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?