One thing that can make me seize up on seeing a proof question is not knowing what the game rules are: what are the axioms, what can be assumed. But I think this instinct of knowing what the question is asking for is something that comes with experience.
For example, I remember puzzling over some questions in Binmore's
Mathematical Analysis, only to turn to the answers in the back and find that he'd taken for granted certain preliminaries that I'd thought would have to be part of the proof. By contrast, Spivak's
Calculus covers some of the same material, at a similar level, but makes explicit what axioms can be used. I think the latter approach motivates me to try harder to solve the problem, knowing that it
can be solved given what I know, and that it will be satisfying to have the thing proved right back to the foundations. Whereas Spivak gives all the axioms, Binmore gives only some rules and says something along the lines of the usual rules of arithmetic are assumed - even though some rules one might consider usual are among the things to be solved. For this reason, a book like Spivak's, which makes the task clear, may be the best starting point in attempting proofs. That said, Binmore's answers are often very detailed, so they give plenty of good, worked examples of proofs, which is useful too.
Of course, a more advanced book on a topic will rely on elementary results; then the reader has to use their own judgement about what can be reasonably assumed. Books will sometimes set an easier or more casual exercise with the words, "The reader should convince themselves." In other words, assume whatever seems obvious to you; just consider what's new and not quite obvious until it becomes so; if it seems to make sense then, you've probably got it.
In his
Real Analysis lectures, Francis Su advises students to write their proofs in a way that assumes as much knowledge as someone two or three weeks behind them in the course.
It's interesting the contrast between
micromass's answer and that of
I like Serena.
micromass emphasises the instinctive aspect (we're
writing proofs all the time in our heads without knowing it), whereas
I like Serena stresses the need to study mathematical logic as if it was a foreign language. Not a contradiction, just an interesting difference in emphasis... Does anyone have any recommendations for a book on proofs or mathematical logic? Am I right in thinking that the official name of this language is
first order logic, or is it
predicate logic in Wikipedia's more general sense? Or is it predicate logic in the sense of their second paragraph: the idea which can be formalized as first order logic (although first order logic is not necessarily the only way to formalize it)?
