The first rule of style is to have something to say.
I wish I had read How to Solve It when I was a fairly young student of mathematics. I wish also that I had read it when I was becoming a teacher of mathematics. It has been recommended many times, and now I will recommend it also. Really first rate stuff. Everyone’s excited these days about Thinking, Fast and Slow, but if you’re moving into real problem-solving that necessitates “slow” mental work, it’s Polya who really has something to say. His comments extend beyond the core problem-solving theses as well. I include here some quotes of particular note.
Mathematics is interesting in so far as it occupies our reasoning and inventive powers.
Can our knowledge in mathematics be based on formal proofs alone? … It is certain that your knowledge, or my knowledge, or your students’ knowledge in mathematics is not based on formal proofs alone. If there is any solid knowledge at all, it has a broad experimental basis, and this basis is broadened by each problem whose result is successfully tested.
Definitions in dictionaries are not very much different from mathematical definitions in the outward form but they are written in a different spirit.
The writer of a dictionary is concerned with the current meaning of the words. He accepts, of course, the current meaning and states it as neatly as he can in form of a definition.
The mathematician is not concerned with the current meaning of his technical terms, at least not primarily concerned with that. What “circle” or “parabola” or other technical terms of this kind may or may not denote in ordinary speech matters little to him. The mathematical definition creates the mathematical meaning.
Teaching to solve problems is education of the will.
Another “problem to prove” is to “prove the theorem of Pythagoras.” We do not say: “Prove or disprove the theorem of Pythagoras.” It would be better in some respects to include in the statement of the problem the possibility of disproving, but we may neglect it, because we know that the chances for disproving the theorem of Pythagoras are rather slight.
If the student failed to get acquainted with this or that particular geometric fact, he did not miss so much; he may have little use for such facts in later life. But if he failed to get acquainted with geometric proofs, he missed the best and simplest examples of true evidence and he missed the best opportunity to acquire the idea of strict reasoning. Without this idea, he lacks a true standard with which to compare alleged evidence of all sorts aimed at him in modern life.
[Not all mathematical theorems can be split naturally into hypothesis and conclusion. Thus, it is scarcely possible to split so the theorem: “There are an infinity of prime numbers.”]
That last is one place where I may disagree with Polya. The split is easy and natural: “Assuming our usual definitions (hypothesis) there are an infinity of prime numbers (conclusion).” It’s easy to forget that what we consider natural, as “the natural numbers” and so on, is just a made-up mathematical system when considered formally. Where mathematics comes from is a good exposition on this sort of thing, I think. And here, in a bit of a contrast, is perhaps my favorite quasi-philosophical quote from Polya’s book:
Do not believe anything but doubt only what is worth doubting.