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This is a question about learning mathematics outside of the standard undergraduate/graduate education.

The following is a quote from Thurston's On Proof and Progress in Mathematics:

My mathematical education was rather independent and idiosyncratic, where for a number of years I learned things on my own, developing personal mental models for how to think about mathematics. This has often been a big advantage for me in thinking about mathematics, because it’s easy to pick up later the standard mental models shared by groups of mathematicians. This means that some concepts that I use freely and naturally in my personal thinking are foreign to most mathematicians I talk to. My personal mental models and structures are similar in character to the kinds of models groups of mathematicians share—but they are often different models

The above quote suggests that Thurston's non-traditional mathematics education was key to his unique insight and contribution to mathematics. I know other examples of very successful mathematicians who benefited from learning outside the standard mathematical canon.

I am trying to incorporate this into my own learning, but I am struggling to get started. For one thing, the standard canons of mathematics are often written by masters of the respective fields who are also masterful writers (e.g. Thurston, Milnor, Serre, Stein). On the other hand, there are many lesser known texts which are poorly written and offer much less insight compared to the canons (even if they are idiosyncratic). Thus, one obstruction to pursuing the above "independent and idiosyncratic" approach to mathematics seems to be in judging what books are truly useful.

My Question. How specifically does one learn mathematics in an "independent and idiosyncratic" way?

Subquestions that may help answer the above:

1.) What should one look for when choosing a textbook? Again, I am looking for nonstandard ways of learning mathematics.

2.) Are there thing to keep in mind when one thinks about problems, theorems, etc.?

3.) What are the sociological obstructions to achieving the above goal? (e.g. as an undergraduate, there is an obvious push to learn mathematics in a certain way. Can this be an obstruction to achieving independence and idiosyncracy?)

I think the answer to my question depends on the audience, so for this question, let's restrict our attention to upper level undergraduates or beginning graduate students. Also, I am asking this question on MathOverflow (as opposed to stackexchange) because my goal is to become a successful research mathematician (as opposed to, say getting better grades in a math class).

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    $\begingroup$ Easy answer: read physics texts. As a math undergrad, take courses in classical/quantum mechanics, fluid mechanics, relativity etc. Physicists have very different mental models for many familiar math objects, and it is a 'superpower' if you can have both. $\endgroup$ – Piyush Grover Sep 6 '20 at 21:10
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    $\begingroup$ I would say your goal is a priori misguided. Mathematics are a certain way most likely because it works. Doing something different for no reason other than to be different is akin to reinventing the wheel. Only do things differently if you think it's better, and not simply because you desire to be distinct. $\endgroup$ – Stanley Yao Xiao Sep 6 '20 at 22:26
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    $\begingroup$ There is an implicit assumption of causality in your question. Thurston learned in an independent and idiosyncratic way; and Thurston was a great mathematician. I suspect it's more likely the causality went in the opposite direction: Thurston was a great mathematician; and therefore had the ability to read independently and idiosyncratically and build his own helpful but different mental models. $\endgroup$ – Anthony Quas Sep 7 '20 at 5:41
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    $\begingroup$ You may find Miles Reid's "Obituary" homepages.warwick.ac.uk/staff/Miles.Reid/Personal/obituary (a sort of autobiography) worth reading, both to see a non-standard path and its pitfalls. Caveat emptor: the causality issue mentioned by Anthony may well apply here also. $\endgroup$ – Balazs Sep 7 '20 at 7:18
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    $\begingroup$ Having your own questions you would like to see answered, questions that aren't currently being asked in mathematics. That would be a way to start down an independent and idiosyncratic road. Of course, these questions would have to come from you, not some textbook. $\endgroup$ – Ryan Budney Sep 8 '20 at 3:58
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If you really want to learn mathematics in an independent and idiosyncratic way, then asking other people for advice and following that advice is precisely the wrong way to go about it.

Thurston thought about mathematical problems and structures largely on his own, and in his own way. He didn't post on MathOverflow asking people how to pick a textbook. If you want to be independent and idiosyncratic, then you should think about things your own way as much as possible instead of reading and following what others are saying.

Now, you also said that you want to become a successful research mathematician. You should be aware that this goal is somewhat in tension with your goal of being independent and idiosyncratic. There's a saying in chess, that to be original is to lose many games. Math isn't chess, but it's still true that following a conventional route is going to give you better odds of success. So you have to decide what's more important to you—worldly success, or an independent mind?

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    $\begingroup$ I disagree with this. I attempted to do this when younger, simply picking books off the shelves that I found interesting. I fell into the trap of the expert beginner: I learned what I was comfortable with, not what I struggled with. Education requires structure and guidance. OP should get a mentor, not just be told to go wild in a library. $\endgroup$ – Akshat Mahajan Sep 9 '20 at 1:50
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    $\begingroup$ Note carefully that I said "if you want to learn mathematics in an independent and idiosyncratic way." I didn't say that one should learn mathematics in an independent and idiosyncratic way. That is for the OP to decide. $\endgroup$ – Timothy Chow Sep 9 '20 at 2:29
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    $\begingroup$ I would somewhat disagree with this answer. There’s a huge middle ground between “learn in the most conventional way” and “work primarily alone and avoid listening to what others say”. The latter is very hard to be successful with — I’ve met many more cranks who’ve tried it than good mathematicians. By contrast, some of the most individual and idiosyncratic mathematicians I know are voracious readers and good listeners — they just analyse and re-interpret things according to their own worldview and tastes. $\endgroup$ – Peter LeFanu Lumsdaine Sep 9 '20 at 8:30
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    $\begingroup$ @PeterLeFanuLumsdaine : You're right, of course, but I felt it was important to point out the irony in the OP's request, which I'm not sure the OP was aware of. Independence of mind is about learning how and when to trust one's own instincts, and not about following Ten Easy Steps to Becoming an Independent Thinker. (I'm reminded of magazines for how to live a simple life. My first advice for living a simple life would be to cancel all magazine subscriptions.) $\endgroup$ – Timothy Chow Sep 11 '20 at 19:54
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Rather than aspiring to think outside the box, I would recommend working to have a bigger box. I would say that many of the greatest mathematicians, e.g. J.-P. Serre, learned and took in mathematics very broadly, and thus had a great set of tools at their disposal when trying to crack a question, and thus were ready at the right time to have `revolutionary' ideas, and develop new fields. A great thing about trying to take in lots of mathematics is that this can continue throughout your life!

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There is a general sense among educators and inspirational speakers that the best learning happens by making mistakes. (I would love to find some science that backs this up!) At the very least that means that doing exercises is as important as reading texts. I do think (having had the privilege of spending some time with them) that the most idiosyncratic and independent mathematicians are the ones most willing to think for themselves and produce their own mistakes.

For example, this can mean that you stop at every point where a statement is unclear and try to fill it in yourself. This is not always easy, but can be really rewarding. It seems harder to learn an entire subject this way; I rather think of it of straying in the woods, and occasionally having to find your way back to the main path. I certainly don't have a recipe for this (nor in fact do I claim that I master this myself).

Mathematics has this strange habit of not documenting mistakes and dead ends as well as scientific principles tell us (how often have you seen a paper whose main topic is "we tried a thing and we got stuck for the following reason"?). It is unique among the sciences (if you can call it that) in not systematically documenting the methodology; rather only the final result is presented. If you really want to understand how great mathematicians work, it would be great if they had documented the mistakes in their thinking.

Some of this is actually available in print! For example, the Serre–Tate correspondence documents a lot of the developments of group cohomology and many other topics as they were evolving, and it is a fascinating read for exactly that reason. It gives that rare insight into the minds of the mathematicians rather than merely the final product that comes out of it.

Let me end with what Goro Shimura said about Yutaka Taniyama (in the Horizon documentary on FLT):

He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I've realized that it's very difficult to make good mistakes.

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    $\begingroup$ The type of learning that you're alluding to falls under the umbrellas of "constructivism" (as opposed to something like "behaviorism") and "situated cognition". If you want to go down the research rabbit hole related to this, check out en.wikipedia.org/wiki/Constructivism_(philosophy_of_education) and en.wikipedia.org/wiki/Situated_cognition $\endgroup$ – MetaSean Sep 10 '20 at 16:34
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    $\begingroup$ I vividly remember, in the PBS documentary about Wiles's proof of FLT, Shimura saying of Taniyama, "he made mistakes, but they were mistakes in the right direction." My (non-mathematician) mother was watching with me, and I think that was as close as she ever came to understanding the nature of the practice of mathematics. $\endgroup$ – LSpice Sep 10 '20 at 17:33
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    $\begingroup$ (I'm sorry that I got so excited to relay that quote that I missed that it was the last paragraph of your answer. I'll leave it there because I still enjoyed the way it let me communicate the experience of my job with my mother.) $\endgroup$ – LSpice Sep 11 '20 at 0:19
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If you intend to learn mathematics in an "independent and idiosyncratic" way, do so with a certain amount of intellectual humility. For every Thurston who can successfully pull something like this off, there are several mathematical cranks who have convinced themselves that e.g. they have discovered an elementary proof of Fermat's Last Theorem. I've dealt with my fair share of them over the years. Thinking outside the box can be helpful, as long as the box in question isn't the box of careful reasoning.

A mathematical crank is not necessarily a person who is incapable of good mathematical reasoning. The most interesting crank I have personally dealt with is a person who had independently hit upon the idea of a factorial number system and worked out many of its properties. For example, he realized that in this system it wasn't obvious that a number can be represented in only one way, and had successfully worked out a good argument for the uniqueness of representation. All this came from an idea that occurred to him while listening to his high school math teacher talking about numerical bases other than 10. I was genuinely impressed with this and told him so. The transition to crankery came when he insisted that his scheme could be used to compress all data, with the idea being that computers used standard number bases and that his system allowed for the expression of all numbers using less "digits". He seemed impervious to simple pigeon-hole arguments about the impossibility of a universal compression scheme. There is no question that he had learned a nontrivial amount of mathematics in an "independent and idiosyncratic" way. Unfortunately, despite some genuine talent on his part, his idiosyncratic way was prone to error.

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    $\begingroup$ While otherwise agreeing, I think that "for every Thurston, there are several cranks" is off by several orders of magnitude. $\endgroup$ – LSpice Sep 10 '20 at 17:32
  • $\begingroup$ @LSpice While otherwise agreeing with your comment, the subset of independent and idiosyncratic people who are interested in learning mathematics rather than independently doing research using whatever they happen to already know is probably less prone to crankery than a randomly chosen independent and idiosyncratic person. $\endgroup$ – John Coleman Sep 10 '20 at 17:54
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There are many many many masters recent and old. They've written not just their most famous books, but other monographs and research papers. Nobody is reading all of them. Part of the point is not that what you learn has to be quirky, but that when confronted by problems in some domain, you have some tricks up your sleeve that others looking at the same problem may not.

So for starters, you could add to your list some masters who wrote wonderful books on varied subjects: in no particular order, Hilbert(-Courant), Weyl, Gelfand, Arnold, Kolmogorov, Hardy, Gromov, Bott, Chern, Siegel, Poincare, Gauss, Euler, Sternberg, Whitney, E. Artin, E. Cartan, Knuth, Novikov, Lefschetz, Wiener, Pontryagin, Alexandrov, Weil, Shafarevich, Manin, Hirzebruch, Atiyah, Donaldson, Ahlfors, Thom, Mumford, JH Conway, Coxeter, Berger, F. Klein.

...to say nothing of many others whose research papers may be of great use, including some that are very accessible. Someone in a different field could write a largely disjoint set of masters who wrote great books.

On top of many excellent mathematicians who wrote lovely books, both standard and idiosyncratic (hot tip: William Burke).

And many many many others from before the 20th century; I mentioned only a couple whose work I've read enough to make a firsthand claim that it's accessible and still of use. There are many nice volumes with original sources, that are chosen for accessibility and often have nice commentary.)

Also: besides individual "masters" there are many "traditions"--Russian representation theory, Polish topology and functional analysis, Hungarian combinatorics, etc. etc.

And to add to a comment: not just physics, but computer science, mechanical engineering, signals and systems, and other fields where people are using mathematics at a high level with different models than pure mathematicians.

And in any case, it's all what you make of it. Any of these sources or the authors you mentioned wrote works of incredible depth. You could take one volume of Milnor or Serre or Weyl and go off to the forest for a year, and you won't exhaust it. I went to a lecture by Steven Chu once in which he mentioned that in college, he took the minimal courseload he could and just read and reread the Feynman Lectures. You could do worse than that.

Or find a nice topic you enjoy and go deep, and you'll find it connects to many many things. Follow the connections in all directions, and presto! You understand something important from a dozen perspectives. Nobody else will have precisely the same toolbox.

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  • $\begingroup$ I know that computer science subfields (e.g. algorithms, machine learning, and theoretical CS such as combinatorics and graph theory) and signals & systems (e.g. compressed sensing) use a lot of cool mathematics. What are mechanical engineering subfields that can be useful for mathematicians? $\endgroup$ – user676464327 Sep 7 '20 at 4:43
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    $\begingroup$ @user676464327 The history of mechanics is intertwined with math going back to Newton. Research in continuum mechanics, vibration, fluid mechanics, robotics all use sophisticated math ideas ..take your pick. $\endgroup$ – Piyush Grover Sep 7 '20 at 11:14
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To effectively learn "independently" (and perhaps "idiosyncratically"), you need to first learn to critically question the material being presented. Every instructor or author will teach/write in their own "idiosyncratic" method. Learn to differentiate the standard material with the idiosyncratic, then reorganize what you learned into your own "idiosyncratic" style. This isn't (nor shouldn't) be about taking shortcuts. It is actually harder. But if you are not satisfied with the "standard" presentation, you need to ask "why?" you are not satisfied.

In other words, don't tell someone to turn off their flashlight unless you know your flashlight is better.

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If you are serious about alternative methods I would suggest you try Lewis Caroll's "Alice in Wonderland" -- it is full of mathematical ideas, and very entertaining.

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    $\begingroup$ Dodgson (Caroll) was a third-rate mathematician; not sure how I feel about this advice. $\endgroup$ – Charles Sep 10 '20 at 19:46

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