proof methods

[Timothy Y. Chow]

Buday Gergely wrote:

> Do you know some reference that writes about proof methods, giving a
> taxonomy of them?
>
> I think of basic proof methods like indirect proof, proof by cases,
> proof by symmetry, induction
>
> and advanced ones like transfinite induction or forcing.
>
> Some of these have formalization in mathematical logic, some others don't.
>
> Some of these connections are direct and easy, like reductio ad
> absurdum, some others like symmetry have a detailed theory, namely,
> nominal sets.
>
> What I look for is what working mathematicians use in their publications.
>
> Is there a survey paper or book on this?

Once you start talking about "more advanced proof methods" then I think 
that it's virtually impossible to provide what you're asking for.

For example, in the following paper, Terence Tao tries to give a taxonomy 
of the proof methods employed by *a single mathematician*, namely Jean 
Bourgain:

https://arxiv.org/abs/2009.06736

If you think about how much expertise and effort went into creating a 
taxonomy for the methods used by just one person, I think you can see that 
creating a taxonomy for the methods used by all mathematicians is a 
completely hopeless task.

Now you could try tackling the problem one subfield of mathematics at a 
time rather than one mathematician at a time.  For example, a really 
beautiful book that describes the fundamental proof techniques in 
transcendental number theory is "Making Transcendence Transparent" by 
Edward Burger and Robert Tubbs.  But not every field has such a nice 
textbook account of all the main techniques, and even if they did, your 
"survey" would comprise hundreds of graduate-level textbooks.

Another approach would be to try to create a big flowchart or "expert 
system" that codifies the approach that a research mathematician takes 
when confronted with a new problem.  Here's an example by Scott Aaronson, 
explaining how he tries to upper-bound the probability of something bad 
happening:

https://www.scottaaronson.com/blog/?p=3712

Tim Gowers attempted to spearhead something called the "Tricki" which, if 
successful, would have been a far more comprehensive 'flowchart' of this 
type.  But you can read about the difficulties it ran into here:

https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/

Tim