One of my mathematics teachers, a long time ago, once objected to statements of the type

Let

XandYbe two compact topological spaces. ThenX x Yis also compact

on the ground that the use of *two* implied that the statement did *not* apply to the case of *X x X*, whose compacity would need to be stated separately, as it was not, strictly speaking, an application of the given statement.

His favored solution was to drop the *two* (or, in French, to replace *Soient X et Y deux espaces…* by

*Soient X et Y*….), with the idea (I presume) that making a grammatical mistake (using a plural form like

**des**espaces*when, sometimes, there is only one object, if*

**des***X=Y*) would be less important than a mathematical one.

Strangely enough, I still sometimes remember this, and I have modified various sentences to try to go around it, although the whole thing seems quite absurd really… I wonder if others have heard this type of rules, and if there’s a mathematically and syntaxically correct way to phrase things without being absurdly formal?

I think that X and Y should be viewed as variables, not as constants. So there is no problem to say “two”, and then substitute the same “value” to the two variables.

I agree that X and Y are in some sense variables. Furthermore, in most constructions of this type the case X = Y is possible. If X and Y can’t be the same space, then this should be explicitly stated, for example by saying “let X and Y be distinct topological spaces”.

How about “Let both X and Y be a compact topological space”?

I don’t even know if that is correct standard English, but I feel comfortable using similar constructions in conversation. And I can’t think of a way to express the same thing in French.

Yes, it seems reasonable to interpret X and Y as variables, but then there must be some “formal language” hidden which dictates the rules of use of these variables, and this is rarely made explicit; it would probably become “absurdly formal”, to work correctly with the natural language which provides the framework for the statements (unless of course the whole statement is entirely expressed in a formal language…)

I wonder if one can concoct paradoxical statements but abusing slightly this interaction.

As a matter of fact, I must admit that I am not certain if the actual French plural (or English, or in any other natural language) is really a strict plural, and not a plural-allowing-singular construct.

Similarly, I think there exist languages with a special “dual”, expressing situations where there are two objects exactly; things must become a bit tortuous then…

I thought the following quote was appropriate. Gian-Carlo Rota writes in his book “Indiscrete Thoughts” (p. 19, in “Fine Hall in its Golden Age”) about Solomon Lefschetz:

He liked to repeat, as an example of mathematical pedantry, the story of one of E. H. Moore’s visits to Princeton, when Moore started a lecture by saying “Let a be a point and let b be a point.” “But why don’t you just say, ‘Let a and b be points!'” asked Lefschetz.” “Because a may equal b,” answered Moore. Lefschetz got up and left the lecture room.

I think historical linguists believe most languages have an ancestor which had a dual number, and that some words in living languages, like “both” in English, are relics of this past. From that point of view, if one objects that “Let X and Y be topological spaces” implies that X and Y are distinct, then I see no reason that “Let both X and Y be a compact topological space” would be any better.

Personally, to my ears it’s perfectly acceptable to regard the English plural as a plural-allowing-singular construct, but it may be that I’ve trained myself to hear it that way in order to parse statements like “let X and Y be topological spaces” and allow X=Y. I’m sure I’ve had students ask explicitly about whether X=Y is allowed by such a statement. I wonder what linguists would have to say about this issue.

Ah, but in the original statement, one could take Y to be canonically isomorphic to X (but not X), so it’s all ok.

#5: the quote of Lefschetz/Moore is very nice! For some reason, I was convinced this type of pedantry was a consequence of certain very formalistic features of the French education system for high-school teachers…

#6: I don’t know any linguist myself, but I would be very happy to hear what they think of this type of questions.

#7: Taking Y isomorphic to X is one way out… But what if they’re supposed to be in some set/category where the only canonical isomorphism is the identity?

Actually, this makes me think of the following semi-paradoxical type of statements:

(Which is a reasonable type of mathematical statement).

If the plural in the beginning of the sentence is supposed to mean “distinct”, then the conclusion is impossible!

I once used the following phrase (where I actually wanted to draw attention to the fact that both `variables’ could refer to the same object is):

Let v and w be vertices in G (possibly with v=w), (…)