Was there a person made the quote about poetry that Poincaré responded to?

Question

It is claimed that Henri Poincaré, a well-known mathematician, said this:

Mathematics is the art of giving the same name to different things.

It is also claimed that that quote was in response to this quote:

Poetry is the art of giving different names to the same thing

^{Source: Quotations by Henri Poincaré}

According to this comment on Math Overflow,

This idea is in his "Science and Method" with an extensive discussion, and he does not mention anything about this being a response to something about poetry

So was there actually a person said the quotation about poetry, Poincaré heard this and responded with his own quote, or was it his quote at the beginning, and the poetry one was just added later?

<sup>FYI:
- Literature: Is poetry the art of giving different names to the same thing?
- Mathematics: How correct is the quote that "mathematics is the art of giving the same name to different things"?</sup>

Answer 1

Poincaré did say the one quote:

We have just seen, through an example, the importance of words in mathematics, but I could cite many more cases. It is scarcely credible, as Mach said, how much a well-chosen word can economize thought. I do not know whether or not I have said somewhere that mathematics is the art of giving the same name to different things. We must so understand it. It is meet that things different in substance but like in form should be run in the same mold, so to speak. When our language is well chosen it is astonishing to see how all the demonstrations made upon some known fact immediately become applicable to many new facts. Nothing has to be changed, not even the words, since the names are the same in the new cases.
_{The Future of Mathematics, 1908}

(Here is the original French version.)

There is no evidence that it was a response to the poetry quote. In fact, the evidence points to nearly the opposite being the case:

Of the many pithy sayings of Henri Poincare perhaps none is better than this:
"Mathematics is the art of giving the same name to different things."
I once quoted that mot to a poet, and got the quick response:
"Poetry is the art of giving different names to the same thing."
_{Mathematics as a culture clue: and other essays, 1947}

This is the earliest result for the poetry quote in Google Books, so it seems like the source. Note that Poincaré is pretty clearly not the "I" in the quote.

Answer 2

The original citation by Poincaré is from Science et Méthode (translated in English as Science and Method), chapter II “L'avenir des Mathématiques”.

Nous venons de voir, par un exemple, quelle est l’importance des mots en Mathématiques, mais j’en pourrais citer beaucoup d’autres. On ne saurait croire combien un mot bien choisi peut économiser de pensée, comme disait Mach. Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes.

My translation:

We have just seen, through an example, how important words are in Mathematics, but I could cite many others. One can hardly believe how a well-chosen word can provide economy of thought, as Mach said. I do not recall whether I have already mentioned somewhere that Mathematics is the art of giving the same name to different things.

This confirms the citation by Poincaré. It is also strong evidence that this was not in response to another quote. Poincaré presents it as an original thought and formulation. In the previous sentence, he uses a concept formulated by another person, and he attributes it (Mach is Ernst Mach, mentioned earlier in the chapter). It would be uncharacteristic of him not to similarly attribute a source for the next sentence.

This, of course, does not guarantee that he didn't withhold a source of inspiration, either deliberately or through forgetfulness. Another piece of evidence is the lack of publications in French with the quote about poetry, or for that matter the tranposed quote in general (“X is the art of giving different names”). There are no hits on Google Books (or for that matter on Google Web) for “est l'art de donner des noms …” or “est l'art de nommer …”, which are the two plausible ways to start the sentence in French. While Poincaré may have read or heard it in a different language, given how much historiography there has been about him, it is implausible that there would be no translation of the original citation if one had ever been identified.