A Mathematician's Apostasy Sujith Vijay

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!

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