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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.

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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.

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.

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.

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.

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...

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.

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

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

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?

You're right, the above posters generalize way too much. They are either CS or Maths students and as such are already well-prepared for self-study! If you study Maths you already know what all the symbols like epsilon, e etc. mean, you can just gloss over a mathematical text and get the gist of it. Same goes for many (but not all!!!) CS-students, some unis lean a lot on algebra, some don't do much maths after the first two courses.

As such, I think it's ridiculous to go up to anyone and tell him/her to just "study by yourself", I could give a maths book to a biologist and that person would understand absolutely nothing without guidance.

Well, you could certainly do it when you start with "Maths 101" or something like that.

I was addressing the money issue and tacitly assuming that they could do the work. For just a Bachelor's and then just a Master's in engineering, they have a good chance of being able to judge correctly if they can do the work.

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.

Basically nothing you said actually made a case against anything I said, or for the thesis that it is easy to self teach. Just stating this for the record.

Could you explain briefly why this is the case? I don't understand.

Well, what was his argument? Based on his second reply, he seems to think that because it only depends on your ability to reason that it should be easy. But doesn't that trivialize the matter? As long as we have mathematical models, as we do in physics and in chemistry, then it should be just as easy to learn physics and chemistry.* So then what does he consider hard to learn? Are the social sciences hard to learn? The 'it is of the mind' is a non-argument to me. And I don't think he addressed anything I said.

Anyway, this is almost irrelevant to what I was saying. Even if you assume every person can reason well, I'm saying you could still be in error unless you seek validation and guidance. It's really easy to think you've given a solid argument for something, but actually be wrong. It happens to everyone.

*By the way, there is a definite trend in physics for math, instead of experimentation, to be leading the way towards discovery.

Well. There are no barriers to entry. If you have any inkling of logical ability, you should be able to tell when a proof is right. All it requires is critical thinking. Presumably humans come with that out of the box.

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).

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.

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