Sciences Participating with Arts & Culture in Education

Interview with Tom Fox

Most people who take a math course at Dawson, and many who do not, know Tom Fox. Depending on who you talk to, you are liable to hear that Fox is the best teacher at Dawson , that he meanders through the halls belting out Russian opera, or that he is a scotch-guzzling, cigar-wielding mathematician who spends half his weekends in jail. He may be both of those things, but he is at any rate a thoughtful man with a number of insights into math, music, and life in general.

The Apprenticeship of Tom Fox

Fox did a bachelor of arts at Oakland University, a small university near Detroit. He did his graduate work at McGill. Oakland was a “fruity, artsy fartsy kind of place,” so they had very loose requirements. Fox did a major in math, which requires nine math courses over four years. He took nineteen, including masters’ courses. He also took music, philosophy, and history, but no science at all. During his last term, the registrar called and said, “You haven’t fulfilled the science requirement.” The philosophy at Oakland was that you had to do a little bit of everything. Fox said, “I’ve taken 19 fucking math courses! What do you want from me?” They insisted that he take at least one science course, so he tried to take physics 001, physics for elementary school teachers. His plan was to never go to class and show up for the final. The advisor wouldn’t let him take the class. Instead, he phoned the physics department, asked for advanced physics courses for physics majors, and told them to sign Fox up. So he got stuck doing quantum mechanics to fulfill the science requirement! From his point of view, it was absolutely pure math. “If you ask me how to roll a ball down an inclined plane, I have no fucking idea. But if you ask me about relativity, that I can do. Kinetic energy could be a third m v to the ninth for all I care. What do I care? I don’t know any chemistry. Malted barley and spring water make scotch, that’s the only chemistry I need to know.”

Math or Music?

Fox has been singing since the age of six. When he was 17, he had to choose between mathematics and music, and as we all know, he chose mathematics. Why?

FOX: I only realize it when I look back. Certainly, in high school, I found math boring. I could do it, but it wasn’t interesting. It was mechanical, stupid. I did well in a math competition. I went for the free food and there was a guest speaker. He showed us stuff that was just beautiful. Just beautiful. Challenging, but beautiful. He showed us the philosophical side, the non-mechanical side.

M: Do you remember what he showed you?

F: That the real numbers are uncountably infinite. When all you’ve seen is mechanical shit, it’s beautiful. Fox occasionally gets paid to sing, but prefers doing it for fun. “I’ve done concerts of Italian opera, Russian love songs, spirituals, a tribute to Paul Robeson – one of my heroes. He was maybe the first, you know, like Bono? Social activist/singer. One of the world’s greatest singers and a big time shit disturber.”

More on Music

F: I’ll tell you an exciting thing that happened to me musically. The CBC French, Radio-Canada, is doing a documentary on what happened to American draft-dodgers that came to Quebec. Now see, I left the States in 69, and after I had arrived here I got my draft notice for the American army. And I wrote back to them and told them I was busy studying mathematics, so fuck off. So I’m in that sense a draft-dodger. I couldn’t go back to the States for many years or they’d arrest me. Then they gave up on that and just said “Forget it, not interested in you anymore.” But, anyway, so French CBC is doing a documentary on what happened to American draft-dodgers after they came here. So they gathered together a bunch of us and interviewed us in French, so I got to trot out my great French. It’s actually appalling, but still…

M: “N’est-ce que pas?” [quoting Fox from class earlier in the day]

F: Actually, it wasn’t as bad as they feared. Anyway, so they interviewed me in French and on camera. It’ll appear on a documentary some time this summer. When I was talking to the guy, he said “Do you have any hobbies?” and I said “Well, I sing.” And he said “Do you sing in French?” and I said “Well, you know, there are very few songs that I know in French, or that I’ve ever performed in French.” One of the songs, in fact the first song, that I ever learned in French, is a very famous song by Boris Vian called “Le déserteur.” The deserter. It’s an anti-war song written in 1954. It’s very, very famous. The guy I was talking to immediately knew the song and he said, “Oh, can you sing that? On camera?” I said “Sure.” And so a couple of days later, I and a pal of mine who plays the guitar whipped down to French CBC and they filmed me singing The Deserter in French. And that’s going to be on the documentary this summer. So I’ll be on French CBC, singing “Monsieur le president.” Great song. Absolutely great song.

M: Is it true that you play the saxophone too?

F: Nope. I don’t play any instrument whatsoever.

M: There are a lot of lies about you around Dawson.

F: I would never call myself a musician. I’m not. I mean, I can’t play any instruments, my music theory is minimal, uh…but I can sing! So I’m a singer, not a musician. Everybody in my family played piano. They also sang, but they knew music stuff that I never did.

Fox at McGill: Research

F: I came as a graduate student in 1969. I already more or less knew what I was interested in. I was interested in abstract algebra and, more particularly, category theory, which is an abstract algebra, an abstract abstract algebra. When I was at Oakland a teacher had given me a book on the subject that I of course had never heard of and just did some extra work to keep me busy.

Category theory was a very new subject in the late 60s. It was kind of the new hot topic and it was something people had just started working on. And he gave me a book when I was an undergraduate and I found it very interesting. Category theory makes an attempt to unify different branches of mathematics, to look for common themes between topology and algebra and geometry and different pieces of math. I found that very nice. I liked the idea of some kind of grand unified theory. One of the stars of the subject was here at McGill and there were some other people at McGill who were very well known in abstract algebra. When I got to McGill, I more or less knew what I wanted to study, so I did a master’s and then a PhD.

My master’s was a mixture of logic and category theory. And my Ph.D. was more [pause]…I got off onto category theory and another subject called deformation theory, which was a subject that begged for unification. It started out as a subject in analysis, and then there was something called algebraic deformation theory, the deformation theory of algebraic structures. It was suggested by my thesis adviser that there was a unified way of looking at these so that it was all one big subject, and category theory provided the tools for unifying these different branches. A few years after I did that, somebody else started worrying about “Is there a deformation theory for Hopf algebras?” which nobody had done, and I could say “Oh yeah, I in fact already have this machine that will do deformation theory for anything.” And so the machinery’s already there. That wasn’t what it was designed for. It was designed to unify things that were already known. But it also can be used if you have something new that you want to do. I did a lot of research in these things for many years.

M: So what kind of a common theme can you bring out between, say, algebra and topology?

F: [sigh] That’s kind of hard for me to explain. It gets a bit technical. In some ways, it goes back to group theory. Groups are a powerful tool in many branches of mathematics: in number theory, in topology, and of course in algebra itself. And there’s a subject called homological algebra or homology theory, which takes different mathematical structures…There’s a topological version of this, where you take a topological space, some geometric object, and find a group that describes it. Then there’s the algebraic version where you have some algebraic structure and you find a group that describes it. The common thing is these groups.

Before I did my thesis, somebody else found a way to unify these homology theories, so that they are in fact the same machine, which can be applied to a geometric object or an algebraic object or a topological object and give you information about the thing you’re looking at. And somebody before I did my thesis had figured out a way of building the big machine, so that you could drop in a topological space and get a group out, or drop in a chicken and get a group out, or drop in anything you want and get a group out. And that group will tell you about the chicken that you dropped in.

That machine had already been built. And it was thought that a similar machine could be used for something called deformation theory, which at the time was another hot topic. And so my thesis was on homological algebra and deformation theory and how these things could be unified. And it just turned out that a few years after that, people started working on something called quantum groups in abstract physics and it became a very hot physics topic. And what they needed was a tool for doing deformation theory of bialgebras. And by God, I could say “Oh yeah, deformation theory for bialgrebras, all you do is apply this unified theory to bialgrebras,” and Booom and it works. And all of a sudden, I looked like a fucking genius even though I had no idea that that was going to happen. For a while, my brilliant work – it looked very brilliant, because I had foreseen it. Of course, I hadn’t foreseen it at all! But anyway, it kept me in research grants for quite a while, because people were very impressed that this actually was good for something. Although I did it because I thought it was beautiful. I thought it was a nice idea that there could be a grand unified theory that would do everything. That’s why I did it. That’s why I do any math, it’s because I like the beauty of it; it’s just a nice idea. Instead of having five different topics that you do in five different ways, there’s one way that applies to them all.

Teaching versus Research?

F: I’ve always taught. I was a member of the future teachers’ club when I was thirteen years old, in junior high school or whatever it was.

M: So it was never a decision between teaching and research.

F: Yeah. In fact most people that came into the CEGEPS as teachers right after their Ph.D more or less stopped doing research because it just takes too much time and energy. But after I’d been here about five years, I did manage to do some research. I wrote several papers during that five years. And then the province started a program of supporting CEGEP teachers that were doing research. And so for a number of years, I got time off from teaching to do research, so I would teach half of the year, and do research the other half of the year.

But then eventually, the research angle got too hard. For one thing, I answered some questions that I’d been working on for many, many years. And maybe just old age or laziness, maybe my brain is just fried. It’s just too hard. I spent ten years thinking about A PROBLEM. That does awful stuff to your brain after a while. And once I answered that question, which I eventually did, I think I just didn’t feel like starting it again. You know? [That was] about ten years ago now, so I was fifty. I’ve more or less stopped. Since then, I’ve written a couple of things that are just expository chit-chat.

But I enjoy teaching. I’ve always loved teaching. Teaching’s a pleasure. I get to spend time yakking about stuff that I’m really interested in, convincing students that it’s beautiful. And I really think it’s beautiful. You know, what do you want? I get to spend my time yakking about something I love. And I get a paycheck every two weeks for doing this. How bad is that as a job? It’s fabulous!

Eight Beauties of Mathematics

A former Dawson student asked Fox for readings about the non-technical, philosophical side of math. Fox didn’t find exactly what he wanted to give her, so he wrote “Eight Beauties of Mathematics,” for her. What did he want to convey?

F: Why is math beautiful? Because that’s the point. It’s not about the mechanical shit, grinding out numbers and equations. Say you want to be a ballet dancer. Sure, you have to work out and build muscles, but that’s not the point. The point is the beautiful thing you can do with it. In school, all you learn is the mechanical stuff. But no mathematician does that! Except maybe Fournier [Richard Fournier, another Dawson math teacher]. Fournier and I have different attitudes…”

M: I was reading your “Eight Beauties of Mathematics” booklet and I saw special thanks to Diana Dubrovsky and Chaim Tannenbaum [respectively, math and philosophy teachers at Dawson]

F: Well, because, they’re two people that I talk a lot to about mathematics as they both, especially Chaim, have a very philosophical interest in mathematics, as I do. We’ve discussed things and sometimes he’s asked me questions of how to look at something or another. He knows math. He knows tons of math, but I spent a lot of time thinking about not only what the facts of math are, but why it’s important or why it’s beautiful, which is what I try to explain in that thing I wrote last summer about the eight beauties of mathematics. Not only that these things are true, but why is that wonderful that that’s true, why is it an amazing thing that somebody thought of it, and why is it beautiful? And that’s very much Chaim’s approach to mathematics, from a more philosophical point of view. And so Diana and Chaim, you know I often talk to Diana about mathematics and Chaim…I talk very little with Fournier about mathematics. Because, as I say, he and I have a very different approach to the entire subject. I mean, if I have some technical question, I can ask Fournier. He knows everything. He’s very, very powerful, but he doesn’t know why, he doesn’t care why antidifferentiation is the same as…addition. Or adding chickens or something. And I do. It sounds stupid, but it’s true. I think about these things. I think about what subtraction means. Fournier doesn’t. It’s not the kind of thing he worries about.

M: I told him about your idea that the complex numbers are too beautiful to be real, and he said [hissing noise followed by poor francophone accent]“Chaaaaaauh…When Fox tells you stuff like that, tell him you don’t care about poetry, you want to hear about mathematics!”

F: That’s right. Exactly. Well, that’s the difference between Fournier and me. I have some poetic view of the entire thing and…I hope I know some technical mathematics too, because you have to back it up, right? I mean, that’s one thing that you can’t do is do math and just bullshit, anymore than you can do music without knowing how to actually hit the notes. It’s very nice to have a loving appreciation of music, and appreciate the beauty of it all, but if you actually want to create things and understand it, you also have to have some technical competence. It’s similar to math in that respect.

Beauty and Math

F: There’s so many beautiful things in math. My minor as an undergraduate was philosophy. As an undergraduate I took tons of philosophy courses. I took no science courses whatsoever. Zero physics or chemistry or biology or any of that stuff. I took music, philosophy, mathematics. That’s always interested me. So I’ve always been more interested in the more misty, philosophical side of mathematics than the technical grinding out formulas and stuff. This is a way in which Fournier and I are very different. He has some function, he writes down the power series for it and then proves a theorem that the ninety-seventh coefficient of the power series is less than a third. I don’t give a flying fuck what the ninety-seventh coefficient of the power series is. What does that tell you about…life? I’m not saying it’s easy at all. It’s extremely hard. But it doesn’t give me the philosophical satisfaction of saying something about fundamental things in geometry or logic or…

M: Yeah. Maybe the argument that you use to say something about the ninety-seventh coefficient is –

F: THAT might be interesting

M: – but the statement isn’t.

F: That’s right. If you had some new way of viewing the entire problem, and I don’t just mean some nice computational crap, that might be interesting. But just some long, complicated computational junk and then you come out with a third, and I say “Engh!” That doesn’t mean it’s easy, by any means…But Richard and I are different in that way. I have an artistic view of math and he has a more technical view of math.

Math as Art and Science

M: What’s your reaction when people talk about the difference between math as an art and math as a science?

F: Well, I mean there is a huge difference. People, some people, many mathematicians, I don’t know what the division is, maybe fifty-fifty, think of math as…you have some specific engineering problem or a physics problem or some computational thing that you want the answer for. And then there are other people that view math as some sort of discovery process where you’re building universes that are pretty. They may have no applications whatsoever to anything, but they’re just beautiful. And that’s why you do math, because it’s just beautiful to see these patterns, and discover universes that other people never even thought about.

In some sense, of course, the amazing thing is that these things become useful. I remember when non-Euclidean geometry was invented in the beginning of the 19th century. Non-Euclidean geometry? Space is curved? A world where there are no parallel lines? Or where there are too many parallel lines? I mean this is just abstract…this is like, you know, you’re smoking too much dope. You know, it can’t possibly have any use for anything, because the real world is so obviously Euclidean. But of course it’s not! As it turns out, it’s not!

All the stuff that’s done in linear algebra, matrix algebra, linear transformations – it was all invented in the middle of the 19th century by Cayley and people like him, and people just said, “Yeah, this may be nice, it may look cute, but it’s utterly useless. What do you care about transformations of four-dimensional space? That can’t possibly have any applications to anything that’s actually useful.” AS IT TURNS OUT, of course, not only do you need that for modern physics, you also need that for economics. I mean, some slob won a Nobel Prize in economics for applying linear algebra to solving economics problems. You think of an economic situation as being a multivariable space where the variables are quantities of chickens and coal and wood and then you write that down as a list. But that list is just a vector, right? If you have ten items and you write them in a list, what are you looking at? You’re looking at a ten-dimensional vector, if you think of it abstractly. And if you apply the abstract ideas of linear algebra to that, and the mechanics of linear algebra to that, you can discover things about supply and demand of chickens, coal, and wood. So in fact, these things turned out to be very useful, although that’s not why they were made. They were made just ‘cause they’re pretty…Leontiff input-output models, a very famous thing. It’s very useful in probability theory, where you think of Markov processes as transformations of a multidimensional space. You know…It’s nice! It’s beautiful! And properties of matrix multiplication turn out to be things about mice running around in cages. But that’s not what I do…

A Mathematician’s Apology

F: Have you read A Mathematician’s Apology?

M: Yeah-

F: It’s a beautiful book –

M: - I found it on the internet [http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf].

F: Okay. Great book. THE great book about why one does mathematics. And of course, Hardy was an absolute, on the art side of mathematics. Although what he did was terrifically technical.

Fox and Philosophy: PHILOSOFOX!

Fox: I don’t read much philosophy anymore. In the old days, I used to read lots of things. I read the Greek philosophers and lots of Bertrand Russell. I had a lot of affinity with Bertrand Russell because of course he was also a mathematician and a logician. Not that I’m putting myself in the plane with Bertrand Russell, but I’m sort of a mathematician with a philosophical outlook and he was really a philosopher with a mathematical point of view, and so I read tons of his stuff. But in recent years, I haven’t been reading much philosophy. I read history books. I learn Russian opera. Keeps me busy, especially since I don’t speak Russian. It’s hard work learning…

Fox and Language

F: I’ve done whole concerts of Russian songs, singing for Russians. And they say [excellent Russian accent], “Oh, it’s very good, speaking excellent Russian. Where you learning speaking Russian?” I say, “I don’t speak any Russian.” I know the songs, and I know what they mean, but I can’t actually speak Russian. It takes a lot of work.

M: Yeah. All your old students say that you have…I don’t want to say a hatred for learning Russian…

F: NOOO! I LOOv…I mean, I wish I COULD. I’ve always been totally incompetent at learning foreign languages. You know why? ‘Cause they don’t make any fucking sense! Language basically doesn’t make any sense. Right? There’s no reason that “dog” should be “dog” instead of “rug”, right? I mean, why? I dunno. ‘Cause one’s a dog and one’s a rug. But why? No reason. Mathematics of course, has to be the way it is. Makes sense! So it’s easy to remember. Because you don’t remember the facts, you remember WHY. You remember the reasons. And then it’s inevitable. Those facts, you don’t have to remember them, because you know why. But languages, you just have to sit down and memorize ten thousand arbitrary words that just happen to be “rug” instead of “dog”. I can’t do that. Maybe my mind or my memory is no good or something. And grammar has never made any sense to me because the rules of grammar are mostly just arbitrary nonsense. How am I supposed to know that “cat” is masculine or feminine in French? I don’t know. La chat. Le chat. How the fuck am I supposed to know? Couldn’t it be the other one? Is there any reason it shouldn’t be the other one? No.

M: No, just complicated historical reasons…

F: So how am I supposed to remember? Give me the reason, and then I’ll tell you! If there’s no reason, how am I supposed to remember that? I’m very bad at that. I’ve always been bad at that. I flunked almost every language course I ever took. Very bad.