numbers, like 28, for which the sum is just right—not too large, not too small, like the Baby Bear’s bed in Goldilocks. And they called numbers with this hypnotic property, perfect.

Imagine, then, my good fortune as a mathematician in giving a perfectly numbered LaFollette lecture. And it really is good fortune, because, you see, perfect numbers are exquisitely rare. Remember my charge from earlier, to try to find another perfect number? Well, if you started with 2, and then moved on to 3, and then to 4, and so on, and checked the counting numbers, each in their turn, you would very quickly notice that 6 is a perfect number (1 + 2 + 3 = 6). You’d then hit a dry spell until 28. But then. . . nothing. . . until 496 (go ahead, check it; you’ll see that it’s perfect). And after that, well, you’ll need some patience, because the next perfect number after 496 is 8,128. And from thence onward, you’ll need a computer, I’m afraid, because the next perfect number sits silently in its unassuming position far off in the hazy distance on the number line, at 33,550,336, just a tiny, rogue, needle lost in a vast haystack.

For those of you who are still awake, let me point out that we’ve found the only five perfect numbers amongst the first 33 million counting numbers. As I said, exquisitely rare, like precious gems to mathematicians. Perhaps you’re greedy and lust for more? If so, you’re in luck; there are more. Here’s the sixth: 8,589,869,056. And the seventh: 137,438,691,328.

By the way, are you buying all this? After all, maybe I’m just making stuff up in the hopes that no one is going to check me for accuracy. I know, I know; the Asperger’s syndrome, Goodrich Hall crew will take this as a challenge. You may spot them, now, here in the hall, rocking back and forth, eyes closed, checking my computations. The rest of you, though, if pressed, would reach for your calculators (and then, probably to throw at me). I say this just to note, for dramatic effect (and a dash of mystery), that these two perfect numbers, the sixth and the seventh, were discovered by a man named Cataldi, in 1588. That is, he didn’t use a calculator, at least not an electronic one.

Well, I’m not going to give you any more perfect numbers (please, no applause) because the next one (which, mind you, is only the eighth perfect number) has 19 digits, and I don’t know the names of numbers this large (although I suppose I could just make something up).

If we continued this game, eventually, we would come to the largest known perfect number, a titanic, nameless, beast with 909,526 digits. Some of you may not be used to thinking about numbers by counting how many digits they have, so let’s consider a modest number like, say, 124. It has three digits: a one, followed by a two, followed by a four; again three digits. But 124 itself is a lot larger than three. And just so with this largest known perfect number. It has, again, 909,526 digits, but is itself staggeringly larger than 909,526 (which has a mere six digits). And I note, by the way, that it is larger (in fact, almost incomprehensibly larger) than even, brace yourself, the number of particles in the physical universe (which my physicist friends assure me is a number with something like 81 digits).

May I ask you to stop for a moment, and just marvel over this? Our number dwarfs the sum total of all of the particles in the physical universe, from which we can draw the easy conclusion that it doesn’t exist in our universe. So how is it that we are able to know this number? A kind of scandal, if you ask me. Well, let it wash over you; revel in it.

And I shall now immediately intrude on your revelries by playing the role I’ve been cast in (what else can I do?) and note that here we see a crucial way in which mathematics differs sharply from the laboratory sciences. The sciences take as their domains some part of the physical world; mathematics does not. True enough—and, I should add, amazingly enough—mathematics is the language of the sciences. But, remarkably—dare I say, unimaginably—it takes as its domain of inquiry one other than the domain of all sciences. How this should be so has been an occasion of wonder for many generations of scientists and mathematicians. I’m now inviting you to wonder over it, too.

Other disciplines also differ sharply from mathematics. For instance, there are popular threads in many of the humanities and social sciences in which the primary focus is on analysis of profoundly ugly objects and ideas: genocide, holocausts, and so on. I recently sat through a talk devoted to an analysis of sexist, visual images from America’s 19^th Century. The speaker had a much larger scholarly agenda then mere prurience, of course (namely, discovering, and then investigating, deep cultural pathologies as part of the even bigger project of conquering them, a large project indeed). But still, the investigations were centered on profoundly ugly images. There is nothing analogous to this in mathematics. Nothing. Mathematicians pursue the beautiful, as we shall see. Maybe we mathematicians are sheltered and privileged to not have to worry our pretty little heads about the ugly parts of the world in our work; I don’t know. On the other hand, what you see every day—the natural beauty (or not) of where you live, the paintings that hang on your wall, the music you listen to, the objects you devote your professional life to studying—eventually, day after day, year after unrelenting year, I think become reflected in your own soul. I’ll have more to say about this later, too. In preparation, then, let us take up the challenging question of the nature of mathematical objects, a potentially powerful soporific, so I promise not to make a din.

Section two: Mathematical Objects

One of the more pressing mathematical problems to the ancients was to sort out the proper ontological status of mathematical objects, a project that no doubt seems quaint to some of you, dreary to the rest. So let’s jump right in! We’ll start modestly, with something tame, say, the most basic 2-dimensional objects from grammar school.

And what could be tamer than a circle? We all know that a circle is the set of those points in a plane which are equidistant from some fixed point, the center. How do we come to this definition so easily? Because we’ve seen so many circles, right? Nope; there aren’t any circles (indeed there aren’t any mathematical objects at all) in the physical world; remember our comparison with the sciences. But if you do want to argue with me about this, remember that the thin little “boundary” that we think of as the circle, is actually thinner than that: it has no breadth, by definition; it can’t be measured, hence, certainly not seen. And even if you’re especially stubborn in this regard, and insist on trying to measure it, I call on Professor Heisenberg to rebut: the very act of measuring something that small changes it. That is, if you’re unbalanced enough to try, whatever it is that you think you measured in your pursuit of a circle, is now of a different shape than it was before you measured it. And by the way, any pale imitation of a circle you find in our physical world, actually has three dimensions, so it’s really a pale imitation of a cylinder, not a circle. The question, then, I trust, is clear: how is it that all of us seem to know circles so well—we all use the same definition and seem to understand them in the same way—in spite of the fact that none of us has ever seen one? Collective self-delusion?

All right, that didn’t work out so well. Let’s scuttle it and try something easier, say, the counting number two. A mere trifle, right? We’ve known two since we were in diapers, for heaven’s sake! Well, let’s be careful. There’s still the stubborn difficulty that mathematical objects don’t live in the material world. On the other hand, unlike circles, many instances of two, do reside in the physical universe: two chairs, two birds, two beers (hey, I’ve been in Prague); perhaps this will lighten our load a bit. Good, good. So why don’t you go ahead, then, right now, in your seats, and try to formulate a definition of “two”; what is two? [Dramatic pause.]

Not working out? Okay, fine. How about just tackling part of the question; answer this: is two a unity or a plurality? That is, is it one “two-ness”, or is it two ones? [pause] Uh-huh. Cat got your tongue?

Just think about that for a moment. On the one hand, the counting numbers are probably the simplest mathematical objects; even tiny children easily develop a facility with them. And, yet, we can’t even say what they are.

You know how mathematicians deal with this problem, by the way? We say, “Phew, this is exhausting; we’re not getting anywhere with this. So what say we just take them as givens—axioms, first principles—and not get our undies too bunched up thinking about them anymore, okay?” Really, that’s what we do. Here’s Kronecker: “God created the counting numbers, all the rest is the work of man.” Steven Hawking even has a recent book by that name.

By the way, if instead of asking about the nature of a particular number, you think it might be easier to ask about “process” (and what could be trendier in the academy today than asking about process, by the way), which in this case would be “the first move” in number theory, the move from one to two, (i.e., learning to count) well, you’d be wrong. In fact, this might be the most difficult move in all of human thought. But once you have it, things get easier, because you can always add just one more, and thence to the richness of God’s counting numbers, and from there, to all of mathematics. But the first move, from unity to two, ah, how to account for this? A human life might be the only real unity any of us will ever know. The move from unity to two, then, is as mysterious as the philosophical move from solipsism to awareness of “the other”; most of us make both of these moves without understanding them, or even being aware we’ve made them.

Friends, we’ve retreated from grammar school shapes back to preschool number awareness and our understanding has grown thinner. This is not happy news! Another way to put it: a conversation about mathematics with even a modicum of philosophical seriousness stalls very quickly.

How, then, to give even a quasi-serious account of more complicated mathematical objects? There is an equation from Complex Analysis, a course you can take right here at Wabash, that relates the six most important constants in the mathematical universe—the first three counting numbers, 0, 1, and 2; the irrational numbers e and π; and the complex number i—into an elegant, little equation that involves nothing but these numbers. It is called Euler’s equation, and here it is (to the Goodrich Hall Asperber crew: I’m giving it in the version I prefer, so just bite your tongue if you prefer another version): e^2πi – 1 = 0. There it is. My god, how beautiful; one of the most beautiful objects I’ve ever encountered, in this, or any, universe; so simple, startling, important, deep, and yet seductively accessible. But how imperfectly understood! How to even begin talking about it? We didn’t get anywhere with 2, for heaven’s sake. e^2πi – 1 = 0? I might as well just start making stuff up in Czech.

And Euler’s equation, by the way, is a standard part of the undergraduate mathematics curriculum. What about the mathematical objects from the frontiers of research (which in some ways, ought to be a part of this talk)? How do I tell you about my own work on, say, the nilpotency class of a Bol p-loop with abelian inner mapping group? Or the longstanding open problem of the triple argument hypothesis from the theory of commutative Moufang loops?

Well, by moving on to section three, of course, on unimaginable beauty.

Section three: Unimaginable beauty.

One night this past summer, I sat in the backyard with my family after dinner. It had been a hot and humid day, but it was comfortable that evening. Chimney Swifts were buzzing overhead; hummingbirds were zipping about the garden. The sky was ablaze with the improbable colors of the setting sun dancing on a few thin, fleecy clouds. The boys were playing happily in the sandbox. Cathy sat next to me, on a lawn chair, reading a book. I was working on some mathematics, trying to sharpen a theorem I’d proved earlier in the week. There was a mostly empty bottle of wine on the table, and another one waiting in the kitchen, if we needed it. Dinner was finished, but there was still some fruit on the table, some pistachios, too. We could just barely hear the music I’d left playing in the living room, through the open windows. The air was heavy with the good smells of summer—flowers from the garden, the neighbors grilling out, chlorine from the pool next door. And sounds, too, such happy sounds—the squeals of the boys in the sand, a squirrel barking softly up in the Walnut tree, the neighbors laughing and splashing in their pool, the rustling wind in our birch tree (that graceful tree that anchors our entire yard), a Gray Catbird softly mewing in the Locust tree. A simple enough scene. I don’t remember ever being happier.

Cathy had put her book down, and been looking over my shoulder at my work for some time before I noticed her, lost in concentration as I was. “It looks so beautiful,” she said, almost sadly. “I wish you could tell me about it.”

And, if you’ll indulge me, it really was beautiful. A partial solution to a long-standing, famous open problem, an elegant proof that I was sharpening. But even the lean curves of the exotic symbols and letters on the page, typeset so neatly, spare black markings on clean white paper, with, to Cathy, no meaning beyond that, like Japanese calligraphy, except maybe for my obviously happy concentration with it. But for me, for me, it was something else entirely.

It’s hard to describe beauty, especially when it’s mathematics. Would it help to tell you what I was working on? I’d found an elegant proof that the smallest Moufang loop with nonnormal commutant would have to have order at least 729 (that’s three to the sixth power); assuming such a loop even exists. So no, of course it wouldn’t help. But something in the scene had moved her; I had to say something.

In some ways, it’s strange that I would have to. The desire for beauty is understood, it doesn’t have to be accounted for, in most settings—music, poetry, painting, architecture, nature, a beautiful face, and so on. But it’s less clear how mathematics participates in this kind of beauty, because in some ways, at least, its participation is so radically different. You look at a painting or a building or a sunset. Even music, probably the most abstract of the arts, is firmly grounded in the sensory world: you hear a string quartet; 1/π doesn’t come up in music (or when it does, it’s as a mathematical curiosity, not as part of anything we can actually hear). Poems, words, are a bit trickier. But I note that poems are usually about—they’re reflections on—experiences in the physical world.

Mathematics is never like this. Never. As we have seen, its objects are not part of the sensory world, and its mode of expression is an especially abstract kind of language, every bit as tricky as the language of poetry, in some ways even more so. The beauty in mathematics, then, is unimaginable, that is, you cannot call forth an image of it (as you can of, say, a pink elephant); you can’t see or hear or smell or taste or touch mathematical objects. They are, then, literally, unimaginable. You can only conceive of them. By the way, learning how, or conditioning yourself to, do this is demanding; it takes a lot of time. Very crudely: like most mathematicians, nearly all of my motivation for doing mathematics is its beauty; but when you’re first learning mathematics, this beauty is anything but apparent; so why continue trying to learn it? I ask this as a genuine question; that is, I don’t know the answer.

To go much further than this, we mathematicians are usually reduced to awkward analogies and groping metaphors. I hope mine, stretched as they might be, are at least unexpected.

Young men play sports for a number of complex and subtle reasons that they probably don’t fully understand themselves, but none greater than this: thymos, a kind of spirited pursuit of honor. Again, there are other reasons (it’s fun, they want to be part of a team, and so on), but thymos is the main reason. Ah, but it’s much simpler for us old men: we watch sport for only one reason: because its beautiful.

There is, of course, a kind of physical beauty attendant to sport—the grandness of a stadium, the crispness of favored formation, and so on. I’m thinking here of the first time I saw a game at Lambeau, my first Formula One grand prix, and so on. But this is mere preamble, idle chatter, compared to where the deepest, most penetrating beauty in sport lies.

In sport, in every sport, there are rules, strong and clear rules, and penalties for violating these rules, and (excepting NASCAR) the competitors and the fans know all of these rules before the competition begins. You simply may not hit the receiver while the ball is in the air. You are forbidden from exceeding the speed limit on pit road, in Formula One, at least.

And these rules apply to the competition that unfolds within a further rigid structure. If you do not move the ball at least 10 yards after 4 consecutive plays, then you forfeit possession of the ball to your opponent. If you have to drop a new engine into your car before the old one has completed two races, then you will start the next race from the back of the grid.

Moreover, the competition takes place in stadiums and arenas, and it uses equipment, that adhere meticulously to open standards and conventions. A football field must be exactly 120 yards long and 53 and 1/3 yards wide. The tread width on the front tyres of a Formula One racecar may not exceed 270 mm.

And most importantly, all who dare to compete are bound by these rules; there are no exceptions for anyone (here is the origin of our political euphemism “level playing field”), all in pursuit of a single, transparent goal: to score more points in the allotted time than your opponent; to be the first man to complete 73 laps.

One of the happy, and perhaps paradoxical, payoffs for enduring all of this rigidity is that it often gives birth to a startling creativity. Think of it this way: when two or more especially skillful and well-matched opponents compete in the sort of contest I’ve just described, in order to win, a special creativity must emerge. And this is, I think, the real beauty in sport. It’s why grown men invest so much emotion in mere games.

Of course, there are other reasons, too; creativity in sport is often accompanied, hence enhanced, by other noble virtues—sacrifice; courage; intelligence; endurance of pain; physical prowess; or even better, much better, in fact: a great spirit that overcomes modest physical gifts. But note, these are all in the larger service of the struggle to win, which so often culminates in a moment of beauty. For instance, I suppose it takes courage to dash in front of a train, but no one cares about the idiot who’s unhinged enough to do this. And similarly for the flagellant with the bum knee who jumps up and down on it in the middle of the town square screaming in pain. Sure, he can take the pain; but he is merely an idiot (although I suppose Kafka might call him an artist). But now take the aging quarterback with the bum knee and two cracked ribs, pounded all day, uniform covered in turf and blood (and that weird grease trainers slather all over players); and now, down five points in the fourth quarter of an important divisional game, skillfully (but just barely) hobbling out of the grasp of the 350 pound brutes half his age who want to ruin him, and somehow finding a way to complete a desperate last second, improbable, but amazingly graceful, touchdown pass to win the game and secure a spot in the playoffs. Well, that bit of creative magic, informed as it was by other virtues, and unfolded as it was on a level field of competition of the very highest caliber, that, my friends, is a thing of genuine beauty.

Complete freedom within the constraints of strict rules, an opportunity for something beautiful to emerge from a moment of creativity.

Mathematics is like this. Hey, don’t laugh. I’m not comparing myself with Joe Montana, although if you want to think of me in this way, I won’t try to dissuade you. But seriously, mathematics is like this. A few clear, but rigid rules. If you thought Bernie Ecklstone was rigid when he fined Maclaren $100 million for stealing secrets from Ferrari, you outta try our unbreakable rules, all of them basically variations on the single unflinching rule of logical necessity, just tarted up a bit in the finely articulated ways that we mathematicians find amusing. But, as with sport, within the simple but strict bounds of those rules, there is complete freedom. In fact, there is only freedom and creativity. True, we mathematicians don’t have to prove our theorems with cracked ribs (although some of us have done that). And I didn’t get a bonus when I signed with Wabash (note to the Board). But a lean beauty that results from the most creative of acts, which emerge on a level playing field in a rigid, rule-bound universe, is common to both mathematics and sport (and not many other activities, by the way).

And the beauty in both, at least to the insider, someone who understands the technicalities and the subtleties, is intoxicating, even overpowering. It’s why crusty old men—whose emotional palettes usually run the spectrum form “pass the ketchup” to “where the $#%# is the remote?” (I’m not sure why everyone’s looking at Professor Warner) —are reduced to tears when Kirk Gibson, sick and badly injured, barely able to walk, hobbles to the plate to pinch hit in the bottom of the ninth with his team down by one run in the opening game of the 1988 World Series, takes the count to 3 and 2—against Dennis Eckersley, no less—grimacing in agony with each swing, and now somehow masters his pain and creates a moment of effortless beauty as his smooth-as-silk swing takes the final pitch over the right field wall, for a walk off, game-winning homerun; his legs are so injured they barely carry him around the bases.

But, as with mathematics, you really do have to be an insider to appreciate this beauty. For instance, in 1971 when Johnny Rogers uncorked his dazzling 72-yard punt return to give Nebraska the opening lead against the revolting Oklahoma Sooners (I know that at least Mr. Wheeler understands me, here) in the Game of the Century (a play still widely regarded as one of the greatest plays in the history of college football), in our small living room in Norfolk, Nebraska, my father yelled with excitement, danced around the room in celebration, and then wept openly, so beautiful was the play, while my mother looked up from her book and said, “Oh, did we score a point?” To my mom, it was just a bunch of brutes, posing as college students, beating the hell out each other; but to my dad (and to me), it was beautiful. God, it was beautiful.

There is, I think, a deep sadness that anchors our yearning for beauty in ways that we’re often not aware of. Or maybe we are aware of them, but only in inchoate, even frightened, ways.

When we encounter the beautiful, it’s hard not to notice our own wounds and imperfections and the ugly tyrannical ambitions we harbor deep in our own souls, in comparison with the beautiful, which is none of these things. And so we are ashamed. The beautiful, then, seems fragile and rare compared to the crudity and profanity of what is common, the crudity and profanity that each of us, at least in part, is.

This is why grizzled old men cry when they watch the magic of a skillful quarterback guiding his team to a fourth quarter, come-from-behind victory (or when they hear the Queen of the Night’s aria, as you all did as you were filing in today; why’s everyone looking at Professor Warner again?): it’s a piercing awareness of the vast distance between who we are and who we wish to be. It’s part of the sadness that runs through each human life. We cry for ourselves. But I hope you see, then, that an encounter with the beautiful can be (it doesn’t have to be, but it can be) an occasion for opening oneself up to the possibility of self-purification. There are no ethical laws in the beautiful, a heretical thought in some circles. But sometimes, encountering the beautiful can make you want to be a better man or woman. And maybe that’s enough.

As you probably know, a LaFollette lecturer has guidelines to follow, namely, and I quote, “to address the relation of his or her academic discipline to the humanities broadly conceived.” And so now I’d like to change gears and address the relation of mathematics to that humanity with which it is most closely allied: psychology, specifically abnormal psychology. This may sound like frivolity, and indeed, it may be. But you see, mathematicians offer tempting and fertile ground for these sorts of inquiries. So on to section four:

Section four: Mathematicians are Peculiar.

My dear, long-suffering wife can now, after being married to one for nearly 19 years, easily pick out the mathematicians at a party. We’re the badly dressed, socially inept geeks. I myself have gone, on many occasions, the entire day with the buttons on my shirt unaligned [here, look down on my shirt], only to learn the terrible truth at the end of the day when my horrified, but unsurprised wife discovers it as I walk in the door from work. I know, I know; you’re surprised, even stunned, but it’s true! Soy un perdador, baby.

One of my professors in graduate school would occasionally show up at work with shaving cream still on most of his face; interrupted by a mathematical thought while shaving, he simply forgot to resume his morning grooming before heading out the door to work.

Because the burdens of selecting clothing for the day are very great indeed, one of the most prominent mathematicians in my field wears the same outfit everyday, day in, day out, regardless of the weather, regardless of the occasion—turtleneck, khakis, tweed coat, work boots—so as to be relieved of this burden. For most of you, your response to this is the commingling of amusement and alarm. But for me, and dare I say, for most of the Goodrich Hall, Asperber crew, our response is, “Dang, that sounds like a great idea!”

Ladies and gentlemen, I have seen horrors far worse than these. But my wife made me remove the section I’d written on, ahem, psychopathologies and planimeters.

Complementing this, some might say compounding it, we mathematicians also have a collective sense of humor that is so quirky it must rightly be compared to the secret languages that develop between idiot-savant twins. My students no longer even pretend to humor me by laughing at my math jokes.

How can you tell when a mathematician is extroverted? He looks at your shoe instead of his own when he talks to you.

Okay, I’m entitled to this next one, as I grew up in rural Iowa and Nebraska. What’s trigonometry for farmers? Swine and coswine.

Yes, exactly. You see, like Michael Jackson in his Thriller video from the early 80’s, we mathematicians aren’t like the other boys and girls. I have a serious point. . . which I promise I will eventually get to, but first, what’s the response you typically get when you meet someone for the first time, and you tell him or her what you do for a living? I’ll bet it’s something like this, “Ah, you’re a banker (or a librarian, or a school principal, whatever), fine, very fine.”

You know what it’s like for us mathematicians? “Nice to meet you, Englebert; what line of work are you in?” “I’m a mathematician!” “Oh my God, Englebert, I hate math; I’m terrible at it; always have been. I almost didn’t graduate from high school because I couldn’t pass algebra. You math types are real sadistic bastards.” Sometimes we just get disbelieving stares. And occasionally, we even get shrieks of horror—“Hi, I’m Englebert; I’m a mathematician.” “Ahhhhhhh!” This has happened to me; I’m not making this up. And I’ll bet it doesn’t happen to plumbers or medical doctors or secretaries or chimney sweeps or professional badminton players or even pet detectives.

My all-time favorite response, though, came at a dinner party for the parents of the students at the shi-shi, private school my wife taught at in San Francisco. The fathers were all gathered at the bar, so, naturally, I joined them there, and we made pleasant small talk. I should point out that this was a very expensive school (Joe DiMaggio’s great-grandkids were students there, as were the children of many of the Bay Area’s professional athletes and celebrities; my wife taught the daughter of our congresswoman,), so need I note that these dads were not from my social stratum? But they didn’t know this; they all assumed that I was just another rich father of a student at the school.

Eventually, though, they discovered that I was not one of them; I was, alas, just the husband of one of “the help.” The lead dad, then, quickly asked me about my background and what line of work I was in. So I told him and his rich dad posse that I’d grown up in rural Iowa and Nebraska; I’d just finished my Ph.D.; and was now a mathematics professor at a local college. The dads stared at me incredulously. Firstly, I don’t think they’d ever actually met anyone from Iowa or Nebraska. They were, I suspect, surprised that I wasn’t wearing overalls, and that I only occasionally used the words “reckon” and “fixin’”. I have to confess, by the way, that I may have encouraged them a bit in their confusions in this regard by telling them that the two things I reckon I most enjoyed about living in California were the indoor plumbing and not having to worry about Indian raids come harvest time.

But secondly, and more importantly, I don’t think they’d ever met a mathematics professor before that moment. It really was such a scandal to them, that they could only stare and gasp. Indian raids and no running water were more conceivable. Eventually the lead dad politely broke the silence by saying, “Well, that’s a nice thing to do while you’re young, and still figuring out what you really want to do with your life.” His posse nodded approvingly.

Okay, the serious, but modest, point I’ve been leading up to with this “humor”:

I know of no other discipline about which otherwise intelligent and well-adjusted people so freely and proudly express ignorance and disdain.

In fact, it’s hard to even imagine in most disciplines. “Ah, reading. I never was any good at it; God, I hate books.” “Eck, history, what good is it; who cares who our president is?” The most cultured of the cognoscenti—folks who would never dream of expressing ignorance about art or music (“Ah, Chopin; over-rated nocturne writing hack”)—proudly confess their ignorance about mathematics. Even physics, the discipline closest to mathematics in the Ack!-factor, fares better than we do. Most folks, most bright folks at least, know something about relativity and quantum mechanics; for heaven’s sake, they’ve at least heard the words. But who knows the Riemann Hypothesis or the classification of the finite simple groups from mathematics? Axtell, put your hand down.

The bizarreness of mathematicians, as I trust I myself have amply demonstrated, surely doesn’t help. But I think there’s more to it than this. Firstly, I note that the distance between the mathematics you study in high school and the mathematics you study at college is probably greater (in fact, much greater) than is the analogous distance in any other subject. Put another way: your experience in high school mathematics classes gives you almost no indication whatsoever of what college mathematics classes will be like; the disciplines hardly resemble each other.

And secondly, the distance between college mathematics and mathematical research is also very great, greater than in many disciplines, I think. For instance, many of my colleagues in other departments here at Wabash give courses that are related to their own research. This is almost inconceivable for most mathematicians. The gap is just too great.

May I echo an earlier point by saying, then, that there is a kind of loneliness in a mathematical life. Again, it’s hard for us to have even a basic conversation about our work with nonmathematicians, a lonely feeling indeed. I used to take careful pains to try to explain my profession on those rare occasions when people didn’t flee from me upon hearing what I do for a living. It didn’t work. So now I usually just make stuff up: “Most days,” I tell them, “I occupy myself by multiplying really big numbers together. By hand.”

Even students, colleagues from other departments, and the folks who sign my paychecks, have no idea what I do. I am on sabbatical this year. In the months leading up to my sabbatical, when people on campus asked me what I intended to spend the year doing, I’d look at them with a bit of surprise and say, “well, my research, of course.” And they’d look at me as if I were daft (or worse) and say, “Research? In mathematics? Come on, really, what are you going to be doing?” Now I find it easier just to say, “Well, most days I’m drunk by noon.”

In preparation for the next section, I should say that were I given free reign to talk about whatever I wished to in this talk, those of you who know me well certainly know that I would talk about the liberal arts. Or maybe birding. Or fly-fishing. Or Formula One auto racing. But as I’ve already mentioned, a LaFollette lecturer has guidelines to follow, guidelines which I shall now mostly ignore as I launch into a brief section, section five, on the liberal arts.

Section five: The Liberal Arts.

First, a sincere apology to those very many of you who have no doubt heard my musings on the liberal arts before, and are rightly sick of them. So instead of forcing you to listen to me, yet again, decry the ways in which the pursuit of disciplinary expertise oftentimes gets in the way of the broad and expansive aims of genuine liberal education let me, instead, use the words of another scholar, Anthony Kronman, in this regard, as a nod toward local freshness.

“Those who teach in our colleges and universities are nearly all graduates of Ph.D. programs, in which they learn to measure success in higher education by the standards of the research ideal. From the vantage point of that ideal, the question of life’s meaning—of what I should care about, and why—is too large, too sprawling, too personal, to be a subject that any specialized scholar feels comfortable tackling. The research ideal has squeezed this question from the field of respectable topics.”

At this point, some of you must be wondering what a mathematician might possibly have to say about any of this, about the liberal arts, which is, I think a good indication of just how rare genuine liberal education has become. You see, the quadrivium, those first four liberal arts of the ancients, were mostly mathematics: arithmetic, applied arithmetic (e.g., harmony), geometry, and applied geometry (e.g., astronomy). These were, of course, appended in the Middle Ages by the trivium: rhetoric, grammar, and logic. So of the seven traditional liberal arts, three to five of them, depending on how you count, belong to mathematics.

Let me also remind you that “mathematics” itself is a Greek word—not surprisingly, given its centrality in liberal education. It means something like “that which is learnable.” To the Greeks, then, mathematics was the academic discipline in which you could actually learn something; you just traded opinions in the others.

I say that in a gleefully polemical tone because the Greeks didn’t believe this, exactly. In fact, “The Greeks” weren’t a single, unified voice agreeing on much of anything, contrary to the patronizing way we in the academy so often think of them, and other older cultures. The disagreements between Plato and Aristotle, for instance, make Webb v. Morillo look like a love-fest. Sorry for that disturbing image.

Plato, as opposed to “The Greeks,” privileged mathematical education for many reasons, but none more than this: he thought it the best propaduetic to philosophy. We all know what was written on the arch above the entrance to his academy: “Let no one ignorant of geometry enter here.”

It’s like this at Wabash too, isn’t it. You won’t be admitted here unless you’ve taken quite a bit of mathematics in high school. And once you’re here, you’re required to take a course or two in mathematics. Science and economics students are required to take a few more.

Why do you think that is, by the way? I guarantee you that unless you become an engineer or a physicist (or something similar), you will never have to use trigonometry. Never. It will never happen that you’re on, say, a plane that’s lost its engines, and the pilot will come running through the cabin in a panic screaming, “For the love of God, does anyone know the sine of 30 degrees?” So why do we make you learn it?

Well, one reason is to train you in argument. And as childish (even tyrannical) as devotion to argument often is (more on this later), properly wielded, it can also be salutary, of course. Arguments (like jokes, by the way) can alert you to absurdity by focusing your attention on the unsavory conclusions that can be drawn from faulty premises, premises that otherwise might seem irresistibly seductive. In fact, without argument, one will only come to know these unsavory conclusions—and hence, the infelicities of the premises that spawned them—the hard way, as they say. Argument, then, can liberate you from the oppressive (and often painful) tyranny of foolishness. This is one of the chief reasons that mathematical training in argument plays such a central role in liberal education.

Speaking of both liberation and pilots, have you ever noticed, when you’re taking off in an airplane and you watch the horizon way off in the distance in all directions, and receding further with each passing moment as the plane rises higher and higher, how you suddenly feel, really feel, the vastness of the world, and it hits you how utterly insignificant your tiny little part of it now seems? What felt like a complete universe down there, is just a tiny dot from way up here in the plane, and getting tinier by the second, while at the same time, the entire world is opening up before you. Oh captain, my captain.

The liberally educated person takes as his or her domain of inquiry (as opposed to the process of mere data acquisition) everything, from the smallest microscopic part of the physical world to the galaxies and stars beyond, to the largest part of the conceivable world,