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

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/