Saturday, September 18, 2004

Mathematical Intuition and Millennium Problems

I have been busy juggling these past few weeks. But now the most recent paper deadline is met; the presentation materials for Tuesday are in hand; my computers are stable and mutually consistent; my desk and my home are both clean and organized; and I lack neither staples to attach my papers nor staples to make my meals (man does not live on ramen alone). I've even managed to deal with most of my technical support mail. So this evening I sit with pen and pad and a pot of tea to finish a thought from the start of the month. (As an aside: I wrote most of this, including the previous sentence, on Thursday night -- but I'm just now typing on Saturday morning.)

At Half-Price books two weekends ago, I found Keith Devlin's book The Millennium Problems sitting neglected on a corner shelf. I have a special place in my heart for popular writing by mathematicians like Ian Stewart, Ivars Peterson, and Martin Gardner -- I had their books checked out from the Bel Air library half the time when I was in middle school, and I continue to find them entertaining and interesting today. I know a mite more mathematics now than I did then, but the writing often focuses on history, applications, intuitions, and puzzles centered around some mathematical idea or set of ideas; and so even when I'm familiar with the mathematical idea, I still enjoy the reading. For the same reason, I enjoy reading articles on science written for a popular audicence. Devlin, who may be most familiar as the Math Guy on NPR, has an engaging writing style, and his topic -- the Millennium Problems, seven great unsolved mathematical challenges with million-dollar bounties on their heads -- is something I wanted to know more about. After reading the introduction, I bought the book in happy anticipation of a good read.

Besides, the man has an extinct species of possum named after him. What's not to like?

As I read, I found exactly what's not to like -- at least, I found points in the presentation that I very much disliked. Devlin undertook a monumental task: not just to explain the history and the impact of the Millennium problems, but to give a flavor for the problems themselves -- assuming a reader with only a pre-calculus grounding in high school mathematics. As Devlin puts it,

Even achieving a layperson's knowledge of what [the Millennium problems] are about takes considerable effort. I believe the effort, however, is worthwhile. Aren't all pinnacles of human achievement of interest?

But the challenge was too great. Devlin explained the problems at a lay level better than I could have done; but then, I don't know how someone could think about some of these problems without knowing -- really knowing, not just having some vague familiarity with -- some group theory, elementery number theory, and the basics of differential calculus. Devlin gives a creditable introduction to some of this background, but it is not enough, and by the end he is reduced in his frustration to section titles like The Hard Stuff, Made as Easy as I Can and The Hodge Conjecture: Not for the Faint-Hearted.

For most readers, the description of cohomology classes in the last chapter would probably be impenetrable. For me, it was frustrating. After all the effort of sketching in the technical background, an effort which will likely go unappreciated by most of your readers, why not finish the sketch for me? Just a little more technical detail? Somehow I ceased to react to the book as the popular exposition it was meant to be, and started to treat it as an inadequately fleshed out technical work -- which it was certainly not meant to be.

I was irked. I was sufficiently irked that I wrote a letter to a friend which fell somewhere between a lecture and a tirade, in which I tried to describe what I disliked in Devlin's book. I wondered almost as soon as I sent the letter whether I hadn't been unduly harsh -- but Yi shared an office with me for a while, and had plenty of opportunity then (and since) to become accustomed to the flavor of my unprompted lectures. So I expect she will take it in stride.

After reading Devlin's book, and after spending some time grumbling, I began to think more about the nature of mathematical intuition. In the first chapter, Devlin has a section entitled Why Are the Problems So Hard to Understand? in which he argues that the level of abstraction from everyday reality makes mathematical ideas harder to explain to a lay audience than are ideas from any area of science. At the same time, as he says near the end of the book,

... a trained human mind that has thought long and hard about a particular problem frequently develops intuitions that prove to be correct.

The problem, then, is not that there is no intuitive picture of the ojbects of higher mathematics -- just that such pictures are inaccessible without a lot of patient thought. I disagree with Devlin's assessment only in one point: I believe there are concepts in modern physics and chemistry which require abstractions which are fully as inaccessible to the lay reader as any of the Millennium problems (which is probably why two of the Millennium problems -- mathematical understanding of the Yang-Mills field equations and the solution of the Navier-Stokes equations -- come directly from modern physics).

So why is popular science still so much more popular than popular mathematics? And why does modern math have such a reputation for impenetrability, particularly when -- in contrast to the experimental sciences -- intuition can be had for only the cost of time and thought? Part of the trouble, I think, is educational. Think when the scientific method was first mentioned to you in an elementary science class -- and then think when the idea of a mathematical proof was first introduced. Do you know more names of physicists from the past two centuries than names of mathematicians? Difficulty alone cannot account for the difference in perception; nor, I believe, can the relative distance from everyday reality.

I don't have good answers to the questions why is it so hard to understand? and why should science seem any easier? But I'm still thinking on it. Maybe one day I'll gain enough psychological intuition to understand how people gain intuition -- but I doubt it. Mathematics is easier to understand than people are.

Works considered

This started off longer, and got shorter as I realized I wasn't sure I agreed with my own ideas. Consequently, this list is not a collection of works cited, but rather writings that I looked at during some version of the above.

  1. Nature's Numbers (Ian Stewart); Islands of Truth: A Mathematical Mystery Cruise (Ivars Peterson) -- Peterson's book was definitely one that I kept checking out from the library. I forget which of Stewart's books the library had, but Nature's Numbers is a good one in any case.
  2. Mathematical Association of America -- The MAA does a lot with mathematics education and popularization. Both Ivars Peterson and Keith Devlin write regular columns for the MAA, which are available from the web site.
  3. The Honors Class: Hilbert's Problems and Their Solvers (Paul Yandell) -- Perhaps a natural complement to a book on the Millennium Problems. The Millennium Problems were deliberately introduced 100 years after Hilbert introduced his list.
  4. The Man Who Loved Only Numbers (Paul Hoffmann); A Beautiful Mind (Sylvia Nasar) -- Biographies of two modern mathematicians. I thought both books were fascinating. No, not all mathematicians are so nutty.
  5. Men of Mathematics (E.T. Bell) -- Biographies of some major mathematicians up through the end of the nineteenth century. There are more accurate historically accurate books on the history of mathematics and on historical mathematicians; but it is hard to find another book which is so enthusiastically and charmingly written. Mathematicians were Bell's heros; and he made a few of them my heros, too.
  6. Fermat's Engima (Simon Singh) -- I have intended to finish reading this book for years now. Two of them, actually. But I particularly wanted to mention a quote from the dust jacket, which I repeat here in full:
    Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room and it's dark, really dark, and one stumbles around bumping into the furniture. Gradually you learn where each piece of furniture is, and finally, after six months or so, you find the light switch and suddenly it's all illuminated and you can see exactly where you are.
    -- Andre Wiles
  7. The Value of Science: Essential Writings of Henri Poincare (Henri Poincare; ed. Stephen Jay Gould) -- Poincare has much to say about the role of intuition and experiment in mathematical inquiry. A profound and prolific mathematician, Poincare turned late in life to writing for a popular audience. His publications were deservedly successful, and are worth reading as a model of accessible exposition, as well as for the ideas they contain on the nature of mathematics and of science.
  8. Linear Differential Operators (Cornelius Lanczos) -- Really, any of Lanczos's books would serve as well, but Linear Differential Operators has a few wonderful paragraphs at the beginning in which he describes the goal of his expository style. In contrast to the very formal presentations of N. Bourbaki (Bourbaki was actually the nom de plume taken by a school of French analysts), Lanczos concentrates less on the details of the analysis and more on high-level intuitions. The Variational Principles of Mechanics is another excellent text in the same style; it was one of my primary texts in learning about classical mechanics.
  9. Catastrophe Theory (V.I. Arnol'd) -- Usually Arnold writes more technical books, and when I found this on the shelf at Black Oaks, I was delighted to find that it was written for a general audience. It's short, interesting, and full of both dry humor and intuition.