On checking a proof

In many respects the most difficult part of mathematics lies in communicating your work. That is not to say that proving results is easy; but the proof is just the tip of the iceberg (even though it often feels like the principal goal). The proof is of no value or interest unless you can communicate it. And to communicate it, you have to be able to write it down accurately. I’ve not found any notes on how best to do this, so here are a few thoughts of my own.

When I look back at some of the proofs I wrote when I started work on my PhD, I realise how much I have learned. My supervisors – who were very gracious, very helpful, and very dedicated – used to cover my early work in red ink. I then learned how to write a proof through an iterative (and very painful) process, in which I would write something, receive the red ink, fix those problems, receive further red ink, and so on. I became very familiar with red ink. Very, very familiar.

An early example of the great value of red ink

In this note I’d like to comment on how one might spot problems oneself, rather than depend on one’s supervisors in this way. This is not a trivial task, but a really important one. Perhaps I can offer a few pointers which might be of help.

Let’s suppose you have proved a result. You’ve written it all up to your own satisfaction, and wish to share your  achievement with your fellows. I began to make a list of the things you should do, but it was very long, exceedingly tedious, and all boiled down to the word check. Which is a bit boring. So let’s try the following, which is less prescriptive if possibly less all-encompassing. It’s just three words. How hard could that be?

First forget. In developing your proof you, no doubt, came up with all sorts of ideas and intuitions and implications and pictures. You have to (somehow) now lay these all to one side. Your reader will not have any of this in front of them, so you have to be sure that none of your work now depends or uses anything other than the words in front of you. (Incidentally, the best way to do this is to put your proof to one side for a few months, and then come back to it. You’ll be astonished how terrible it will look).

Second focus. Focus on the words in front of you, and what they say. This is easier said than done; because you expect your words to say one thing, you will tend to interpret them in that way. Try not to. Look at what is written and nothing else.

Third check. Read what you have written, word by word, sentence by sentence, and ask yourself the question “why on earth does that follow?” Notice the negation; if you expect things to be wrong you are more likely to spot mistakes than if you expect them to be correct. In my personal experience they are probably incorrect.

I could probably make a list of common mistakes, but it really is hard to make that interesting. So I will highlight just three (three is a useful number here):

The word “clearly”: It is very easy to make the mistake of writing “clearly XYZ” when what you mean is “XYZ seems pretty darned obvious to me but I can’t quite work out why”. If you can’t work out why XYZ is true, chance is that is isn’t.

Things that are true but don’t actually follow: This is a very easy mistake to make; you write something like “Since X, then Y” and assume it is OK because Y really is true. But you are not asserting here that Y is true, and that is not what you need to check. You need to check that Y follows from X and nothing else!

Failure to satisfy all necessary conditions: If you use another result (maybe a book result, or a lemma of your own from earlier) you need to be sure that all the conditions are checked. This is especially true of a book result – if that says something like “If A, B, C, D and E, then F”, then there is no chance to use this result if only A, B, C and D are true.

Yes, this is all amazingly tedious. Yes, this is a very lengthy process. No, there is no alternative (apart from asking a friend to check). Yes, you will be a better mathematician when you can do all this. No, I do not claim to be able to do this all the time myself. Yes, I welcome feedback and other suggestions.