Writing Better Proofs Part 5: An Exercise for the Reader

This post first appeared 24 May 2021.

This one is easy:

Never explicitly leave part of the proof “up to the reader”.

If you are writing a proof, you must convince the reader that the result holds. Asking the reader to show intermediate results or perform calculations just goes towards convincing them that you don’t have a complete solution1.

There are some easy phrases you should avoid:

That is not to say that some things are not “clear”. For example, it is clear that there are infinitely many prime numbers2, and you can claim that without proof.

However, I once had a student claim that the entire result they were asked to prove was a “quick calculation”. This, of course, would not earn them any marks. In a research paper, it would at best waste a few hours of a graduate student’s time. At worst it might hide a fundamental flaw.

Why this happens

Here’s where I stray from the realm of giving advice to that of making idle musings. Why do students and researchers alike leave so much up to the reader?

I think the reasons are three-fold:

  1. Firstly, it saves space and time. When you are writing exams or a paper, you may be conscious of how long it takes you to write out every last detail or how it affects your page count.
  2. Secondly, it allows you to gloss over subtleties. By deferring them to the reader, you can ignore all those tricky edge cases and exceptions that have to be dealt with specially.
  3. Finally, I think it comes down to how we teach maths. Leaving details to the reader is a common tactic both in textbooks (where exercises are actually useful to the reader) and in lectures (where minutes are useful to the lecturer). Many people try to emulate how they are taught in how they do maths, not realising the shortcuts that don’t translate from “teaching” to “doing”.

In the end, if you are reading these posts, by the very title, the burden of proof is on you. Asking the marker or reader to tell you why you are wrong is just… wrong.

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.

  1. There is a caveat to this. If you are giving a summary of a proof or a presentation, you may (and should) omit all the “boring” calculations so you can explain the “shape” of the proof faster. ↩︎

  2. Unless that was the question. ↩︎