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Firstly I'd like to apologize if this is too much of an opinion-based question. I do not know where else would be appropriate for such a post, if there is such a place please let me know.

I am very interested in mathematics, particularly mathematical physics, but I currently do not have access to an institution or any professors. As a result I have mainly been self-teaching myself through textbooks. My usual routine is to very carefully go through a chapter in a book, making sure I understand every step in every proof, and then move onto the exercises. Going through the proofs this way helps me measure which theorems/results are more important, and also helps me learn common methods of proof. This part of my studying is going fine and I'm able to finish the chapter and understand most of the material, sometimes with the help of other books or online resources such as math stackexchange.

The problem I have been having is with the exercises. I use these as a way to reinforce the material and test my understanding, but also as a way to develop my problem solving skills. I very rarely am able to solve a problem in any graduate math textbook I have read (which I have already finished 2-3), and have trouble even getting started with them. Eventually if I make no progress I open a question on stackexchange, or try to find a solution online which I usually am able to follow but I would not have thought of myself. I want to get to a point where I can actually do these exercises and not just be able to understand the solutions. I am beginning to question if I'm just not cut out for it or if my approach is wrong.

In summary, I am able to read about these topics and understand proofs written by others, but I struggle on actively doing the math myself. I am looking for some advice on how to develop these problem solving skills and do better on the exercises. I have taken a look at similar questions on this site such as:

The advice given in the intersection of all these posts seems to be more practice and to get help/hints from your professor. As someone who is self-studying and often gets stuck on where to even begin, I am not sure how to apply this advice and I would love to hear what working mathematicians would advise. For example, when I don't know how to start or I am stuck what should I do? I often think about the problem for some time but get nowhere, only to find a clever trick or small lemma should have been applied.

For some reference on what kind of books I am studying, I am currently going through the following:

  • Methods of Modern Mathematical Physics Vol 1. (Functional Analysis), by Reed & Simon
  • Mathematical Gauge Theory by Hamilton
  • Partial Differential Equations by Evans
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    Generally questions about the content of specific fields are outside the scope of Academia.SE, and I think this question basically boils down to "how do I learn math" which is too broad for any Stack Exchange site. It is however very normal for problems in graduate-level textbooks to be extremely difficult, so do not be discouraged!
    – Nate Eldredge
    Jun 17 at 20:03
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    One general tip though - when you do solve problems, with or without help, make sure you are learning not only the result but also the techniques. When you see a "clever trick" for the first time, think about how and why it works, how it generalizes, and try to develop intuition for it. Then you will have a better chance of seeing where to use it next time.
    – Nate Eldredge
    Jun 17 at 20:06
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    Compared to those books you mentioned, how is your experience with solving (proof-based) problems in 'less advanced' topics such as real analysis or linear algebra?
    – Jochen Glueck
    Jun 17 at 20:21
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    When I read "making sure I understand every step in every proof," I got the impression that you might be looking so intensely at details that you're missing a bigger picture. It's good to understand all the details, but I'd suggest that, after carefully reading a chapter as you've been doing, and before moving on to the exercises, you should go through the chapter again to get a broader picture of what's going on. Perhaps just review the theorems (skipping the proofs) and try to understand how they fit together and what the key ideas are.
    – Andreas Blass
    Jun 18 at 0:37
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    As other contributors suggest, it is extremely important that you are able to write a proof for any considerably less advanced statements / exercises. In my opinion, there is no point in studying functional analysis if you can't provide epsilon-delta proofs when half asleep.
    – Thomas Bakx
    Jun 18 at 10:15

1 Answer 1

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I expect that what you lack is appropriate mathematical experience and maturity. Graduate-level textbooks often assume a level of mathematical maturity that is different than the "logical prerequisites". You're also hampered by the lack of mentors/classmates to discuss things with.

My suggestion is to begin by going through easier (e.g., undergraduate) books in your area of interest, where you can do many of the problems on your own. That will help you naturally improve your problem solving abilities, and make it easier for you to understand more challenging books later.

You can also look on Math StackExchange (e.g., questions with the [book-recommendation] tag) for recommendations about books for either your particular interests, problem solving, proof writing, etc. And if you don't find exactly the the kinds of suggestions you want, ask a new question there.

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    I'd strongly second the observations of this first paragraph: it's not just having the "formal, logical" prerequisites, but also having enough experience with those prerequisites so that they are now "second nature", rather than requiring conscious thought or recollection. Somewhat parallel to the difference between knowing the rules to a sport, and playing it... much less playing it well. :)
    – paul garrett
    Jun 19 at 17:26
  • Thank you for your answer and @paulgarrett for his comment. Based on your suggestions, I took a look at that tag and came across the book Berkeley Problems in Mathematics, which according to the book description, is a collection of problems to help students review undergraduate mathematics in preparation for preliminary exams. This seems to be exactly what I need. I'll try doing a few problems from it everyday as I also study the more advanced material.
    – CBBAM
    Jun 19 at 19:30
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    @CBBAM I get a bit of a sense from your comment that you're attempting to "skip" undergraduate-level material. An undergraduate level of understanding can't simply be rushed past, and you should make sure to work through dedicated undergrad textbooks in a subject first. You can't do an end run on mathematics this way. Your problem is a symptom of this strategy. You do not lack knowledge per se, but the aforementioned mathematical maturity. Cramming with that Berkley book is to continue with the same flawed methodology. I will gently suggest that you have not yet internalized Kimball's answer.
    – Jerome
    Jun 21 at 19:56
  • @Jerome Thank you for your comment. I agree with your assessment that I do not have the mathematical maturity I need to tackle tough problems and this is what I would like to develop. For more on my background, I actually have done my undergraduate in mathematics but it wasn't a great program and left me with a very shallow foundation. This is why I only truly learned things such as real analysis and linear algebra while self-studying more advanced topics.
    – CBBAM
    Jun 21 at 22:50
  • @Jerome So far I have only gone through the first 10 problems of the Berkeley book and I have been able to do about 7 of them without too much trouble. Should I instead go through an undergraduate textbook such as Rudin and treat it as if I've never seen the subject before? Would that be better than directly tackling problems from the Berkeley book?
    – CBBAM
    Jun 21 at 22:51

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