Writing Better Proofs Part 1: Why Bother?

I’ve done a fair share of reading mathematical proofs. It comes with the territory. As an aspiring professional mathematician I expect to spend a substantial chunk of my remaining lifetime reading (and hopefully writing) proofs.

During my PhD, I’ve taught scores of undergraduate students and marked hundreds of problem sets. I’ve seen it all, from rote copies of the results from lectures (complete with errors verbatim), to proofs that were unintelligible, to submissions that made me want to read them again because they were just that good.

But almost everybody I’ve seen could do better. There is almost always some way to improve your proof writing¹. No argument is truly static. New terminology or frameworks may cast old results in new light. Long arguments can sometimes be shortened by key observations. Untidy scrawls on rough paper can be typeset into beautiful edifices of logic.

So here are some reasons to write better proofs:

1. To be a better mathematician. It is all too often that one finds a proof in a published paper that really could be better with just a small amount of effort. Anything that is technically incorrect or makes the reader have to work harder to understand you is a failing on your part. If your reader has to work hard, they are less likely to read your work.
2. For marks. If you are a mathematics student in a school or a university, writing better proofs will get you more marks. That much should be clear.
3. For grade boundary marks. Even if your proof is “correct”, making it better is more likely to make the marker more sympathetic to you in those cases where a mark hangs in the balance. Sometimes that single mark is the difference between a grade boundary.
4. To be proud of your work. Good mathematics is an art. A carefully crafted proof is valuable in its own right², and you should be proud of the value you create. Each recounting of a proof (new or otherwise) is a fresh piece of work and deserves care and attention from its creator.

But there is one reason that really is the crux of this whole matter.

Mathematics is at least 50% communicating things you know to other people.

Most of that communication comes in the forms of proofs of theorems and lemmas. The greats may be able to leave conjectures scattered about their work, or make bold claims that are “left to the reader”, but us mere mortals must back up our assertions with proofs.

We might as well make them good ones.

This is part of a series on writing better proofs. The pages in the series are not static and will be updated and improved as time goes on. If you have comments or suggestions, please email me. I’d be particularly interested in hearing if you disagree with me or have suggestions for topics to cover.