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There are plenty of questions around here, which are proven to be right or wrong in various ways. I wonder, what one can learn from these differing ways of how to prove something, despite the fact that: The more proofs, the better.

Let's say a statement is something like a way $A\to Z$. One proof might then break this down to $$A\to B\to \cdots \to W\to Z,$$ while the other proof takes another route $$A\to \beta \to \cdots \to \omega\to Z.$$

Is there way to morph between the various ways and by that learn something about the general structure?

EDIT What is it worth to have plenty of proofs for the "$\Rightarrow$" direction, if a have only one proof for "$\Leftarrow$"?

The question is very general. Examples, are welcome.

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    You may be interested in this and this. – Bruno Stonek Feb 19 '12 at 23:33
  • I think the question is too general. – lhf Mar 23 '12 at 00:58
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    @draks Different proofs might be related to different areas of math. The origin of various proofs is either genious (like Gauss' MO) or different people that do maths their way. I personally prefer a variety of proofs since you might not get one and understand another, or because they provide different insights. Think about $$1 = \cos(x-x) = \cos(x)\cos(-x)-\sin(x)\sin(-x)=\cos ^2 x+\sin ^2 x$$ It is a purely analitical proof a the Pythagorean Theorem, which I like the most over any other. – Pedro Mar 23 '12 at 01:32

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If you look at this from a proof-theoretic point of view, then each proof yields certain kinds of information which ideally facilitate the extraction of computable realizers or things in this fashion.

An interesting field of study is the topic of proof mining which concerns the extraction of computable realizers or uniform bounds from (possibly non-constructive) proofs. Ulrich Kohlenbach has written an extensive book on the topic [1].

Based on Gödel's Dialectica interpretion and a negative translation of formulas, one can show that if a sentence $\forall \vec{x}\exists\vec{y}A_0(\vec{x},\vec{y})$, where $A_0$ is quantifier-free, can be proven in weakly extensional Peano-arithmetic using only quantifier-free choice and some universal axioms, one can extract realizers (computable functionals) $\vec{t}$ for $\vec{y}$ such that it is constructively provable that $\forall\vec{x}A_0(\vec{x},\vec{t}(\vec{x}))$.

Therefore, it is at least for some cases possible to extract general information from proofs, independent of the actual form of the proof (however, it is important that the axioms used are in a certain set of allowed ones).

NB: When it comes to bound extraction, the quality of the bound might well depend on the actual proof. For example, there are different proofs by Euclid and Euler for the proposition "There are infinitely many prime numbers" which yield different upper bounds on the $(r+1)$th prime number. I think that this also motivates the construction of different proofs for the same theorems.

[1] Kohlenbach, Ulrich. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Heidelberg: Springer, 2008.

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