Transition to advanced mathematics

Question

I need help understanding the layout of proofs, I find myself lost after hours of trying to work it either if I know how the problem needs to be worked or not. I am taking probability and transition to advanced mathematics. I am struggling in both classes and just nervous for the worst!

Answer 1

So, you don't technically ask a question, but you imply a question which I will attempt to answer.

The best thing I could recommend to you will be really inconvenient, and that's how it goes. There are many books about proofs (eg. Velleman's "how to prove it", and you should start by reading them in addition to your classwork. Then, try to prove everything, even if it's not assigned. Practice, and you'll get the hang of it.

Second, if you are having trouble writing down your proofs "in math", start by writing them down in english. It is easier to write down your thoughts completely and translate them, especially while learning, than it is to try to put it into math the right way first.

But, seriously, practice. This stuff doesn't get easier by ignoring it. Yes, it's a lot of work. Yes, it's hard. But, it will pay off in the long run.

Answer 2

There is far more to be said about "layout of proofs" than anyone can say here. Nevertheless, in addition to @atomic's good points...

Do not think of "proof" as something different from ordinary language, ordinary persuasion, ordinary logical thinking. Nevertheless, as one can easily see in the popular press, much of what passes for "persuasion" is really just bullying or scare-tactics or "good graphics" or touching upon conditioned reflexes or cultural traditions. The idea in mathematics is that one should "rise above" any of those too-highly-conditioned contexts... In the last 150+ years people have also observed that "physical intuition", while obviously guiding us, is not "proof".

Nevertheless, the spirit of "proof" is that one simply explains what one "observes" in some quasi-Platonic very-real world. There are not "sacred" motions to go through to form a proof. Appeals to dubious "intuition" are not legal, although, at another level, appeals to "common knowledge among experts" (an improved "intuition") are in fact routine.

A sad aspect of "proof" in "elementary" mathematics is that it too often emphasizes proving artificial abstracted things in "the null context". So one should not be surprised if one has no traction!!! That is, when the "game of proof" is turned into a nearly content-free game of manipulation of symbols, small wonder that we can't get a grip!!!

Again-nevertheless, there are some benefits to be had from being able to play the symbol-game, just as there are benefits from being able to manipulate Hindu-Arabic numerals. (Well, anyway, there once were.) But this is not the same as more intuitive understanding of the real things.

After the highly non-trivial issue of "clear writing", which is indeed the true obstacle for many, the next issue is "feeling and expressing quasi-physical intuition". These two issues are wildly different from each other, and, often, I think people have troubles because they entwine the two...

But, back to another point: in any case, practice. Familiarity. There's no magic.