[stiff]

I self-studied some amount of pure mathematics, having only an undergraduate CS degree and while having a 40 hour a week day job, and while it's definitely harder and slower than learning it at a university, it's not unrealistic. If you look hard enough, there are books with proof-based exercises with answers in the back (example proofs) that you can use to check your understanding (Spivak's Calculus, books from Springer UTM series). For many other books, you can find course pages online with homeworks+solutions, e.g.

http://www.cs.berkeley.edu/~oholtz/teaching.html

has homeworks for Rudins "Principles of Mathematical Analysis", and Halmos "Finite-Dimensional Vector Spaces". Finally most problem books (again Springer has a nice selection) have very detailed solutions. Yes, there are proofs that can be done in a number of different ways, but in my experience diverging too far is not very common and in most cases some core ingredients have to make it in in the end anyway. It's impossible to go through a lot of exercises "writing complete nonsense solutions" like that, when you are precisely checking your solutions against the given ones. For many types of exercises there are also simple ways of validating your solution, for example, in probability theory you can often do a computer simulation. In the end that's what anyway has to be done in real world and in research work.

[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...

[stiff]

I learnt only a bit of discrete math in my CS undergraduate degree, I had calculus and linear algebra but barely passed it by memorizing how to solve concrete problems and by having merciful professors - I was already working full time and had 4 or 5 courses going on, I just didn't manage to find enough time to study properly. If you have bigger gaps in your math knowledge and can't go to an university you just have to start at a lower level, there is a wide selection of "intro to higher mathematics" books meant for people like that, and if that level is still too high you might need to review high school math, e.g. Serge Lang has a good "Basic Mathematics" review book, or you can use Khan Academy videos etc. I myself had to review a lot of high school math when I was starting.

MIT OCW has excellent courses for discrete math, calculus and linear algebra and in some cases videos from the recitations are included.

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab...

I also stressed a few times already I don't think it is "the same kind of experience". But as long as you make an effort you will make progress and not everyone can manage to fit a university degree in their schedule.

[graycat]

Nice. A better response than I gave.

Halmos, Rudin, and Spivak 'Calculus on Manifulds' were at least at one time the main references in Harvard's Math 55 as at

     http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers

and are some of the best stuff for a ugrad math major.

Your last sentence is on the center of the target of reality.