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Alexander Grothendieck is known to have revolutionized several areas of mathematics. His insights were very deep, original and revolutionary. Time after time he showed that he could see in ways that no one else could.

And that brings up the question. How could he do that?

  1. Is it because some people are more intellectually gifted? I have not seen anyone producing tens of thousands pages of abstract deep mathematics in 1-2 decades. If you divide by time, it comes down to something like 5 pages of new highly abstract mathematics every day (a conference paper!).

  2. Or is it because he took a special approach that most of us don't take?

  3. Or was it because he had an anarchist upbringing?

I would like to better understand what makes some people great scientists. I look forward to your comments.

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A person's life and behavior are always shaped by a number of factors, not (in general) just one. I think it highly unlikely that any one of the three points you bring up is responsible for Grothendieck's success. At the same time, all of them may have contributed to his life.

The point that I find singular about Alexander Grothendieck is his overarching approach to mathematical research. As Allyn Jackson writes in Comme Appelé du Néant—As If Summoned from the Void: The Life of Alexandre Grothendieck (all quotes from this work unless otherwise noted),

Grothendieck changed the landscape of mathematics with a viewpoint that is “cosmically general”, in the words of Hyman Bass of the University of Michigan.

. . .

He had an extremely powerful, almost other-worldly ability of abstraction that allowed him to see problems in a highly general context, and he used this ability with exquisite precision.

Grothendieck had an incredible capacity for abstraction, even more so than other leading mathematicians who studied in the same fields as he.

Allyn contrasts Grothendieck with John Forbes Nash. Nash went after specific problems that he found interesting, while Grothedieck searched for a more general framework:

If Nash is an ideal example of a problem-solver, then Grothendieck is an ideal example of a theory-builder.

However, one clearly can't use this facet to argue that Grothendieck's approach was better than others. Nash's work in game theory revolutionized many disciplines. Grothendieck's strategy and use of abstraction simply worked for him.

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  • $\begingroup$ Would the downvoter mind explaining? I was a bit unsure of how to present the points here, so constructive criticism would be awesome. Thanks. $\endgroup$ – HDE 226868 Nov 2 '15 at 22:13
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This citation, form Grothendieck himself, to me shows a little bit why, a part from him being exceptionally gifted, his approach to problems was radically different. He describes the process of solving a math problem as that of opening a nut:

*The ... analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet it finally surrounds the resistant substance.”

In this paper you may find some description of the intellectual environment and the ideas that allowed such approach to problems to work incredibly well.

http://www.cwru.edu/artsci/phil/RisingSea.pdf

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  • $\begingroup$ Thanks. I am well aware of that article. It's a good one $\endgroup$ – eli Dec 19 '15 at 18:29
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"The way to understand a mathematical problem is to express it in the mathematical world natural to it -that is, in the topos natural to it. Each topos has a natural cohomology, simply taking the category of abelian groups in that topos as the category of sheaves. The cohomology of that topos may solve the problem. In outline:

1) Find the natural world for the problem (for example, the Etale topos of an arithmetic scheme).

2) Express the problem cohomologically (state Weil's conjectures as a Lefschetz fixed point theorem).

3) The cohomology of that world may solve your problem, like a ripe avocado bursts in your hands."

The rising sea, Mclarty.

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