Can I "denote X by Y" in a mathematics paper?

Question

It comes as no surprise that the word "denote" appears quite frequently in mathematics, mostly in the context where succinct notation is being introduced for 
some object.

As far as I understand, it's uncontroversial to say "Let N denote the set of positive integers".

However, that does not quite convey the intended meaning (at least for me): It's the speaker(s) who assign the meaning "the set of positive integers" to the symbol "N". This tempts many (myself included) to instead say:

"We denote the set of positive integers by/with N".

This is often criticised as incorrect, but I've also heard it being defended. So, is the latter usage correct?

JeremyC · Answer

In my field, statistics, working in the UK, it is quite usual to use 'We' throughout papers. I happen to have a PhD thesis in front of me as I write this and I find, opening it more or less at random: "We define... to be x...". "We" is then used three times more in the same paragraph to define notation, after which we find  a different way of using 'we': "Since we are ignorant about... we must consider...".

I could have picked almost any paper at random from my bookshelf to find similar examples.

That is not to say "Let x denote..." is in any way wrong. The same author uses it frequently, as do I. If there is a difference at all, I would say that "Let" tends to be used in the more mathematical sections of papers, for example where it is necessary to give a mathematical proof.

You refer to criticism of the use of "we". I have seen none, but one does occasionally find people recommending that papers be written in the passive, as in "Observations were taken at...". Is that what you had in mind? In my field it is now rare to see passive throughout a paper, and it seems very stilted.

Can I "denote X by Y" in a mathematics paper?

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