Job Seeking

A kind person has contributed an entry about me to Wikipedia. There I am in the categories 'living person' and '21st century philosophers'. I must say it's rather humbling to be mentioned alongside such august scholars, especially since were another category 'academics who have never had a permanent contract' to be formed, my name would unfortunately be included there too.

I had hoped to be able to report today that this condition would be very temporary. However, yet another interview has been and gone, and the signs aren't looking promising. The draining part of the job seeking process is that, with posts so extraordinarily thin on the ground, while you know your chances are slim, you still need to think yourself into the post and imagine integrating it into your life story. This one would have involved a significant shift of focus to my non-mathematical interests. I would have had to rechristen the blog 'Philosophy of Real Mathematics and Mental Health Ethics'.

There are plenty of us out there with a considerable range of experience of being interviewed, not that I feel in my case it makes me perform any better. Perhaps this experience could be put to some use. To reverse the tables if only briefly, isn't it about time that the interviewers and their chosen methods be assessed? When I talk to people in other professions they are flabbergasted to hear that an employer intending to take on a person for possibly 30 years, limits the contact time with a candidate to a 20 minute talk to faculty and a 20 minute interview with half a dozen people. This is the minimal British package interview. There are more elaborate ones, and I can recommend the History and Philosophy of Science department in Cambridge for also requiring candidates to pass between pairs of faculty for brief conversations. In the category of 'panel which had best prepared to ask good questions about your work', the winner for me by a stretch is Notre Dame, Indiana.

A New Bogdanoff Affair?

In a paper on the ArXiv today, A Comment on "On Some Contradictory Computations in Multi-dimensional Mathematics", we read that a serious journal Nonlinear Analysis has allowed to be published an article full of nonsense. Apparently, Carvalho, L. A. V., On Some Contradictory Computations in Multi-Dimensional Mathematics, Nonlinear Analysis 63, 725-734 (2005), claims among other things that "multi-variable mathematics is inconsistent with arithmetic (1 = 0) and also auto-contradictory as calculus is part of this theory".

So, hoax or incompetence? Is this another Bogdanoff affair, also know as Reverse Sokal? The slightly odd thing is that the authors of the debunking give references for previous debunkings. But one of these, reference [2], has authors Carvalho, A. L. Trovon and Rodrigues, W. A., Jr.. I wonder how common the surname 'Carvalho' is.

Update: I think my suspicions were wrong, and that they are different Carvalhos. LAV Carvalho has 23 papers listed on MathSciNet. In the paper concerned, it has an ominous "There will be no review of this item." A. L. Trovon de Carvalho has clashed swords with Myron Evans.

Reply to dt

In his comment to this post of mine, dt asks some questions about how I see philosophy, to which I promised replies. A quick orientation to what philosophy is about can be found on Wikipedia. A very useful web resource is the Stanford Encyclopedia of Philosophy.

Now, philosophy as toolkit or tradition. It's worth making the point that the range of ways of writing philosophy is extraordinarily broad, so much so that some philosophers have difficulty seeing other philosophers' work as philosophy. This found one of its most dramatic appearances in the passionate objection of Cambridge University's philosophy department to the proposal of the English department that Jacques Derrida be awarded an honorary degree. Some varieties of philosophy have given the appearance of being the search for tools, usually logical ones, to resolve what are seen as thorny problems. E.g., if a theory appears to commit you to the existence of entities you don't believe to exist, then rewriting that theory in some formal framework may free you from this commitment. Hartry Field attempts this kind of thing when he rewrites physics without mathematics (assuming logic is not mathematics). Some would find this exercise completely pointless.

But despite these very large disagreements as to method, there is still a unity, or perhaps connectivity, provided by our relation to a tradition. In some sense, you can't be a philosopher without being able to position yourself iu relation to some of the questions considered by Plato and Aristotle.

dt also asks about why I take such notice of category theory. I've given a couple of reasons here, but there are plenty other reasons to like it. It breaks down what I see as an artificial distinction between the framework of a language and what the language talks about. And I'm very much enjoying its rise to prominence in contemporary physics. As to what I would have done 60 years ago, well do what all philosophers of X ought to do, namely, make apparent the 'constellation of absolute presuppositions' (to use a phrase of Collingwood) of X, and the tensions within it, understanding this constellation to be an historical entity.

Cognitive control in mathematics

Jukka Keranen's interesting PhD thesis is available here. By a careful comparison of an 1826 proof of Abel and a 1940 proof by Artin concerning the solvability of polynomials by radicals, taken as characteristic of their respective centuries, Keranen articulates what has been gained epistemically as we pass from the nineteenth to the twentieth century.

I was led to conclude that while many of the resources characteristic of 20th century mathematics do in fact allow us to understand some range of mathematical facts better than do the corresponding 19th century resources, this is typically not the fundamental difference between them. As I will argue in this essay, typically the fundamental difference is that the 20th century resources allow us to attain better cognitive control over our mathematical epistemic processes such as proving theorems and solving problems: they make it possible for us to attain a higher grade of a certain kind of rational mastery over what we do in mathematics, one I chose to call “cognitive control.” (p. 3)

I motivated my characterization of cognitive control by noting that there are three basic types of epistemic challenges we need to be able to negotiate in typical mathematical epistemic processes: identify a terrain of facts to be examined, find a theoretically productive way of representing that terrain, and examine the appropriate locations in that terrain so as to extract features thereof that are directly relevant to answering the question driving the process. The basic idea was that, depending on one’s epistemic resources, one may or may not be able to negotiate these challenges in a rationally orchestrated manner. (p. 224)

Pages 225 to 229 then give a neat synopsis of his findings.

As someone who adheres to the idea that the aim of mathematical practice is improved understanding, this thesis is very welcome. I think one should not take the first of these passages to be in conflict with my position, since I take it that there's more to understanding than the understanding of facts.

Recalling MacIntyre's comments about the master craftsman, good mathematicians don’t just know facts like people at a pub quiz, they know how things behave, they sense promising directions. They communicate a vision of how things might be. This is surely why mathematics exam questions go a certain way. State a result, prove it, then apply it in a novel situation. What is being tested is fledgling understanding. (How Mathematicians May Fail to be Fully Rational: p. 10)

I went on to describe MacIntyre's account of the Aristotelian-Thomist view of understanding as the adequacy of the mind to its objects (see p. 12). I would want to argue that Keranen has provided us with some conceptual resources with which to clarify this adequacy in the case of mathematics. I also recommend Colin McLarty's account of Grothedieck's philosophy of mathematics - The rising tide: Grothendieck on simplicity and generality - which will appear soon as an article. Grothendieck had cognitive control if anyone did. See also the quotation in the fifth comment to this post, and those in the September 30 post.

Setting out your stall

An example to all graduate students, Jeff Giansiracusa has on his website a research statement and a research proposal to search for a functorial relationship between Poisson manifolds and von Neumann algebras. I like this point from the latter, which allows for a certain hedging of bets if the desired functors don't appear:

Incomplete analogies have played, and will continue to play a prominent role in mathematics.

This is reminiscent of the observation Weil makes in the first column of page 6 of this letter to his sister that when an analogy is fully worked out, the 'majestic beauty' of the resulting theory 'can no longer excite us'. I wonder what's happening with the analogy between 3-manifolds and algebraic number fields. In the addendum to TWF218, Kevin Buzzard suggests that genuine analogies are to be distinguished from two instances of the same thing. But then maybe it's just a question of time.

Mathematical miracles

Kenny Easwaran has a post on the essences of mathematical concepts. As I commented there, the next step is to think about natural kinds and laws in mathematics. A further step would be to think about miracles. What could be meant by a mathematical miracle? Well, the term has already been used. Frank Morley's result that

The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle

was apparently deemed so surprising that it was called Morley's miracle. (See Connes' proof of it on p.6 of this.)

When this topic was raised during my talk at the IMA n-categories workshop. André Joyal suggested that the fact that the complex numbers are algebraically closed after merely adjoining the roots of one equation (x² + 1 = 0) to the reals is a miracle. However, someone else pointed out that there are more general results concerning algebraic closure of fields. Perhaps the physical parallel to this may be surprising consequences of physical laws, such as that helium balloons in planes move forward on take off. Indeed, it is very common for people to present seeming miracles, and then explain them away. Speaking about a state sum model for 3-manifold invariants, using quantum 6j symbols to label edges of a triangulation, John Baez points out that the symbols compose just right for the resulting partition function to be independent of the triangulation, saying "See week38 for an explanation of this seeming miracle: it's actually no miracle at all." Perhaps Joyal's intuition, however, is that it is still surprising that the reals just need an extension of degree 2 to achieve closure. This is not an instance contradicting a mathematical law, but then in our secular age we've also given up on controventions of physical laws, unless we mean to overthrow them.

A seemingly inexplicable surprise seems to be the modern, non-theological, sense of miracle. By any sensible measure, most of the truths expressible in a formal mathematical language are true for no good reason. But we wouldn't want to call them miracles, just as there are unmiraculous brute physical facts - 'a uranium atom within a mile of me decays within a picosecond of a solar neutrino passing through my big toe' or 'the distance between the sun and Pluto now is x metres'. To be surprising there must be theoretical considerations which tend to count against the occurrence of the result. Once again, mathematics and physics don't seem to be so very dissimilar.

Selling mathematicial concepts

On page 1 of Introduction to Coalgebra: Towards Mathematics of States and Observations, Bart Jacobs writes: This chapter is devoted to “selling” and “promoting” coalgebras. Do the scare quotes suggest a certain unease that commercial terms are needed to talk about this kind of activity? Employing humour, Miles Reid has a section entitled 'Commercial break' in his textbook Undergraduate Algebraic Geometry (CUP) where he tells us:

Complex curves (= compact Riemann surfaces) appear across a whole spectrum of math problems, from Diophantine arithmetic through complex functions theory and low dimensional topology to differential equations of math physics. So go out and buy a complex curve today. (p. 45)

As readers will have gathered, I'm all for mathematicians promoting their ideas, concepts, and theories. In chapter 9 of my book I discuss the range of ways in which mathematical concepts are viewed, from useless, through mildly and very useful, right up to essential. Promoting a concept is about trying to shift its rank upwards so that it receives its due.

Mathematicians and philosophy

This survey of mathematicians' philosophical reading habits was a little small, but interesting nonetheless. It's hardly surprising when someone in their early twenties who is wondering whether to become a mathematician is drawn to read philosophies addressing lived experience. Norbert Wiener's I am a mathematician and Paul Halmos's I want to be a mathematician indicate how much this choice gets taken up into one's identity. Alexandre Borovik on his home page writes "I am a mathematician and that is what makes me interesting."

Now, as I have been insisting recently, a philosophy of mathematics ought to recognise this dimension of commitment to a tradition of rational enquiry. As Alasdair MacIntyre puts it:

…critical rational enquiry is not itself the kind of activity that anyone can undertake on her or his own. For the same reasons that I cited, when I argued that we are able to become and to continue as practical reasoners only in and through our relationships to others, we are able to engage in critical enquiry about our beliefs, conceptions, and presuppositions only in and through relationships to others. Rational enquiry is essentially social and, like other types of social activity, it is directed towards its own specific goals, it depends for its success on the virtues of those who engage in it, and it requires relationships and evaluative commitments of a particular kind. (Dependent Rational Animals, Carus Publishing 1999: 156-7)

...to introduce the Thomistic conception of enquiry into contemporary debates about how intellectual history is to be written would, of course, be to put in question some of the underlying assumptions of those debates. For it has generally been taken for granted that those who are committed to understanding scientific and other enquiry in terms of truth-seeking, of modes of rational justification and of a realistic understanding of scientific theorizing must deny that enquiry is constituted as a moral and a social project, while those who insist upon the latter view of enquiry have tended to regard realistic and rationalist accounts of science as ideological illusions. But from an Aristotelian standpoint it is only in the context of a particular socially organized and morally informed way of conducting enquiry that the central concepts crucial to a view of enquiry as truth-seeking, engaged in rational justification and realistic in its self-understanding, can intelligibly be put to work. (p.193) ‘First Principles, Final Ends and Contemporary Philosophical Issues’ in Kevin Knight (ed.) MacIntyre Reader, pp. 171-201.

All the same, at some point philosophy must address what mathematics is about. This all too small survey reflects what I know from elsewhere that very few mathematicians read contemporary philosophical treatments of the nature of mathematical entities. The greatest appreciation seems to be directed toward their fellow mathematician Reuben Hersh's 'mathematics as the study of mental objects with reproducible properties'. If they were tempted to read about mathematical entities as structures, they'd sooner read Barry Mazur's
'When is one thing equal to some other thing?' than anything by Benacerraf, Resnik or Shapiro. It seems unlikely to me that we'll beat them at this game of clarifying their working language, although we might play the role of gadfly there.

Perhaps we might find richer pickings in answering a challenge posed by Michael Harris in "Why Mathematics?" You Might Ask that philsophers "...have a duty, it seems to me, to account for terms like “idea” and “intuition” — and “conceptual” for that matter — used by human mathematicians (at least) to express their value judgments." (p. 17). Take the term idea:

Nothing in the life of mathematics has more of the attributes of materiality than (lowercase) ideas. They have “features” (Gowers), they can be “tried out” (Singer), they can be “passed from hand to hand” (Corfield), they sometimes “originate in the real world” (Atiyah) or are promoted from the status of calculations by becoming “an integral part of the theory” (Godement). (p. 14)

Harris makes the very useful point that my own use of the term is liable to a certain slippage:

Corfield uses the same word to designate what I am calling “ideas” (“the ideas in Hopf’s 1942 paper”, p. 200) as well as “Ideas” (“the idea of groups”, p. 212) and something halfway between the two (the “idea” of decomposing representations into their irreducible components for a variety of purposes, p. 206). Elsewhere the word crops up in connection with what mathematicians often call “philosophy,” as in the “Langlands philosophy” (“Kronecker’s ideas” about divisibility, p. 202), and in many completely unrelated conections as well. Corfield proposes to resolve what he sees as an anomaly in Lakatos’ “methodology of scientific research programmes” as applied to mathematics by

a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas… (p. 181)

He sees “a kind of creative vagueness to the central idea” in each of the four examples he offers to represent this shift of perspective; but on my count the ideas he chooses include two “philosophies,” one “Idea,” and one which is neither of these. (p.16)

Point taken. I'll see what I can do.

The history of quaternions

If this paper of mine is to be believed, we should expect the best of the action in the philosophical study of mathematics to take place between what I call there genealogical and tradition-constituted accounts of mathematical rationality. An example of a paper I'd classify as genealogical is Andrew Pickering's 'Concepts and the Mangle of Practice: Constructing Quarternions' (in Mathematics, Science, and Postclassical Theory, Barbara Herrnstein Smith and Arkady Plotnitsky (eds.), 1997, pp. 40-82), which tells the story of Hamilton's work on quaternions. Genealogical histories of a practice tend to delight in bringing contingency to centre stage - things could have been so very different. What is very noticeable in such histories is that often the very early days of a practice are treated. This gives the genealogist the advantage of only needing to study a handful of people with all their idiosyncracies. The underlying thought is that if so much could have been so different while the course of a practice was being set, how different things could be decades later. And, if we can find a sharp change of direction away from the original pioneer's intentions late on our story, so much the better. Most of the original thinking guiding the practice will be revealed to be just a story. Any number of stories might have governed at that time, leading mathematics in very different directions.

So, in Pickering's paper, with the pace of research so slow, we can dwell on Hamilton's idiosyncratic metaphysical views, and we can tell the story of the quaternions as having "mutated over time into the vector analysis central to modern physics." (p. 45). Hamilton had failed to reach his original goals, only achieving "a local association of calculation with geometry rather than a global one. He had contructed a one-to-one correspondence between a particular algebraic system and a particular geometric system, not an all purpose link between algebra and geometry, considered as abstract, all-encompassing entities." (p. 59). The quaternions could not form the required calculus for reasoning about entities in three-dimensional space. Even after Hamilton had considered multiplication on just the imaginary part, where the product of two lines could be an ordinary number or another imaginary, "...the association of algebra with geometry remained local. No contemporary physical theories, for example, spoke of entities in three-dimensional space obeying Hamilton's rules. (p. 60). "It was only in the 1880s, after Hamilton's death, that Josiah Willard Gibbs and Oliver Heaviside laid out the fundamentals of vector analysis, dismembering the quaternion system into more useful parts in the process. This key moment in the delocalization of quaternions was also the moment of their dintegration." (p. 60). From this an innocent reader might take it that, by and large, that was that as far as the quaternions were concerned.

So what do we make, then, of this paper, which documents the use of the quaternions and allied algebras in physics to recent times in the form of an analytic bibliography of 1300 references? With so many man-hours devoted to the extraction of whatever can be found to be useful aboout quaternions, and their relationships with other mathematical entities, do the first few decades of their lives tell us very much? Although it makes for engaging history, do we learn so much about the ways in which mathematics operates at its highest level of organisation from the quixotic quests of individuals, rather than from an account of droves of workers, most of whom must necessarily remain largely faceless?

Responding to a journalist's account of the 'Mathematics and Narrative' conference, in his letter to the Independent (British national daily) - Mathematicians struggle for truth - Ronnie Brown stresses that mathematics is not the work of a handful of individuals, but rather "a world-wide collaborative effort involving tens of thousands". But, to raise again the kind of question of an earlier post, how can a history of the application of the quaternions, the work of hundreds of people, be written?

Mathematicians reading philosophy

In the comments to this post, John Baez talks about reading Wittgenstein and then later Heidegger. I'd be very interested to know which philosophers other mathematicians/mathematical physicists are glad they have read, even if not recently.

Joyal and 'vulgarisation'

John Baez mentioned in his comment to my last blog post the expectations the higher-dimensional algebra community have for the awaited book on quasi-categories by André Joyal. These omega-categories with weakly invertible j-morphisms for j>1 are already being used in very important work of Jacob Lurie on elliptic cohomology.

I recently came across an interview with Joyal - Entretien avec André JOYAL - in which he says things with which I am very much in agreement. I won't bother to translate:

L'accès aux mathématiques, au sens perdu
La difficulté des mathématiques est en grande partie une difficulté d'accès. C'est pas une difficulté intrinsèque parce que c'est relativement simple, on s'en rend compte quand on a compris (rires). Mais justement ça, on pourrait comprendre même si ça demeurait complexe. Comprendre c'est pas nécessairement réduire à des éléments simples.

Très souvent, la difficulté se trouve dans une sorte de déchiffrage: il s'agit de comprendre ce qu'il y a au-delà d'une certaine écriture qui est purement algébrique alors qu'en fait le contenu est géométrique. Et le contenu géométrique est totalement absent lors du développement algébrique alors que dans la tête de l'auteur il était présent, ou encore il y a des développements heuristiques qui ne sont pas donnés, les méthodes heuristiques sont très très importantes ...
Il y a des choses bizarres comme ça, il y a des résultats mathématiques qui ont un sens extrêmement simple et le sens est comme perdu.

- N: Ce qui vous intéresse c'est de retrouver cet aspect perdu, ce qui est caché derrière.

- J: Oui, c'est-à-dire que c'est un aspect qui est perdu un peu à cause de la culture actuelle ... C'est quand même une culture qui dure depuis assez longtemps ... je ne sais pas comment, mais ça pourrait changer, ça pourrait être autrement ... Les connaissances sont accessibles surtout aux spécialistes, il n'y a pas d'effort de synthèse, il n'y a pas d'effort de véritable vulgarisation. Il y a quelques efforts mais ils ne sont pas suffisants ...

So, 'could do better' is the verdict.

Maybe it's because I haven't worked so hard at it, but I do find that *vulgarisations* in noncommutative geometry tend to start smoothly enough, but then race up through the gears too rapidly. Now it appears that this is not a necessary feature of the subject matter. Pierre Martinetti has put on the ArXiv a very nice piece about distances in noncommutative spaces, where he bothers to work out some simple examples.

Mathematical retrodiction and prediction

In this discussion over on Urs Schreiber's blog, The String Coffee Table, Greg Kuperberg writes:

Consider, for example, how category theory might guide you to define quantum groups. As a first try, you might consider group objects in the contravariant category of non-commutative algebras. This is a natural proposal, but it isn’t the most fruitful one. A much better idea is to use Hopf algebras. Hopf algebras are a non-cocommutative generalization of group algebras, but they are not group objects. They have their own category-theoretic motivation (in terms of monoidal categories), but it takes hindsight to see the way in which category theory is important here.

John Baez in some recent lectures shows how (involutory) Hopf algebras can be generated from groups (see p. 53), providing a category theoretic account of their nature. Kuperberg's point, however, is that this is reconstruction rather than construction. He also admits that category theory can play a suggestive role, providing targets for future research.

I was reminded during The String Coffee Table discussion of a debate which raged for a time in the philosophy of science as to the credit a theory deserves for retrospectively accounting for observations - retrodiction - as compared to the prediction of unseen phenomena. The consensus seems to admit that, to some extent, you should think your theory more likely to be accurate in the future if it has retrodicted an observation, so long as that observation wasn't used in the construction of the theory, by, say, fixing values of parameters. But there were sharp disagreements on what one could say about permissible uses of the observation. One of these days, I shall write up what statistical learning theory has to say about this topic. The key idea is that if you start with a class of hypotheses that isn't too rich, then relying only on training data you can say how probably accurate a selected hypothesis is. And richness isn't measured by the number of parameters.

All things considered, though, there's nothing like a bold new prediction. Now, higher-category theory performs something like this task too. Say you are interested in a certain construction, such as Hopf algebras. You then formulate the construction in a way that suggests features of a 'categorified' version, i.e., sets become categories, categories become bicategories, functions become functors, functors become natural transformations, product operations on elements become monoidal products on objects, and so on. Or perhaps you look to internalise constructions within a category, by, say, representing a group as a diagram, and realising that diagram within a category. You might end up then with a good idea of what your categorified entity, say, a Hopf 2-algebra, should look like. You predict that there should be an important concept which possesses specified properties. This is higher category theory in its guiding, predictive role, pushing you in promising directions. But there is no magic bullet. There's still very likely a huge amount of work to be done in getting the construction right, and in finding rich examples of it. Sometimes these examples already exist, but recognising them as a cases of concepts which fit into some hierarchical pattern can be extremely important.

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.