How to write two different proofs for the same theorem?

Question

I am writing a research paper in mathematics. I am proving a theorem. I have two proofs for the same statement i.e., the theorem. I want to embed both proofs. How should I do this?

Do I make it two different theorems in the paper, keeping the same statement? Or, should I state it just once and, under the proof section, mention Method 1 and Method 2?

For example, we maintain the following procedure:

\begin{thm}
(statement of the theorem)
\end{thm}

\begin{proof}
(proof of the theorem)
\end{proof}

Should I repeat the above syntax twice, keeping the same statement both times but with different proofs? Or, should I follow the below procedure?

\begin{thm}
(statement of the theorem)
\end{thm}

\begin{proof}
Method 1: (proof here with 1st method)
Method 2: (proof here with 2nd method)
\end{proof}

Is there another way to do this?

This is not a latex question. This question is based on the orientation of mathematics paper that includes a theorem with two different proofs.

Answer 1

First, make sure you have a reason for doing this. For the validity of the theorem itself you only need one proof. But, often enough the value of a proof is that it gives some insight into the inner workings of things and that insight can lead to other advances.

So, I suggest the second method, provided that both proofs have some unique characteristic, but, in that case, you should also say at least a few words about why you think this proof has value. You don't have two theorems.

But if both proofs use known and standard methodology, I'd only give one of them. Two different "standard" proofs don't really add anything.

Note that special insight can come two ways. One is an insight that leads to this particular proof (insight first). The other is an insight generated by the proof (insight last).

Answer 2

Personally, I would present one proof directly below the theorem, and then later something like this:

\section{Alternative proof of Theorem~\ref{thm:main}}

Here we present an alternative proof because... (state reasons 
why you think this is useful)

\begin{proof}[\protect{Alternative proof of Theorem~\ref{thm:main}}]
(Write your second proof here.)
\end{proof}

I used Latex syntax also to show how this can be done: the optional argument to proof changes the label printed before the proof, and I think protect is needed here, which may be tricky if you've never used it --- and the Latex error messages you'd get otherwise are horrible to read.

(This assumes that you are using the standard packages amsmath,amsthm --- but I have seen journal styles that use their own abominations instead of them.)

Answer 3

It really depends on how the rest of the paper (results downstream from your Thm, applications etc) relates to the ideas contained in the two proofs. If both are about equally indispensable, then by all means use any of the double-stacking methods suggested above; as long as you signpost clearly that the proof is done and the next one begins. Else choose one, preferably the one that makes the other stuff in the paper easiest to understand. The alternative proof can be (1) relegated to an Appendix; (2) Sketched if and when the ideas are needed in only one place; (3) mentioned (or briefly described) in the Discussion; (4) saved for another publication (I am not suggesting you publish the same Thm twice; I am suggesting that the basic idea of the alternate might be more appropriate within the context of a related but genuinely distinct paper).