Fermat’s Enigma, by Simon Singh
My Brain Is Open, by Bruce Schechter
The Man Who Loved Only Numbers, by Paul Hoffman
Men of Mathematics, by E.T. Bell
History of Pi, by Peter Beckmann
e: The Story of a Number, by Eli Maor
The Nothing That Is, by Robert Kaplan
An Imaginary Tale, by Paul Nahin
The Penguin Dictionary of Curious and Interesting Numbers, by David Wells
What Is Mathematics, Really?, by Reuben Hersh
The Night is Large, by Martin Gardner
John Kenneth Galbraith remarks in one of his books that if you’ve ever worked on a farm nothing else ever seems like work. Those of us who have studied mathematics at university level can make a similar claim. If you’ve ever grappled with advanced math, the study of other subjects seems like a joke. How we used to scoff at friends pursuing subjects like English! “What, they’re giving you a degree just for reading novels? Hooo-hooo!” And it was a stock comment among math students whose girlfriends were non-mathematicians that when we went to their rooms we browsed their books; but when they came to our rooms they never tried to browse our books. The study of higher math makes you a terrible intellectual snob.
My own math snobbery, though I shall not deny its existence, was softened and muted by the awareness that I am no good as a mathematician. I loved the subject, and still love it; but alas, it is an unrequited love. I hit the wall with a topic called Functional Analysis, of which let it be sufficient to say that if you feel fairly sure that three dimensions of space are quite enough for anyone to cope with and four dimensions a concept properly belonging to the realm of science fiction, try developing theorems in a space with an infinite number of dimensions. Yet still I cannot shake off the old affection. I love to browse my college textbooks and am a keen consumer of books about math aimed at a general readership, of which there is something of a flurry at the moment.
I think it started with Andrew Wiles’s proof of Fermat’s last theorem, which made newspaper headlines in 1993. (It later turned out that the proof was flawed; but Wiles fixed the flaw, and the theorem can now be taken as definitively proved, after 360 years of concentrated effort by the best minds on the planet. You think math is hard? You don’t know the half of it.) Simon Singh wrote the whole business up in Fermat’s Enigma, which attained the well-nigh oxymoronic status of a math bestseller two years ago. Singh’s history was a bit wobbly — he had the Dark Ages ending in the seventeenth century — but his math was excellent and it was refreshing to see my old flame on the bestseller lists.
At about the same time the death of the number theoretician Paul Erdős occasioned no less than two biographies: Bruce Schechter’s My Brain Is Open and Paul Hoffman’s The Man Who Loved Only Numbers, both published last year. I thought at the time that this was a bit over the top for a man to whom pretty well nothing happened, who had no interests whatever outside math, who fell asleep when not in the company of mathematicians and whose most intimate letters went like this: “Am in Sydney. Next week Budapest. Let p be any odd prime …” Neither of the authors was very sure-footed with math, either. Hoffman betrayed a gross misunderstanding of the concept “transcendental number” while Schechter thought the cables of a suspension bridge form catenaries. Really! Still, either book (I cannot see who would need to read both) is worthwhile as a portrait of the extremes of human oddity.
Anyone who executes the biography of a mathematician labors in the shadow of E.T. Bell’s 1937 classic Men of Mathematics . I have mislaid two or three copies of this book on my travels but always ended up buying a new one. For any person who wishes to have any acquaintance whatsoever with the Queen of the Sciences, Bell is simply indispensable. He not only knew his math, he succeeded in what I, having mixed a good deal with mathematicians, would have thought an impossible task: he humanizes his subjects. There is, for example, the heartbreaking story of Évariste Galois. Challenged to a duel, Galois sat up through the night before the encounter:
All night … he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin “I have not time; I have not time,” and passed on to the next frantically scrawled outline. What he wrote in those desperate last hours before the dawn will keep generations of mathematicians busy for hundreds of years.
He was killed in the duel, of course, and — as Bell remarks, barely restraining his own feelings, “buried like a dog.”
Still, as interesting as biographies of mere human beings can be, there is nothing to beat biographies of numbers. I possess three of these: Peter Beckmann’s History of Pi, Eli Maor’s e: The Story of a Number, and — in an advance copy from the publishers, who will have the book out this fall — Robert Kaplan’s The Nothing That Is, subtitled A Natural History of Zero. The first of these is not as much fun to read as it should be; the second is very nearly a textbook — masses of footnotes; the third I can recommend on content, though not on style. Still making its leisurely way to my door from Amazon.com (what exactly does “usually ships in 24 hours” mean, Mr. Bezos?) is Paul Nahin’s An Imaginary Tale, which tells the story of i, the mathematical symbol for the square root of minus one.
For a more, shall we say, social approach to the secret lives of numbers, a must-buy for anyone who has, or wants to acquire, a feeling for the little devils is David Wells’s The Penguin Dictionary of Curious and Interesting Numbers . You might not think there is anything very curious or interesting about 371, or 5282, or 111,777; but Mr. Wells will set you straight. (371 equals the sum of the cubes of its digits; 5282 is the number of ways you can place 8 non-attacking rooks on a chessboard; 111,777 is “the least integer not nameable in fewer than nineteen syllables,” a famous paradox — count the syllables in that description.)
Math is not all numbers, of course. Since the time of Pythagoras twenty-five hundred years ago, math has played a part in deeper speculations about meaning, natural laws, and the structure of “reality” (one of the few words, said Vladimir Nabokov, that mean nothing without quotes). The best recent venture into this territory has been Reuben Hersh’s What Is Mathematics, Really? Well, what is it? What, exactly, is the status of mathematical truths? Do they exist out there somewhere, waiting for us to discover them by painstaking inquiry, or stumble upon by chance, or seize by a sudden flash of insight? Or are mathematical truths mere social entities, like marriages, sonatas or declarations of war, which would not exist if there were no human beings? Suppose there were no human beings: would two plus two still be equal to four? Hersh covers the whole territory very readably and offers a clear conclusion. I do not myself agree with that conclusion; but Hersh is one of those writers who forces you to take out your most fundamental assumptions and examine them carefully.
And then, of course, there is Martin Gardner. Having brought up the name I had better stop right here, or I shall fill the Globe with praises of this brilliant and prolific writer, who for many years kept Scientific American on my subscription list (it has since fallen off) with his “Mathematical Games” column. As well as having done more for mathematics than any human being alive, Mr. Gardner is, I can personally attest, an outstanding specimen of that near-extinct species, the American gentleman. Yes, yes, I shall stop … just buy Gardner’s book of essays The Night is Large and encounter one of the most interesting minds of our age.