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Friday, March 03, 2006

Hope for a common ground, and Motifs

The 'hope for a common ground' was Polya's explanation for our faith in analogical reasoning. If we believe proposition A to be similar to proposition B, and we find B to be true, then our confidence in A will increase. For Polya this is due to the existence of a common H bringing both A and B about, allowing increasing confidence in B to feed up to H and back down to A. In Mathematics and Plausible Reasoning, the case of analogy he considered at greatest length was Euler's factorisation of sin x/x to yield a sum of the inverse of the squares of natural numbers. This relies on a likening of sin x/x to a complex polynomial. Polya didn't mention the common ground, which we now call the Weierstrass Factorization Theorem.

In an interesting article What is the motivation behind the Theory of Motives? Barry Mazur quotes from Grothendieck's Récoltes et Semailles:
Contrary to what occurs in ordinary topology, one finds oneself [in algebraic geometry] 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.
Something close to Polya's common ground is operating here, though not so much at the level of a proposition, but more that of a theme. In fact, Grothendieck seems to have had music in mind when he devised the term motif, which in French covers both our motif and our motive. He continues:

Ces différentes théories cohomologiques seraient comme autant de développements thématiques différents, chacun dans le "tempo", dans la "clef" et dans le "mode" ("majeur" ou "mineur") qui lui est propre, d'un même "motif de base" (appelé "théorie cohomologique motivique"), lequel serait en même temps la plus fondamentale, ou la plus "fine", de toutes ces "incarnations" thématiques différentes (c'est-à-dire, de toutes ces théories cohomologiques possibles). Ainsi, le motif associé à une variété algébrique constituerait l'invariant cohomologique "ultime", "par excellence", dont tous les autres (associés aux différentes théories cohomologiques possibles) se déduiraient, comme autant d' "incarnations" musicales, ou de "réalisations" différentes. Toutes les propriétés essentielles de "la cohomologie" de la variété se "liraient" (ou s' "entendraient") déjà sur le motif correspondant, de sorte que les propriétés et structures familières sur les invariants cohomologiques particularisés (-adiques ou cristallins, par exemple), seraient simplement le fidèle reflet des propriétés et structures internes au motif.

Musical terminology also appears in the title of Part 0 of Récoltes et Semailles - Prélude en Quatre Mouvements. Arguably, then, one should really speak of a Theory of Motifs.


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