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I am self studying pure mathematics as a hobby (no connection to my career at all). I have been working my way through a few analysis texts, and have a broad goal of learning analysis at the level of the Stein/Shakarchi books, differential geometry, and topology (inspiration from working through Rudin/Pugh). If I make it that far, I would eventually want to study probability theory and mathematical physics. I have lots of books and know math stackexchange is a great resource as well. One of my challenges in doing the proof exercises, is when referencing solutions, if they exist, my proof may have differences from the official (or unofficial) solutions - and these differences can either be subtle or a completely different approach. While I will oftentimes discover an error in my reasoning when comparing solutions (and gain some intuition in the process), sometimes I can’t reconcile if mine is actually wrong or if it is just another way to prove the statement (even if I am very confident in my solution before comparing it). This has led to some frustration on my end and uncertainty if I am actually making progress in understanding the material.

For those who have had experience and success with self study of pure mathematics (particularly at higher levels), did you have similar difficulties and do you have any advice on how to approach doing textbook exercises and knowing if you are making progress? I ask as these are lofty goals, will require a huge time commitment (already very busy as it is lol), and I wonder if they are at all realistic.

Thanks in advance.

dish197506
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As long as it happens only rarely, this is nothing to worry about. I think asking other people is a good idea. Stack Exchange would be a good place. Proof verification questions are often closed because people just dump their proof without comment, but if you carefully explain why you are unsure, you should be fine.

But usually you should be able to judge for yourself if a proof is correct or not. There are lots of students at your level who don't have that ability (yet), and I can't know if you're one of them.

It's important that you don't guess. If you're not 100% sure that a step in your proof is correct, write it down in more detail until you are. If you find that difficult, chances are you haven't really understood your own argument. If you're only 80% sure, looking at the solution in the hope of getting the remaining 20% is the wrong strategy in my opinion. I'd rather treat the question if the step is valid as another exercise.

I ask as these are lofty goals, will require a huge time commitment (already very busy as it is lol), and I wonder if they are at all realistic.

That depends a lot on your abilities and the timeframe you have in mind. But you might also ask yourself if that is really important. If you don't enjoy the process, it's not worth doing it for what will certainly be years, even if you achieve your goals at the end. And, if you enjoy the process, it doesn't really matter if you achieve your goals or not.

Stefan
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The advice is simple: don't be overconfident and find people that are at least on your level, preferably above, and work with them. Regularly. So that they can verify your ideas. You should not work alone, it is very dangerous, as in: easy to make mistakes that you won't ever notice. And also, it is not necessarily good for mental health.

Also do not fool yourself. Even the greatest mathematicians make mistakes. For example Kurt Gödel published a theorem with a proof in 1933. The proof turned out to be invalid, after 30 years. And another 20 years later it turned out that the entire claim was false (see that: In the history of mathematics, has there ever been a mistake?).

The point is: everyone makes mistakes.

freakish
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    "You should not work alone, it is very dangerous. Even for your mental health." Without personally knowing the OP, you can't possibly know that. It's perfectly natural to have a hobby that doesn't include other people. – Stefan Oct 12 '24 at 17:25
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    @Stefan are your trying to say that isolation is not bad for mental health? I don't have to know OP to understand it is very bad. – freakish Oct 12 '24 at 17:29
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    I'm not saying that, although we'd probably disagree on that as well. But OP is doing math as a hobby while having a completely unrelated career. So math only occupies a small portion of their time, the rest could be full of social interaction. Are you saying that not being around people 24/7 is bad for mental health? – Stefan Oct 12 '24 at 17:36
  • @Joe well, that's not how I remembered the story. Unfortunately I'm not smart enough to understand the paper. So I've modified the answer to refer to something (hopefuly) less controversial. – freakish Oct 12 '24 at 18:00
  • @Stefan I've never said that. I'm only saying that if you are doing something and never sharing this with anyone this is bad. Regardless of whether this is a hobby or not. – freakish Oct 12 '24 at 18:02
  • To be clear, I don't understand much of the paper either. But if you look at say, page 5, you can see a couple of really quite elementary mistakes that you would not expect of a fields medalist. For example, he says that the proof "is by contradiction and this is not accepted as valid in ZF, it does require choice". The writing style also just seems completely off in a couple of places. – Joe Oct 12 '24 at 18:03
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    @freakish I disagree, but I like your edit and completely agree with "easy to make mistakes that you won't ever notice". – Stefan Oct 12 '24 at 18:08
  • Thank you all for your comments. I get freakish’s point about having peers to discuss this stuff with to check your understanding. For what it’s worth my job requires constant social interaction and navigating complicated social situations so I don’t mind the insular nature of my self study hobby lol. At the same time if I have no one to bounce ideas off progress may be slow. Good to know mistakes are normal. I guess as long as I am enjoying what I am learning and the process it’s worth it. – dish197506 Oct 12 '24 at 18:59
  • @dish197506 I know that finding appropriate people to work with is not easy. But I strongly encourage you to search for one. – freakish Oct 12 '24 at 20:22
  • Also, here's a funny story. Once I had an exam during my time at university. After the exam my friend told me about his solution to one of the problems. It was very clever, unorthodox and... completely wrong. Unfortunately the mistake was very well hidden and subtle. And it was essential, the solution was not recoverable. Still he did get max points for it, the examiner missed the error. Even though it was a very experienced professor. – freakish Oct 12 '24 at 20:24
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There are also automatic proof assistants like e.g. Isabelle/HOL https://isabelle.in.tum.de/ or Coq https://coq.inria.fr/ that can help to review formalized proofs. Formalization can be a lot of effort though, i.e. am not saying that that you have to use it (nor does the majority of mathematicians I think) but just to make you aware that this also exists.

Holger B
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