Motives versus Motifs - MathOverflow

Motives versus Motifs

2010-05-15T11:59:20Z

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.

Answer by Xandi Tuni for Motives versus Motifs

2010-05-15T12:09:03Z

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.

Answer by ogerard for Motives versus Motifs

2010-05-15T12:56:34Z

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.

Answer by Georges Elencwajg for Motives versus Motifs

2010-05-15T13:17:38Z

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:

http://www.ams.org/notices/200410/what-is.pdf