Use the most specific word possible

The premise of this essay will seem extremely obvious, but this is a concept too few people have drilled into their head.

Today a student in my Fall 2023 AIME course submitted some solution writeups which contained the following line:¹

We can simplify $x^2 + xy$ into $x(x+y)$.

And the gist of my feedback was,

Instead of saying simplify, it’s better to say factor.

Imagine you are teaching a class to students. And you say “do a thing to this thing and get a thing” instead of saying “apply the function $f$ to the integer $x$ and receive a string”. Very unspecific, right?

In my eyes, “we can simplify” is like that. It could mean a bunch of things: you could be expanding a factored polynomial, factoring an expanded polynomial, cancelling terms, etc. It’s basically the same as “we do a thing to the equation”. What thing?

When the word “factor” is used, it is much clearer.

To be clear, the proof was written very well (which is why I bothered to be nitpicky like this). But this is a good segue into some thoughts I have about proofs and communication, so I’ll use it as an excuse to talk about them.

Being clear in a proof is the most important thing. Sure, it should be reasonably correct, in that it shouldn’t claim anything that’s untrue and should be reasonably² complete. But the primary responsibility of a proof-writer is to make it clear, “Hey this is what I am doing right here and now,” at every step of the proof. Yes, compromises need to be made between being explanatory and being concise. But using the most specific word possible is being more explanatory without being any less concise. In other words, it is an improvement you can get for free, so take it.

Pedantic? Yes. But when you’re proving something, your job is not to say “and the things I said are right”. It is for the reader to say, “I see what you did, I am convinced it is correct, and I get what you’re doing.” So even if being more specific doesn’t make your proof more correct, it is still just as valuable.