Pedantic style

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

Let X and Y be two compact topological spaces. Then X x Y is 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 des espaces….), with the idea (I presume) that making a grammatical mistake (using a plural form like des when, sometimes, there is only one object, if 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?

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I am a professor of mathematics at ETH Zürich since 2008.

9 thoughts on “Pedantic style”

  1. 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.

  2. 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”.

  3. 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.

  4. 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…

  5. 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.

  6. 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.

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

  8. #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:

    Let X and Y be (whatever_s_) such that (whatever). Then X=Y.

    (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!

  9. 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), (…)

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