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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?

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    From the point of view of {logic and prototypical senses of English words}, your sentence is the more felicitous. However, pragmatics demands that we go with the flow (though I've just gone against the flow by adding braces to show a 2-part but 'single' consideration), so your 'Let' sentence (and 'let' is used weirdly outside the maths domain: "Let us pray" – it's a hedge really for "Pray now, folks!"), being widely accepted, is ... acceptable. Your variant is IMHO better (maths, after all, strives for accuracy and consistency). But pragmatics rules if you're going to get a paper rejected. Jul 21 '19 at 10:29
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    @EdwinAshworth It is an exhortation to the listeners but phrased so as to include the speaker in the group being exhorted. The construction "let us + verb" is widely used, although usually in the condensed form such as "Let's go!", "Let's get started", "Let's see what we can do" and so on.
    – BoldBen
    Jul 22 '19 at 1:36
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    There's nothing wrong with 'Let N denote the set of positive integers' but if you want to put it in the first person you can say 'We use N to denote the set of positive integers', or 'We call the set of positive integers N.' Just don't say 'We denote' because it's the symbol N that does the denoting.
    – bof
    Aug 21 '19 at 12:47
  • A search in Google Books for "We denote" shows that this is used in academic writing.
    – DjinTonic
    Jul 27 at 17:12
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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.

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    As an engineer and mathematician, I would say that this form is like "Legal speak" where words, phrases and punctuation have exact meanings making many legal documents are difficult to read for the layman. Likewise mathematical based subjects have developed their own way of expressing thoughts that are well understood by those versed in the art, but which sound somewhat odd in general English. Whilst you can use other constructs, the standard forms mean that everyone (mathematicians) will understand exactly what you mean without ambiguity. Jul 22 '19 at 9:10
  • @Peter Jennings A strong argument that questions about maths-domain jargon should be asked on Mathematics SE rather than ELU. They deal with phraseology over there. Legal jargon is, or used to be (as far as I remember) specifically proscribed here. Jul 22 '19 at 11:23
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    @EdwinAshworth Re legal jargon, quite right, it isn't English as we know it! :} Jul 22 '19 at 23:33
  • @PeterJennings I would say you are more of an engineer than a mathematician at heart. Such usage of "we" is just everywhere in math literature and has nothing to do with "legal speak". Notations and concepts are readily defined and handled differently by different mathematicians. And this is not a mere personal preference, I would say.
    – Lei Zhao
    Mar 29 at 9:22
  • @LeiZhao I never said it was "legal speak", only that it was like legal speak in that it has evolved from ordinary everyday English to have s slightly different or specific meaning which may sound strange to the layman, but is readily understood by other mathematicians. And yes I am more of an engineer, I started out studying mathematics, switched to electrical engineering (telecomms) and ended up writing software. Mar 30 at 10:45

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