Mathematical Habit

Question

Whenever I have to do proofs or understand a concept, I start with examples and generalize. As an aspiring mathematician, I have fears that it might hurt me, especially when we have so many counterintuitive things. It is more convenient to do things this way, when I am proving known facts or understanding those concepts. However, it might not be efficient when I am trying to prove/disprove unknown situations.

Do you have any suggestion as to how to avoid generalization? Is this only my problem? If so, what would be a better approach to proofs?

Thank You.

Answer 1

You are doing exactly the right thing. Generalise to a wider rule, then try concrete examples which challenge this wider rule. Does it hold true for infinite sets? Does it only apply to convex shapes? Is it true if x is a complex number? If it holds true, you may see the general reason when trying concrete examples. Or you might be able to find counterexamples. In this sense it is no different to physical sciences - you test the theory experimentally. Generalisation is good, as long as you test your theory against the new cases.

Answer 2

In general, I think it is a very good strategy to start with examples. It may not always be the fastest way, but I think you understand proofs, concepts better when you first start with examples before trying to proof it in general.

I think many mathematician would call this way of understanding a concept "intuition". And a good developed intuition is a really important thing for a mathematician.

I don't think that Plato proved out of the blue that $\sqrt{2}$ must be irrational. More likely he first tried to find an $m,n\in \Bbb Z$ such that $\frac{m}{n}=\sqrt 2$. After many tries he probably get a "feeling" that this may be impossible. This was his intuition, which gives hope that there must be a general way to prove that $\sqrt 2$ is irrational.

Answer 3

The advise I have seen generally amounts to what you are doing, mathematics is (even though I'll get stoned for saying this) an eminently experimental science. You play around with concepts, try to see how they fit together, stumble on a problem and look for ways to solve it. You find something that looks interesting, you try to prove it is true (by fooling around with it, seeing why it has to be true); you fail repeatedly and begin believing it really isn't (always) true, and start looking for ways in which it can fail.

The only way to become proficient in anything is to dedicate some 10.000 hours total to it. And much of those hours will be "wasted", in that you won't be able to show groundbreaking theorems, or even just correct results, for most of those hours. Just keep at it.

Read (and make sure you understand) Pólya's "How to solve it". There are other books by prominent mathematicians in which they discuss the soft(er) side of the profession, mostly in form of a scientific autobiography of sorts. I'm sure others around here can add a list of suggestions.

It is tacitly assumed that whoever does math is automatically a stellar writer. This just isn't true. Sure, there are people who are naturally gifted, but we the vast majority are mediocre or much worse at it. But it is a skill that can be improved. Look up e.g. Halmos' "How to write mathematics", Knuth et al's "Mathematical writing", there probably are many others worthy of a read.

Much mathematics is financed by teaching... and again, that isn't something many of us are born knowing how to do well (or at all). Try to get some training in that area. I've found out the hard way that doing it right involves lots of rather counterintuitive stuff. Many stumbling blocks for rank beginners are in areas we have so well travelled that we just can't see the snares and pitfalls anymore. Concepts we find simple on closer analysis aren't, or clash so blatantly with the common-sense meaning of the technical terms used to describe them that understanding is next to impossible.

Answer 4

You can't reach the stars without generalization. The problem is how to prove the generalization, which it turns out always follows from the definition of what you're studying some way or another.

If you take the definition of a ring and a definition of a homomorphism, you can deduce the definition of an ideal, and then the isomorphism theorems, quite directly. From there you can prove the spectrum of a commutative ring is a topological space, again quite directly. These are clues that backing up your intuitions is mostly feasible, but that when it's infeasible you're on the way to something deep, such as how your intuition can fail, that that intuition is not a direct reflex of the language used to formulate the concept (syntactically tautological), or that you didn't realize that your intuition had a pleasant syntax (the isomorphism theorems, in application, repeatedly point this out), and hence the meaning loaded in such a theorem. Fermat's last theorem was an unproven intuition.