[gems]

It's unrealistic because I really think you need the social aspect of it: collaboration and criticism. I didn't say you can't learn something on your own sometimes.

Also lots of people have convinced themselves of lots of silly things. You can probably find dozens of papers from people with bachelors in math (or no degree) claiming they have solved P=NP. A lot of these turn out to be completely bogus, but the authors nonetheless thought they were serious attempts.

[stiff]

What are you exactly claiming then, what would it mean that mathematics self-study is "unrealistic"? There is certainly a danger in not seeking external validation of your work (or denying it), and being at a university is very nice for getting that. But with some motivation I think you can get the knowledge equivalent of an undergraduate degree in mathematics by self-studying in maybe twice the time it would take at an university (I am speaking from personal experience and assuming full time job and having some life, and that you don't have kids yet). You can get feedback on the Internet as well nowadays, or seek university-level tutoring. And there is no shortage of people with degrees doing faulty P=NP proofs either.

Again, I agree doing it at the university is more effective way of doing it, and if you have the possibility to do it, good for you! But most of us can afford to dedicate at most 5 years to studying full time, and than other responsibilities kick in and you can't do it anymore. The majority of your life all the new knowledge you get will come from self-study. So you better learn to do it.

[mikevm]

Having an undergrad CS degree is enough to self-study math, I believe. But having no formal education in the sciences/maths is not (unless you are a genius).

For me, merely pushing myself through a theoretical CS curriculum made me see (and write) hundreds of proofs, hear them explained by professors, and see non-trivial exercises solved during recitations. I don't think you can get the same kind of experience by just reading a textbook, even if it does offer full solutions to problems.

Maybe when there will be full video lectures for both lectures and recitations for the basic math (or theoretical CS) curriculum you could self-study by watching those and solving problem sets. Right now, the math courses offered by Coursera don't seem to match college level, and their platform doesn't really work for proof-based courses like Analysis, Linear Algebra (not the applied kind), etc...