April 21, 2015 8:18:27 AM
Take a prime number that when you divide it by four leaves a remainder of one. For every such prime, there is one and only one way it can be expressed as the sum of two squares. The smallest such prime is 5, and 5 = 12 + 22. Another example is 41 = 42 + 52. If you looked hard enough, you could find the two squares that add to 65537 or any bigger prime. Not every such prime has been tested this way, of course, since there are infinitely many; Fermat proposed the theorem, and Euler provided the first proof. I have, ever since learning about this theorem years ago, enjoyed doing the mental arithmetic to disassemble such primes into their squares. It is sort of comforting to know that it works every time, that there is something in the weave of numbers that demands that this be true. I have looked into books about number theory (the branch of mathematics that studies those mysterious whole numbers 1, 2, 3 and so on) to see how the proofs work, but I have not been able to make much of a dent in them; I admire the number theorists who can do such work, but I am not one. I thought of Fermat's theorem because there are references to it in Mathematics Without Apologies: Portrait of a Problematic Vocation (Princeton University Press) by number theorist Michael Harris. Harris did get me a little further to understanding the theorem, but this is a book about doing mathematics, not a book of mathematics. It is a refreshing look into a mysterious field; for me, large swaths of the book were mysterious because the mathematics was dense, but that's the "problematic" part of the subtitle. As a picture of what drives mathematicians, and what it is like being a professional one, this proves an entertaining, if necessarily complex, explanation.
Harris's title is a bow to a famous book by the number theorist G. H. Hardy, A Mathematician's Apology (1940). Harris plays upon the use of the word "apology;" Hardy used it not as a request for forgiveness but as a formal justification. Hardy was writing at the end of his career, and regretted that since mathematics was a young man's game he was spending his time explaining about mathematics and no longer was doing mathematics itself. Nonetheless, he explained the beauty of pure mathematics, and part of what he found beautiful was that it was useless; it was using mental powers for the intellectual thrill of knowing in the abstract, without trying to make something practical of it. This timeless beauty has had to be updated, as Harris accounts. Ironically, the "useless" work of research in number theory includes theorems on prime numbers and on factoring huge composite numbers, and the work has proved invaluable, because every time, for instance, you put your credit card details into a website, they are (one hopes) encrypted using huge primes. "Number theory," writes Harris, "is not merely useful: it is the bedrock of modern shopping."
The question "Why do mathematics?" is coupled with "Why fund mathematics?" Governments should be interested in having people skilled in using mathematics that has practical benefits, and of course much of math has such benefits. But what about pure math done with no goal of application? Part of the reason governments and colleges ought to fund such programs is that no one knows (just like Hardy) what is going someday to have application and what is not. But mathematicians like Harris are not driven by getting something done that is or will be useful. They are interested in mathematical patterns and proofs that are beautiful. There is a long discussion of beauty in mathematics, and even non-mathematicians who can understand the proof, known to Euclid, that there is no largest prime number might get an inkling of the proof's mathematical beauty. Whatever beauty Harris and his fellows are seeking and finding is something like that, though far more complicated. The beauty isn't like the beauty of mainstream arts, which can be far more widely appreciated; Harris does not accept the argument that mathematics should be supported as art is. He ends up with the conclusion that mathematics is a form of creative play. Mathematicians do what they do, solo and as colleagues, because it gives them pleasure. The question of whether they should be funded in such play is left hanging (well, Harris comes down on the funded side). "If it's so enjoyable, wouldn't we do it for free? Well, maybe we wouldn't be so efficient or creative if we had to work for a living - in our spare time, so to speak - at something we presumably don't enjoy so much."
But don't mathematicians have the best parties? Harris shows that mathematics has a surprisingly strong showing in popular cinema, citing Good Will Hunting, Proof, A Beautiful Mind, and Pi as a few examples. He admits that even with such fascination, there is a public feeling that a party of mathematicians would be less fun than a party of, say, "industrialists, Russian Orthodox theologians, or anyone involved with the movies." Harris notes that at one traditional conference, "The often-prodigious consumption of alcohol starts just after dinner and certainly contributes to social integration - and, thus, over the long term to fruitful scientific collaboration; but its direct effect on mathematical creativity is questionable." At champagne receptions in France, the results are similar: "Mathematical notes are compared for the first glass or two, after which conversation reverts to university politics and gossip." There are stories that drugs have proved an inspiration for mathematical invention, but Harris says the evidence is ambiguous. One possible exception was the mathematician Paul Erdős, who maintained his stamina by use of coffee and amphetamines, and it must have worked because he was extraordinarily prolific. A friend bet him he couldn't stay off the amphetamines for a month, and Erdős successfully won the bet, but complained he didn't get any work done. "I'd have no ideas, like an ordinary person. You've set mathematics back a month."
There is good discussion about how mathematicians gain authority among themselves by making significant breakthroughs recognized by others. There seems to be competition, but overall a strong feeling of collegial, communal effort. Harris reflects on the role of mathematicians on the recent financial crisis, reminding us that Aristotle said that philosophers could easily be rich but ought to have other ambitions instead. Nonetheless, computers have allowed some mathematicians to "pile up micropennies by the billions." I didn't understand the mathematics of the idea that came to him in a dream enabling him to work toward a particular solution, but his description of how he worked the dream into something useful is an engaging explanation of a mathematician at work. Harris describes the famous affinities between music and mathematics, but who knew there was an "Oxford Math Rock group" named "This Town Needs Guns," which created and performs 26 is Dancier than 4 which is in 26/8 time. In between chapters is an imaginary dialogue with a performing artist, "How to Explain Number Theory at a Dinner Party," which is a long answer to a question Harris and friends all get asked, "What is it you do in number theory, anyway?" This gets to be daunting; the dinner guest gets an earful winding up with "the guiding problem of the Birch-Swinnerton-Dyer conjecture, which concerns equations of degree 3 in two variables, through problems of increasing complexity...." I admit I didn't come near to understanding all the math in Harris's book, but he is an entertaining writer, drawing from Eastern philosophy, rap music, and (especially) the novels of Thomas Pynchon. If you are stuck with a mathematician at a dinner party, his sometimes rollicking volume will convince you that Harris is the one to pick.