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I am curious if the "Bourbaki's approach" to mathematics is still a viable point of view in modern mathematics, despite the fact that Bourbaki is vilified by many.

Even more specifically, does anyone actively approach mathematics from the more "yielding" point of view famously practiced by Grothendieck? Which, or what type of, research areas are welcoming to (or practicing) Grothendieck's approach to mathematics?

Motivation:

To me, there is a deep question regarding motivation of mathematicians over time which is addressed by this viewpoint. An emphasis on resolving hard technical problems is quite depressing, generally, whereas the idea of finding a general framework which presents a natural and explanatory solution through the development of a vast theory seems very motivating. In such a view, the open problem only serves to motivate a better development of the general theory surrounding the core difficulty, bringing into focus a clearer picture of the essential issue at hand.

It seems to me that carefully developing a general (sometimes axiomatic) theory is analogous to performing scientific experiment. One is not looking to be clever, but instead is filling in data which may, when examined later, reveal clear and natural answers to mathematical questions. Obviously such an approach can be exhausting, in that one must spend much more time to fill in an entire picture than to, at some point, jump to a resolution of a particular question. On the other hand, It may be possible to persevere longer at such a task, as one is not so sensitive to one's loss of quickness or cleverness and can simply engage the task at hand.

Is this viewpoint valid?


[Edited (Dec. 17, 2012) by A. Caicedo, following suggestions here. Question originally asked by user curious1.]

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closed as not constructive by François G. Dorais Dec 15 '12 at 23:14

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

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The formulation of this question is really bad. Is theory-building considered quackery? Is this a joke? Also, the picture of the theory-builder being tempted to take shortcuts and solve a problem is bizarre. I voted to close as "subjective and argumentative". –  Angelo Dec 12 '12 at 21:44
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In my opnion Bourbaki's approach is (in some sense) that predominant (in consderable parts of math) that it is not even noticed anymore as some particular approach. It is simply the way things are done. [To appreciate this one would just need to compare pre-Bourbaki texts, Bourbaki, and any number of current moderately adavanced textbooks.] Voting to close as not a real question. –  quid Dec 12 '12 at 21:50
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Each person should follow what appeals. One way of progress is to find new structure which will express intuitive ideas. Another is to find new ways of using existing structures. One way to tackle known problems. Another is to find new questions. S Ulam suggested to me in 1964 at a conference in Sicily that each person should be encouraged to develop the mathematics most appropriate to them. "There are many ways of skinning a cat!" –  Ronnie Brown Dec 14 '12 at 11:07
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Dear curious, Grothendieck was a very powerful mathematician. I think it is more useful to think of his style as a manifestation of his power, rather than the converse. Among other qualities, his power included enormous, penetrating, mathematical insight. When you write that you "would like to begin to try to do mathematics in this way", this is tantamount to asking "how can I too become a more penetrating mathematician"? This is a hard question to answer, either in the abstract or in the particular, but I think that using this as a criterion for a field is likely to be a mistake. ... –  Emerton Dec 17 '12 at 2:17
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... rather, I think it makes more sense to choose a field based on where your strengths, intuitions, and interests lie, and then work hard to develop your understanding of that field. Regards, –  Emerton Dec 17 '12 at 2:19

2 Answers 2

The following appears in "Reminiscences of Grothendieck and his school", published in Notices of the AMS:

Bloch: I wonder whether today such a style of mathematics could exist.

Illusie: Voevodsky’s work is fairly general. Several people tried to imitate Grothendieck, but I’m afraid that what they did never reached that “oily” character dear to Grothendieck.

I am not completely sure what Illusie meant by "oily", but this seems to be a hint:

Illusie: To him no statement was ever the best one. He could always find something better, more general or more flexible. Working on a problem, he said he had to sleep with it for some time. He liked mechanisms that had oil in them. For this you had to do scales, exercises (like a pianist), consider special cases, functoriality. At the end you obtained a formalism amenable to dévissage.

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I am thinking 'oily' might mean 'slippery', in the sense of being very slow to commit oneself to fixed and precise notions, but to be flexible for as long as possible until, at length, the notions crystallize into a form of satisfying and useful generality and power. I think this is an extremely difficult state for most mathematicians to maintain for any appreciable length of time; it requires an incredible tenacity of inner vision. Working with such a mathematician (I have someone in mind) is both fascinating and frustrating, and 'slippery' is a good word: the ground never feels quite secure! –  Todd Trimble Dec 16 '12 at 0:09
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@Adeel: thank you for the reference of this wonderful paper of the notices, that I had completely missed. @Todd. I don't think that "oily" means exactly "slippery" in Illusie's talk. I'd like to know if Illusie was speaking in French or in English originally, but I am pretty sure that was he had in mind is the french phrase "bien huilé". One image that can explain this phrase is that of the engine of a car filled with clean new oil so that it runs smoothly, without jolt. An other translation... –  Joël Dec 17 '12 at 2:33
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Illusie was speaking in English because the conversation took place in a Chicago apartment in the presence of people not all of whom knew French. "Bien huilé" is exactly what came to my mind too, when Illusie refers to the "oily" character of Grothendieck's mathematics. –  Chandan Singh Dalawat Dec 17 '12 at 4:22
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The phrase "well-lubricated" has other meanings in British English... ;-) –  Yemon Choi Dec 17 '12 at 9:57
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@Joel: thanks. I'd been thinking over my comment recently, and also thought I might have been off (and was inclining more towards what you're suggesting now). I'll keep my comment up there anyway, just because I think the phenomenon of the 'slippery' mathematician (who is slow to commit definitely to fixed notions) is an interesting one. @Yemon: those meanings are known to Americans as well. –  Todd Trimble Dec 17 '12 at 17:21

Jacob Lurie seems the most obvious answer. His publication history (deep books published in his own time rather than a bunch of small articles) is indeed of the sort that you allude to at the end, but fortunately he had no trouble being offered a suitable position (while still quite young).

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