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I was recently reading this, and found it quite interesting. It is basically an account on intuition in mathematics, by Poincaré himself. He provides a cut, dividing the mathematical minds of history in two categories, namely, the "analysts" and the "intuitionists". It is obvious what each of those refer to, so I won't bother you with mindless gab.

My question can be summed up to the following : from your knowledge, what names do come up in mind when thinking of mathematicians that sacrificed rigor for intuition? Riemann and Ramanujan are obvious examples. Who else could there be added to the list?

What account can you provide for Galois? How was he in his proof writing? Was he intuitive or rigorous?

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Well Galois died as a result of a pistol duel due a fair young lady! He was very young. Certainly Ramanujun would make the list. –  Amzoti Mar 18 '13 at 0:32
    
Italian algebraic geometers. –  Lepidopterist Mar 18 '13 at 0:34
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Euler was an intuitionist, in my opinion, but he had a power intuition for analytical methods. The quintessential example is his non rigorous yet "analytical" solution to Basel's problem. –  Pedro Tamaroff Mar 18 '13 at 0:35
    
from what I gather, Euler would have been scolded today. His proofs, generally, where not the most rigorous the world has seen. –  MikhailSchmokloff Mar 18 '13 at 0:39
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It is risking confusion to use the word inuitionist this way, because it already has a meaning that is quite specific in mathematics. en.wikipedia.org/wiki/Intuitionism –  Thomas Andrews Mar 18 '13 at 0:51

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I would say an example par excellence of this type of intuition can be found in the creation of Calculus. Newton and Leibniz were operating on a different level of thought, they had a sense for a certain type of approach to mathematics and physical phenomena, and they were able to implement this approach without the sometimes overwhelming fetters of rigor. They did not supply "modern" proofs for their assertions, but their ideas worked. The duo (although not collaboratively) developed an ideal theory of infinitesimals, a way to measure change as it approaches the "infinite beyond" and the instantaneous moment.

To be more concrete, take what some might term the cornerstone of calculus, the limit. To Newton, this idea sprung forth from intuition; it made sense as an a priori (not in the strictest philosophical sense) idea. One feels that one knows what a limit is, and it is enough work in the practical with specific functions (say, $ lim_{x\to3}2x^2=18)$. But to the modernist, this mathematical object had to be given some precise definition that accords with rigor; it is not enough to couch intuition in a phrase like "as x approaches". And so the $\epsilon-\delta$ definition of the limit was born (some attribute it the idea to Cauchy, but it was brought to its fullest form by Bolzano and Weierstrass).

The point is that Newton, with his fluxions, and Leibniz with his $dx$ and the like, tapped into a whole area of mathematics, a whole area of physical study, without the addition of rigor! But even more that that, their ideas continue to influence almost all of mathematical thought today. This is what comes to my mind when I think of intuition in math.

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