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).
Best Answer
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?