[gems]

You spent a significant chunk of your post suggesting self study. For something like math, this is really really really unrealistic. Most pure math textbooks don't have simple problems you can just check in the back. They're multistep proofs that can be done in a number of different ways. Oh, and you encounter plenty of material where you can easily trick yourself into thinking that you really understand it when you actually don't. Additionally, there is a standard of rigor that you don't experience at bad schools (or with no schooling). You could be writing complete nonsense solutions and not even know it.

Also: how are you going to self-study when you don't live at home? Studying is a full time job.

[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.

[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.

[graycat]

You have a good point, but my post was already at the limit of 10,000 characters. Of course the solution to your point is partly a theorem proving course in high school plane geometry and then, finally, a theorem proving course as, say, a college junior in abstract algebra. For such a course, I did say that the last two years should be at a four year institution; at such a school, a good enough course should be available even if the first two years were in a community college where the calculus teaching was poor. Again your point is correct: To learn how to do proofs well enough to be self-sufficient, need at least one theorem proving course where can get homework and tests graded by a competent mathematician.

Don't worry: I've tried to show that P = NP and know that while I've had some candidate ideas I don't have a good idea or a proof. And, I've nearly never written a bad proof; once catch on to how proofs are done, they are surprisingly easy to check for correctness.

Studying is not a full time job -- I was heavily self taught in math and totally self taught in computing and nearly never studied full time. E.g., I read Nearing on linear algebra, Halmos 'Finite Dimensional Vector Spaces', Fleming 'Functions of Several Variables', yes, with the exterior algebra, and much more while working full time in mostly DoD work around DC. I did the research for my Ph.D. dissertation in stochastic optimal control independently in my first summer in graduate school.

Edit: There's a better answer in this thread in

     https://news.ycombinator.com/item?id=6177643

[Peteris]

This is false. I studied Maths at Cambridge and learned some courses completely on my own.

[jinfiesto]

Seconded. Math is one of the most easily self teachable subjects. The field of study is objects of mind (unless you're a platonist.) Literally no materials required except pen/paper, a brain and maybe a straightedge and compass. The point of math is not to do endless worked exercises. It's to understand mathematical objects and prove interesting things about them. You can generate unlimited problems for yourself by investigating some mathematical object at random.

[gems]

It's false that any of what I said can be true for some (or many) students? How do you determine if you're one of those students?

[agentultra]

I think it is very reasonable.

There is plenty of free access to materials and papers.

It's incredibly cheap and fast to communicate with pretty much anyone in the world.

All that is really required is motivation, discipline, and curiosity.

When there wasn't cheap access to global communication networks and near-zero cost to duplicating data then it made sense to go to university because that was where everyone who you would be interested in talking to would be. That's where the libraries and books were. I don't think that is the only option anymore. And that's a good thing.

You can learn anything you want on your own and still have all of the benefits of a college (access to knowledgeable people, peer review, etc).

[vacri]

I was on a forum where there was a guy who had self-studied maths and physics and was an 'ideas man'. The problem was that he just didn't have the standardised nomenclature to get his ideas across - and also meant that he couldn't understand the reasons why other mathematicians debunked him.

One outstanding example was his method for a 'free energy' spacecraft movement system, that hinged on an arm throwing -foo- into a receiver. The argument was that there is a difference between throwing something and merely releasing it at speed, and he didn't have the understanding of physics to realise there is no difference.

Another one of his ideas was a system for finding prime numbers, which he couldn't articulate well enough for people to figure out whether it was a valuable predictor or merely a sieve.

He was a regular at the forum and respected, and I've never seen so many people patiently explaining physics and maths at such clear lengths before... and he just couldn't grok it, because he didn't have the standard language to get his ideas across.

This anecdata doesn't contradict the GP though, who is talking about doing self-study in parallel with formal study.