Is it correct to use an expression "we denote something by x"?

Question

Consider the following two sentences.

(1) We denote the set of natural numbers by N.

(2) N denotes the set of natural numbers.

Both expressions are very common in math papers. According to the Oxford dictionary, the second expression is correct. See http://www.oxforddictionaries.com/definition/english/denote.

However, I prefer the first expression because of the following. To use the second expression, the symbol N should have appeared before without definition. Instead, I could use an expression such as

We let N denote the set of natural numbers.

But if I use this expression many times, somehow I feel I make duplications.

Now I am afraid if the first expression is incorrect. I googled and found some answers in the following links, but they are not very convincing to me.

http://forum.wordreference.com/threads/we-denote-by-n-an-integer-instead-of-n-denotes.2379959/

http://www.waywordradio.org/discussion/topics/going-against-convention-while-writing-in-your-field-denote-by/

Answer 1

Both expressions are very common and universally understood in mathematical English written by native speakers -- which means that that, as a practical matter they are both right. And if a dictionary claims that "denote" can't be used in this way, that is the dictionary's fault.

Your assumption that (2) requires that the symbol has already been mentioned does not match how these phrasings are used in practice. Indeed, a sentence such as (2) would commonly be found in an early section of a paper entitled "mathematical preliminaries" or something like that, where the author briefly summarizes the symbolism that he's going to use in the rest of the paper.

As a writer your wording would probably sound most natural if you use the bare statement of fact "N denotes ..." for notation that already exists, such that the sentence just reminds the reader what is already true, whereas "We denote ..." or "Let N mean ..." and so forth are for definitions that are part of the original content of what you're writing, and therefore are new to the reader.

However, as a reader you shouldn't try to extract any meaning at all from the difference between these phrasings. The trend I just sketched here is not followed consistently enough for that to work. Go by context instead.

By the way, your third sentence

We let N denote the set of natural numbers.

sounds slightly clunky to me; simply

Let N denote the set of natural numbers.

would be more straightforward.

---

(Note that actually neither of your sentences could usefully be used in a mathematical paper, since there is no real doubt that N and "the set of natural numbers" means the same, whereas differing traditions about whether zero counts as a natural number or not. Defining N to mean the set of natural numbers without revealing whether this set includes 0 for you is just a waste of ink and/or taunting the reader).

Answer 2

We can use denote in the following forms:

1. (We) denote the set of natural numbers by/as N
2. (We) use N to denote the set of natural numbers
3. Let N denote the set of natural numbers
4. N denotes the set of natural numbers

I just searched a random mathematical textbook for denote and summarized its usage as above.

Answer 3

My personal preference is never to use the word denote in written mathematics. Rather I always write something like:

Let N be the set of natural numbers.

I want the reader to grapple with the actual object that I am writing about, whereas using "denote" creates an extra level of separation between them and the object. I'd like to say that this is part of writing in active voice, rather than passive voice, but I am not sure that I can substantiate this.