From Jordan Ellenberg’s article in Slate, Is Math a Young Man’s Game?:

For the notion of the inspired moment of mathematical creation, we have Henri Poincaré […] partially to thank. Poincaré was not only a monumental figure in the mathematics of the late 19th century, but a popular writer on science, creativity, and philosophy. In a famous 1908 lecture at the Société de Psychologie in Paris, he recounted his discovery [...] of a principle underlying the theory of automorphic functions.

Then there’s a longish quote about how he had a sudden flash of insight as he stepped on a bus. Ellenberg quotes, “At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it […].” He felt absolute certainty that he was right, but waited until a later time to verify it. Ellenberg continues,

Poincaré’s story is the most famous contemporary account of mathematical creation. (If we throw open the competition to all time periods, it must take second place to Newton’s apple and Archimedes in the tub, which speak to approximately the same theme.) Poincaré recognizes that mathematics requires both moments of illumination and months of careful deduction. “[L]ogic and intuition,” he writes, “have each their necessary role. Each is indispensable.” But he can’t quite conceal his preference for the intuitive leap over the logical slog.

The youthful genius, the instant of insight: The pictures fuse into a romantic vision of the mathematician as a passive conduit for inspiration. As Carl Friedrich Gauss wrote of one of his own triumphs, “I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved.”

[–––] Poincaré couldn’t have made this conjecture [that three-dimensional shapes that are “simply connected” must be some form, however distorted, of a sphere] absent his years of study of topology, or the earlier theorems he’d carefully proved, or the earlier conjectures on the same theme he’d tried out and found to be false. But neither could he have made such a bold guess without the kind of wild intuition he valued above all. It’s only in the presence of both conditions – deduction and inspiration, long experience and youthful audacity – that new math gets made.

The above was posted to my personal weblog on June 13, 2003. My name is Peter Lindberg and I am a thirtysomething software developer and dad living in Stockholm, Sweden. Here, you’ll find posts in English and Swedish about whatever happens to interest me for the moment.


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