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When I asked a Portuguese mathematician of my acquaintance whether he had any insight to offer me on the subject, he replied, “The foundations of mathematics are full of holes and I never felt comfortable dealing with such things.”
Full of holes. Earlier generations of mathematicians assumed that the stability of the landscape on which mathematical structures were built was guaranteed by God or nature. They strode in like pioneers or surveyors, their task to map the fundamentals and in so doing secure the territory that future generations would colonize. But then the holes—of which the liar’s paradox is merely one—started popping up, and the mathematicians started falling in. Never mind! Each hole could be plugged. But soon enough another would open, and another, and another . . .
Bertrand Russell (1872–1970) spoke for any number of idealistic mathematicians when he wrote in 1907,
The discovery that all mathematics follows inevitably from a small collection of fundamental laws, is one which immeasurably enhances the intellectual beauty of the whole: to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction, this discovery comes with all the overwhelming force of a revelation: like a palace emerging from the autumn mist as the traveller ascends an Italian hill-side, the stately storeys of the mathematical edifice appear in their due order and proportion, with a new perfection in every part.
I remember that when I read George Eliot’s Middlemarch in college, I was particularly fascinated by the character of Mr. Casaubon, whose lifework was a Key to All Mythologies that he could never finish. If Mr. Casaubon’s Key was doomed to incompletion, my astute professor observed, it was at least in part because “totalizing projects,” by their very nature, ramify endlessly; they cannot hope to harness the multitude of tiny details demanded by words like “all,” just as they cannot hope to articulate every generalization to which their premises (in this case, the idea that all mythologies have a single key) give rise. Perhaps without realizing it, my professor was making a mathematical statement—she was asserting the existence of both the infinite and the infinitesimal—and her objections to Mr. Casaubon’s Key hold as well for any number of attempts on the part of mathematicians to establish a Key to All Mathematics.
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David Leavitt (The Man Who Knew Too Much: Alan Turing and the Invention of the Computer (Great Discoveries))