In mathematics, there are not only theorems. There are, what we call, "philosophies" or "yogas," which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement. When I want to understand a problem, I first need to have a panorama of what is around it. A philosophy creates a panorama where you can put things in place and understand that if you do something here, you can make progress somewhere else. That is how things begin to fit together.In Mathematicians, Mariana Cook, PUP, 2009, p156.
motif: A distinctive, significant, or dominant idea or theme; Art. a distinctive feature, subject, or structural principle in a composition or design; in literature or folklore, a particular or recurrent event, situation, theme, character, etc.... . [This word is sometimes written "motive".](Shorter Oxford Dictionary)
motive: A factor or circumstance inducing a person to act in a certain way....
When I was preparing LNM900 for publication, both "motif" and "motive" were in use for Grothendieck's notion, and I had to choose between them. I chose the anglicized form, partly because I'd come across people (Bernstein, Donington) who use "motive" for "motif" in the musical sense. Since then, "motive" has predominated.
In his 1968 article (in Russian), Manin explains the word by quoting Herbert Read (an English poet and critic):
Cézanne's method of painting was first to choose his 'motif' --- a landscape, a person to be portrayed, a still-life; then to bring into being his visual apprehension of this motif; and in this process to lose nothing of the vital intensity that the motif possessed in its actual existence. (A Concise History of Modern Painting).In his review (BAMS 39 (2002), p138), Weibel paraphrases Read. As far as I know, Grothendieck was simply using a fairly common French word "motif", and Weibel's claim (ibid.) that he borrowed it from Cézanne is wrong.
More from Herbert Read:
Cézanne's immediate predecessors, the Impressionists, had seen the world subjectively---that is to say, as it presented itself to the senses, and for each occasion there must necessarily be a separate work of art. But Cézanne wished to exclude this shimmering and ambiguous surface of things and penetrate to the reality that did not change, that was present beneath the bright but deceptive picture presented by the kaleidoscope of the senses.
For a comparison of Grothendieck's and Cézanne's notions of "motif" by Xu Kejian, see here (Chinese).