How to Self-Study Higher Math Without Solutions

Question

I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best reference for gleaning general information about college-level math.

However, recently I've come to realize that few of the major/popular math textbooks tend to have solutions to the exercises contained within them or available for free/online. As someone without access to a professor or peers who can check the accuracy of my answers, working through book solutions becomes very tedious. How do I know if I am getting the right answers and thus understanding the material? I know people are encouraged to post here for help with challenging problems, but it doesn't seem appropriate to post in a forum such as this to simply check the veracity of answers. In addition, certain books (e.g., Pugh's Real Mathematical Analysis) contain hundreds of exercises throughout the text, and it would be neither plausible nor acceptable to constantly ask for help with problem after problem for an entire textbook.

I am not in school nor do I plan to enroll anytime soon, so there's not really an option of waiting until college to talk things over with someone more knowledgeable about mathematics. I am learning math out of pure curiosity and not in preparation for future classes or career goals, and while there are a plethora of resources available to the independent student of mathematics now more than ever before, being able to assess how well the material is being absorbed remains a difficult obstacle to overcome.

Answer 1

I'm kind of in the same 'boat' as you; I've graduated but continue to study in my own time as a leisure activity. I completely feel the same way in that I now self-study but find it difficult to seek critique of my newly acquired knowledge.

These are suggestions and probably will not be a full answer to your question:

1) Participating in this community,

2) Seeking people in the same position as you,

3) Intense 'Googling' and wikipedia,

4) University notes and solutions,

5) 'Crack on' and Critique yourself.

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1) As @omnomnomnom says, you are welcome to use this community to verify and give suggestions to your answers by using the relevant tags. You will find that some people have used the materials that you are referring to and will be able to give you hints/solutions/intuitions/motivations etc.

2) You will find that there are many people who are in the same position as you. And, that there will be a similar number of people who seek support and guidance with their study. Have a look if there are similarly minded people in your community (or even f***book) that you can buddy up with.

3) In my experience, it's easier to look for a list of books, in the field of chosen study, and find whether there is solutions for any of the books on that list. Then you can optimize for the best book against the most resourceful solution-base and usually find almost exactly what you want.

Or even work the other way; You could search the internet for solutions in a chosen-field and then find the book the solutions are for. (I did exactly this for Guillemin's & Pollack's Differential Topology)

A tip for finding books is using wiki; search for a subject that you think you might be interested in on wiki, then look at the references and then check these references with review forums etc.

4) As you said there are plenty of sources for mathematical study but try to use university websites. If they are available to the public then it's fair game. They will often be accompanied by tutorials, solutions and amendments.

5) You'll get to a point where you'll be able to see whether your arguments and solutions are correct, overkill, not quite there or just plain wrong. It takes time. Look for counter-examples, always ask what if (and what if not) and make sure a logical statement means what you think it means. Most importantly, you should try to understand (properly) every new 'thing' before you move onto the next.

Lastly, ask your own questions. And, try to answer them. This way you (sort of) have a feeling of whether your answer is rigorous enough. And always write these down, you may come back to them and want to improve (or re-write) them.

Disclaimer: I offer suggestions based on my experience.