The pair of words "denote by" is ubiquitous in mathematical writing. For instance, one might see:
Let N be the set of natural numbers, 1, 2, 3, etc. Denote by S the subset of N consisting of numbers that leave the remainder 1 when divided by 4.
This phrase "denote by" has long struck me as awkward, perhaps even grammatically incorrect. I haven't yet found a dictionary that includes in the definition of "denote" an example or description that includes the preposition "by" in it. I have two questions:
1. Is it, indeed, grammatically incorrect? Or is it so accepted in mathematical culture that it has become correct for writing in that field?
2. I never see anyone give the following version of the second sentence above: "use S to denote the subset of N…". This seems to me an appropriate usage, but would it be wrong to go against convention by adopting this usage when no one else seems to? Say that I am writing in some formal setting, like writing a research article for a journal.
Generalizing question 2, there are many adjectives in mathematics whose root is the discoverer of the type of object to which the adjective-noun pair refers — for instance, Galois extension, Leibnitz notation, abelian group, and archimedean place. These last two seem always to be written in lower case. I have heard it said that "abelian" became lower case because the concept of abelian group became so important that the A was switched to lower case out of deference to Abel. I don't buy it. Is it wrong for me to go against convention by capitalizing "abelian" and "archimedean" in my own writing?
Oh, this is fun! Welcome, by the way.
1) In my opinion "denote by" is grammatically correct, if unusual. This is true of pretty much any transitive verb; you can say "gunfire killed the victim" or "you killed the victim by means of gunfire" or "you killed the victim by gunfire". You can let S denote the subset of N, or denote the subset of N by means of S, or denote the subset of N by S. Moving the "by" phrase after the verb doesn't change the meaning, it's just what makes it sound awkward to your ear...and to mine, unless the object of the verb is long and involved. Killing the victim by gunfire is pretty straightforward. But denoting the subset of N consisting of numbers that leave the remainder 1 when divided by 4 by S is confusing; better to leave "by S" where your textbook had it.
2) I'm the merest dabbler in math, but in physics when a term loses its capitalization it's usually from lots and lots of use, and eventually they adopt the lower-case version by agreement. They used to say water freezes at 273.15 degrees Kelvin, for instance; but they changed it and now water freezes at 273.15 kelvins. In that case it would be wrong to capitalize it, because the convention was adopted officially and I would regard that as authoritative. But where a commonly used proper noun becomes so routine that people begin to spell it in lower case—"machiavellian", for instance, "wagnerian", "boolean" or "euclidean"—I think you should feel free to capitalize the word or not as the mood and context strike you.
Here's one that should work too:
Let S denote the subset…
The merit of this one is 'let' is one of the most used words in math texts.
I hate to disagree with Bob Bridges -- he almost always gives good advice, and the new one, "bear", seems on point --, but "denote by" is just wrong.
Denote is a transitive verb. It always has an object noun post. "Denote by" is nonsensical.
It is transitive verb alright. It works like this :
Denote by Indirect_Object Direct_Object
Denote Direct_Object by Indirect_Object
The result of either of the above is : Indirect_Object now denotes Direct_Object