Should I create two distinct proofs? [*Soft question*]

Question

This is a soft question, and if it is of poor quality, just let me know.

As a method of improving my proofing abilities, should I make it habit to go about proving something twice. What I mean by this is, should I prove something the way that seems the most simple(exhaustion for example) and then prove it using a 'cleverer' method, or should I sit and consider the problem until I find a concise solution. I will note that I normally have trouble coming up with a concise solution.

Secondary question: Is this method viable in the long term? Will proofing become too difficult to go about doing twice?

Note: I am relatively new and inexperienced in proofing.

Answer 1

First, I want to be clear about what exactly it is you are asking about. If you want to improve your ability to understand and/or remember proofs which you have seen in a course, then I think it is a good idea to go over the proofs that have been presented until you can forget about the details and take a "bird's eye view".

You can think of a proof as a path from one place to another. If you wanted to remember the path from your house to school, say, it would be unwise to try to remember what every house and street along the way looked like. Instead, you would remember some landmarks and maybe some places where a change in the path has to occur.

For proofs it is similar. Remember the major "landmarks" of a given proof (maybe one part employs induction to proceed, then later an invocation of the dominated convergence theorem for example) and just "follow your nose" the rest of the way.

If your question is more about coming up with proofs, then the context matters. If it is in the context of a university level course for instance, then you should be familiar with the proofs you have seen in that course because what you are asked to do on a test will likely use similar techniques with some minor deviation/cleverness depending on the course/prof/university.

If it is in the context of doing research then it is a lot more difficult but essentially the same principle. You will have to know the field well and employ the standard methods there, but things will be a bit messier because test questions are usually selected for their nice answers whereas research questions seem selected to induce madness. You may also be required to come up with new techniques in this context.

I hope that helps!

Answer 2

In my experience, it's typically best to try and prove a theorem in whatever way gets the point across, and then going back to clean up the proof, make it more concise, and think about different ways in which it could be proved.

I don't think that in general every theorem/proposition/corollary/etc can easily be approached with the hope of coming up with two different proofs; there will usually exist two (or more) substantially different proofs, but this doesn't necessarily mean that they will be at all apparent, and just understanding one proof is generally good enough for some of the higher level proofs.

To get better at proofs, it may be better to just try and come up with a single proof for each of a large collection of results, rather than try to come up with a larger number of proofs for each result. It might be better to take a middle ground, however: try and prove a large number of statements, and when you feel that another methodology could be used to prove a statement, try and prove it that way as well.

Overall, the main idea is to try and find a lot of different methodologies, but also in how to approach a problem and begin the path to writing a correct proof. The first is done using the methodology you mentioned, but can also be done by doing a lot of proofs in general, whereas for the second, doing a lot of proofs in general is probably the best approach to doing this.