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I was in Paris recently for a meeting about motives or motifs, and since I'm too jet lagged for real work let me ask the following somewhat frivolous question. The word "motif" is usually translated as "motive" in English. However, I wonder if this is really the best choice. "Motive" has, for me, a primarily psychological meaning, whereas "motif" -- which is a perfectly good English word -- means pattern or theme. I guess my question is which word better captures the intended meaning?

Incidentally, it appears that this usage of "motive" goes back to Grothendieck himself, cf."Standard conjectures on algebraic cycles". So perhaps, one should allow him to have the last word and not question his motives, which have wonderful if unintended consequences.

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I've quite often seen the English word "motif" (meaning A distinctive, significant, or dominant idea or theme...) written "motive", which is one reason I chose as the title of LNM 900 "Hodge cycles, Motives,..." (at the time both "motive" and "motif" were in use in English). I would prefer to say that the French word "motif" has been borrowed three times into English, and twice anglicized. – JS Milne May 15 '10 at 13:32
The meanings of motive and motif as you describe them aren't exactly disconnected; a psychological motive can become a motif in one's actions, or something to that effect. I'm of the opinion that it enriches one's language, even in mathematics, to keep all of the meanings of one's words in mind. – Qiaochu Yuan May 15 '10 at 17:46

Dear Donu, here are Grothendieck's own words:

"Contrary to what occurs in ordinary topology, one finds oneself confronting a disconcerting abundance of different cohomological theories. One has the distinct impression (but in a sense that remains vague) that each of these theories “amount to the same thing”, that they “give the same results”. In order to express this intuition, of the kinship of these different cohomological theories, I formulated the notion of “motive” associated to an algebraic variety. By this term, I want to suggest that it is the “common motive” (or “common reason”) behind this multitude of cohomological invariants attached to an algebraic variety, or indeed, behind all cohomological invariants that are a priori possible"

They can be found in his autobiographical "Récoltes et Semailles", where there is also an allusion to a musical meaning of "motif".

The translation is Barry Mazur's in his article "What is... a Motive?" which is, needless to say, a fascinating short survey (plus bibliography) .Here is the reference:

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Georges, thanks for digging up (the translation) of the quote. – Donu Arapura May 15 '10 at 13:28
You're welcome, Donu. I'm sorry we couldn't talk more in Nice! – Georges Elencwajg May 15 '10 at 13:36
@Georges: I never associated Grothendieck's use of motif with music, but now that you say it, it makes also good sense, as in Leitmotif. – ogerard May 15 '10 at 13:52

Motif in french has both the meaning of english "motive" and of "pattern". It is still actively used in decorative arts and art history "Cette tasse est ornée d'un très joli motif", "This cup is decorated with a very pretty pattern", and so on for tapestry, greek freeze, wallpaper, etc. And still used when describing a police case : "Il a un alibi et n'a aucun motif".

So I believe that Grothendieck was well aware of this ambiguity.

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I see. Thanks for clarifying that. – Donu Arapura May 15 '10 at 13:03
The common point of the two meanings is movement: motive is what moves someone to act, and a freeze motif is running around the edges of a plate, like footsteps of an animal and can also give an illusion of movement when following it visually. – ogerard May 15 '10 at 14:03

If you ask "which word better captures the intended meaning?", the answer is "Motif" - everyone who tells the contrary is just wrong. However, people use "motive" in english, so better stick to that.

If you have moral problems with that: Publish in french.

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Par ce terme [motif] j'entends suggérer qu'il s'agit du "motif commun" (ou de la raison commune) à cette multitude d'invariants cohomologiques différents associés à la variété. [Recoltes et semailles SS16] – Xandi Tuni May 15 '10 at 13:11
So then it appears that either way is correct. – Donu Arapura May 15 '10 at 13:24
@Xandi: I would like to see more non-native french speaker publish mathematical articles in french, but the arrogant style of your answer is certainly not going to help. – ogerard May 15 '10 at 13:56
It firmly remains my opinion that the term "motif", alluding to a pattern, a recurring element in a story or a theme would have been the better of the choices, whereas "motive" sounds forensic to me. The orthographical question, whether "motif" may be spelled "motive" as I take from Milne's comment, is a different one - if one can do so, just write "motive" and never explain what you mean by it. – Xandi Tuni May 15 '10 at 14:04

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