Sometimes. Many parts of math are very difficult to tell whether you're wrong or not, except by reading through the proof extremely carefully. For example, about a hundred years ago there was a proposed proof of Fermat's Last Theorem, but it had a very subtle flaw: it implicitly assumed that unique factorization held in a particular set of rings. This turned out to be wrong in some cases. This error was very common back then, and many people believe this was the proof Fermat had in mind.
So, you see, in mathematics your unconscious assumptions are sometimes wrong, and there's nothing that would tip this off to you, like a compiler. In fact, very few areas of mathematics are the sort where you can easily tell when you're wrong.
I suspect you think of your high school math classes when you think of mathematics. That's really a very poor representative. Mathematics is about proofs, not answers. To be fair, though, the original post makes a similar misconception. Computer programming : computer science :: high school math : mathematics.
That's a good point, but the same occurs with programming. We generally can't catch every corner case. Of course, this is much more devastating in a mathematical proof, i.e. the proof is invalid. However, it is still very possible to develop good mathematical skills without a degree.
Also, this is somewhat unorthodox, but I've read that Godel proposed a new method of doing mathematics based on the experimental method, due to the implications of his incompleteness theorem. If that were the case, then math becomes more like computer science, or perhaps computer science is this new way of doing math.
> However, it is still very possible to develop good mathematical skills without a degree.
Absolutely. I don't think the degree is very useful for developing skills at all. I'm sure there are plenty who would disagree, but mathematics courses don't set a very high bar for how much you have to think about a topic. I learned more in the past month preparing for my algebra qual than I did during an entire year of an undergrad class at Northwestern (of course, I drew heavily on the basic knowledge I had).
I'm having exactly this experience, of a lack of feedback, with proofs. It also seems frustratingly ironic to me that you can never be sure that a "proof" of the truth of a theorem is actually true...
I'm learning COQ (proof assistant), to make some aspects of it automatically checkable.
The feedback of programming is also encouraging and satisfying, and this makes it more easily self-taught, like a (computer) game. Unit testing builds on this.
PS: My proofs are part of my compsci PhD, which began without any maths; so I choose a supervisor has some knowledge of proofs but isn't very experienced in them. Perhaps more guidance with proofs could really help...? Things are finally coming together for, but I keep stumbling on what I think are extremely elementary proof ideas (e.g. that f(AUB) = f(A) U f(B) ). My feeling is that rigorously understanding takes effort, no matter how good the guidance.
Progress in mathematics comes from research. To contribute something new, you often need a lot of background knowledge and university is the place for this. As with all things, there are exceptions - Ramanujan for example.
To be a financially successful programmer, you often just need a few books, courage to step outside your comfort zone, and a good idea. There are lots of examples of this, but for me the embodiment is Ken Williams of Sierra fame who made out pretty well, and I don't recall that he had a degree in CS.
