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I admire Grothendieck's style of mathemtatics and want to do mathematics like his style. For example,

  1. Generalization of things by discarding unnecessary hypotheses which leads to simplification.

  2. Underwater work: making solvent in which solutions or proofs come out by themselves naturally.

  3. Accomplishment of fundamental transform and redefine foundation of mathematics (in particular algebraic geometry).

  4. Creation of new mathematics and achieving revolutionary innovation.

  5. Painting landscape (highway) in which proofs become obvious and natural.

  6. Finding new hidden structures so that the solution would come out by itself.

  7. Rewrite the existing mathematical language into a new language (transformation of mathematics).

  8. Conceptual way of doing mathematics.

  9. Grand unification.

etc..

How can we achive or implement these style / ideal of mathematics ? Does anyone have any experience with this? Is there a simple and clear way/knowhow? By selecting some 'good' problems ( with discerning for cognizing good problem ) and trying to solve it, like the galois theory? An issue that makes me confused is, when I try to approach to this ideal directly, I feel like that my mind is being distracted. Is problem selection and solving necessary, even as a means of focus? Perhaps, do new theories come naturally in the process of solving problems?

Let me upload few quotes about this issue :

  1. "He who seeks for methods without having a definite problem in mind seeks for the most part in vain," - David Hilbert.

  2. "Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.” - Peter Scholze.

  3. On the other hand, there is also quote like next : "Then around 1956–1958 Grothendieck began to work in algebraic geometry, a field with a rich history, and with already many existing theory. He told us that this was because he was planning to solve the Weil conjectures. However, do not interpret this as “problem solving”. On many occasions Grothendieck told us that finding the structure involved was the only essential thing to do and then the solution would come out by itself: “immerse a large nut in a softening fluid, and the nut opens just by itself”. We have seen many instances where going to the very roots and pure thought gave insight and solved difficult problems." - Life and Work of Alexander Grothendieck by Ching-Li Chai.

etc..

This may be ignorant, but I want to do math with 70% theory building and 30% problem solving. I don't know what to do in loneliness. Do I need to interact with other mathematicians?

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    What is your background? Masters student looking for a PhD? Postdoc looking for collaborators? Impossible to answer without some background... Jan 12 at 5:46
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    If you're not Grothendieck, first you need to come to terms with not being as brilliant as Grothendieck. Most people who attempt to be Grothendieck end up coming up with useless junk theories. Jan 12 at 5:47
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    If there were a simple and clear direction, then there would be many people doing mathematics in the style of Grothendieck - certainly many people have expressed a desire to do mathematics in that style. Since there is very little mathematics in the style of Grothendieck, one must conclude that there is no simple and clear direction. Jan 12 at 7:11
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    Why don't you ask the same question in Math:SE?
    – EarlGrey
    Jan 12 at 7:30
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    Grothendieck also famously said: "I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." Let me adapt this quote to your situation: "I never bothered about whether I was doing mathematics in the style of Grothendieck, but just tried to understand - and it always turned out that understanding was all that mattered." My advice would be to follow this adapted quote. Jan 12 at 13:21

2 Answers 2

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Some of these items just form a particular style of doing mathematics, or constitute lofty but not ridiculous ambitions here. This encompasses your items 1,2,5 and 6. Probably the most concrete objective here is the first one: You can start with an existing theorem, and then tinker with it to try and find out what exactly its requirements should be, what concepts should be mentioned in the proof, and what the "proof from the book" for it is. The others are more what is in the far distance, and getting anywhere will require a significant body of work. Also, do keep in mind that many of your goals refer to aesthetic judgment, and thus will require substantial interaction with the broader community.

Goals 4, 8 and 9 are a mix between being ultimately meaningless and far to grandious to ever state about ones own work.

Goals 3 and 7 are bad goals to have. They may be very worthy things to actually do, in certain circumstances, but if you start out intending to transform mathematics, you will fail. In order to convince others to use your new language, you need to make the case that it is a significant, maybe even necessary improvement over the status quo. To able to make this case, you should only seek radical change if the mathematics itself drives you to it. Otherwise, you will just come about as a crank and be ignored.

One important question you need to ask yourself here is whether you actually seek to do mathematics in this way, or whether you are simply captivated by the image of yourself-as-the-new-Grothendieck. The latter would be a very human motivation, but that motivation would also explain why the doing doesn't come naturally to you.

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  • Thank you for your detailed advice. I think this will be an opportunity to reflect on my aspirations. I think my preference system and tendencies are similar to Grothendieck, but I don't think we are the same. The reason I've posted this question - the possibility of implementing it in some style - is fundamentally because I want to unleash my full potential in mathematics. Your answer also gives me some insight into why progression is not natural in mathematics. How can I conduct mathematical research more naturally?
    – Plantation
    Jan 12 at 11:27
  • In my opinion, enjoyment (love) and immersion are the most important things in mathematics. From your perspective on my current situation, how can I become more focused and natural at math while better integrating it with my ideals? Is there a part that I am doing something stupid or overlooking something? Anyway, I will consider your answer carefully. Thank you.
    – Plantation
    Jan 12 at 11:27
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    @Plantation the questions you ask (which should be asked as a separate question) require more focus. Where are you holding in math, what have you done/tried to do in regards to research, what issues come up, etc. Such open-ended questions tend to have relatively useless answers. Jan 12 at 11:59
  • @David Raveh : O.K. I got it~
    – Plantation
    Jan 12 at 12:00
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If you are committed to this vision, then my advice is to:

  • apply to PhD programs in mathematics, and go as strong of a one as you can
  • find an advisor who works on mathematics to your taste, work under their supervision, and listen to their advice
  • talk to your fellow students, attend conferences and seminars in your area of interest, and participate in the community.

Grothendieck was a once-in-a-lifetime talent; very few people would have any hope of accomplish your stated goals working on their own. (Certainly I am not among them.) But many more mathematicians than Grothendieck accomplish theoretical breakthroughs, and the vast majority of these go the traditional route and spend plenty of time talking with other mathematicians. If you put in the time and effort, and consistently listen to advice from those with more experience than you, then you can participate in a community that shares much of your vision, and contribute to its collective efforts.

Good luck.

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  • O.K. Yes. I think it's important to interact with other mathematicians. I will challenge myself and try harder. Thank you.
    – Plantation
    Jan 12 at 12:39

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