proof methods

[Timothy Y. Chow]

On Tue, 10 Nov 2020, Buday Gergely wrote:
> In science, narrowing a topic is a useful method and can lead to a 
> tractable problem.
>
> How about this one: proof methods in university mathematics textbooks? 
> So, what is in the standard toolbox of a mathematician who has an 
> undergraduate degree in pure or applied mathematics.
>
> Is this still too broad?

I feel that I'm still not sure what you're really asking for.

Are you asking for strategies for coming up with a proof in the first 
place?  Or are you focusing on the final writeup of the proof and trying 
to categorize the different kinds of templates that exist?

Are you looking for high-level descriptions that are largely independent 
of subject matter ("to distinguish two objects, find an invariant that 
differs in the two cases") or do you also want to include techniques that 
use specific theorems ("bound your sequence and apply the dominated 
convergence theorem")?

Is your goal to teach students how to come up with proofs and/or write 
them up properly?  Or to teach a computer how to come up with proofs?  Or 
to draw philosophical conclusions about mathematical proof?

If your answer is, "All of the above," then I think your question is still 
too broad.

Tim