A Mathematician's Apostasy Sujith VijayOf the several dozen books I devoured during my first two years of college, few had as much impact as that thin bundle of charming pages with the words "A Mathematician's Apology" on the front binding. People of better judgement have fallen headfirst for G. H. Hardy's classic defence of pure mathematics as a vocation, and I was barely out of my teens. Some of my American friends have complained to me about the cricket references, but to me these references (and those complaints) added to the joy. In any case, the book swept aside my utilitarian pretensions, and it dawned on me that real mathematics was art, after all. But all that was several summers ago.
A clarification may be in order. It has lately become fashionable to criticise Hardy for things he never said, or could not possibly have foreseen. An oft-repeated criticism is six words long — "Hardy said that chess is trivial." Hardly. What he did say was, "Yet a chess problem is simply an exercise in pure mathematics (a game not entirely, since psychology also plays a part), and everyone who calls a problem 'beautiful' is applauding mathematical beauty, even if it is beauty of a comparatively lowly kind." Notice the subtle difference. Then there are those who gloat in the applications of number theory to cryptography, and by extension, to war, as the ultimate comeuppance for the purist. All he said was, "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years." So fine, he was no Nostradamus. A third line of attack alleges that Hardy revelled in the uselessness of pure mathematics. While he did take comfort in the harmlessness of pure mathematics (the year was 1940, after all), he leaves no room for confusion in the words, "If the theory of numbers could be employed for any practicable and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications." Case closed. It seems to me that a lot of critics will do well to read that book, slowly and without prejudice, on a nice afternoon.
Unfortunately the book does seem to have an undercurrent of elitism, a sort of tender hope that the profound mysteries of mathematics will somehow make its practitioners slightly more important individuals than stockbrokers and bookmakers. While one could shrug it off as part of the author's personality, it is likely that some of this elitism, thanks to the tremendous influence of the book, may have passed on to the subject as well. Perhaps this will help to explain why modern mathematics, in spite of having some of the wisest and nicest people, is rapidly becoming an insufferably dull subject. There, I said it.
As W. T. Gowers observes in his beautiful essay on the two cultures of mathematics, there seems to be a division of mathematicians into researchers and scholars, or to use Gowers's terminology, problem-solvers and theory-builders. Fame and glory have favoured the latter camp, but every new theory seems to breed people who spend their entire post-dissertation career admiring the house that Jean built and looking down on disciplines where ideas are not particularly difficult to understand once they are explained. Such erudition makes me tired.
I will not be surprised if the vast majority of mathematicians who are alive today find the vast majority of mathematics unappealing. Ambition and etiquette may restrain these impulses, but let us not be deceived by appearances. Maybe this is just me, and my credentials are a laugh, but I find the fetish for gratuitous generalisation more than disconcerting. It seems to me that we are headed straight towards what C. L. Siegel called "the theory of the empty set".
The joy of craftsmanship is dying, and perverse entertainment is taking over. Classic results are being redrawn and quartered in modern language. Clarity is no longer a virtue, and comprehension is sacrificed for comprehensiveness. We still know next to nothing about Van der Waerden numbers, and they still don't care.
Let the ruling categories tremble at a commonsense revolution. The problem-solvers have nothing to lose but their temper. They have a world to win.
Working mathematicians of all countries, keep working!