proof methods

[Buday Gergely]

11/9/2020 3:36 AM keltezéssel, Timothy Y. Chow írta:
> 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

Dear Tim,

thanks indeed for your thoughtful answer.

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?

- Gergely