## 2 Aug 2009

### Manifolds of Time, Dimensions of Duration: Bergson’s and Husserl’s Different Riemannian Multiplicities

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Manifolds of Time, Dimensions of Duration:

Bergson’s and Husserl’s Different Riemannian Multiplicities

Deleuze tells us in Bergsonism that both Husserl and Bergson developed their notions of multiplicities from Bernhard Riemann’s distinction between continuous and discrete multi-dimensional manifolds.

On Riemann’s theory of multiplicities cf. G.B.R Riemann, Œuvres Mathématiques (French Translation edited by Gauthier-Villers, “Sur les hypothèses qui servent de fondement a la géométrie.”); and H. Weyl, Temps, Espace, Matière. Husserl too gained inspiration from Riemann’s theory of multiplicities, although in quite a different way from Bergson. (122)

Sur la théorie riemanienne des multiplicités, cf. B. RIEMANN, Œuvres Mathématiques (tr. fr. Gauthier-Villers éd., « Sur les hypothèses qui servent de fondement a la géométrie.»”). – et H. Weyl, Temps, Espace, Matière. Husserl aussi, bien qu’en un tout autre sens que Bergson, s’inspire de la théorie riemanienne des multiplicités. (32)

We will try to determine that difference, following Deleuze’s observation. So first we will look at Riemann’s distinction between continuous and discrete multiplicities, as described by both himself and Hermann Weyl.

We begin first with some background from Weyl’s Space, Time, Matter. Here he explains what Riemann means by a dimensional manifold. Space, for example, is a three-dimensional manifold. Length, breadth, and depth all intersect and correlate with each other. Each varying coordinated axis ‘pushes-and-pulls’ values from another axis into its alternate dimension. We might think of a square. Then we construct from it a cube. In a sense, we stretched the square through another axis of variation. This takes it from being a two-dimensional manifold to a three-dimensional one. Or consider when we hear a melody. The tones at different points will vary in pitch. But at the same time, their volume will vary as well. So while the tone is being stretched along an axis of pitch-variation, it is also being pulled through another coordinated axis of amplitude-change. Hence the melody considered in terms of these two axes of variation is a two-dimensional manifold. What I would like to note is that each axis has its own push-and-pull tendencies of quantitative variation. When the square is stretched into a cube, in a sense, we are distorting the square, disfiguring it in a way. Hence manifolds involve these intensive forces pushing-and-pulling things – and as well processes – into various qualitatively-different dimensions that themselves are varying quantitatively. The resulting “shape” or (deformed) form will have certain qualitative properties. The cube will be rectilinear. The sphere is round. They thus have qualitative differences manifesting in the way the thing or process extends along different coordinated axes of variation. But what creates these qualitative differences are the push-and-pull tendencies of intensive quantitative tensions that distort the figure through various varying alternate coordinated dimensions. Hence there is an implicit intensive quantitative basis for explicit extensive qualitative differences. The intensive variations are folded-into the extensive explications. So the extensities envelop (involve) the intensities. But the extensities are the products of the intensive dynamics. Hence intensities fold-out into extensities. So intensities develop (evolve) the extensities.

Weyl writes of Riemann’s manifolds:

The most fundamental property of space is that its points form a three-dimensional manifold. What does this convey to us? We say, for example, that ellipses form a two-dimensional manifold (as regards their size and form, i.e. considering congruent ellipses similar, non-congruent ellipses as dissimilar), because each separate ellipse may be distinguished in the manifold by two given numbers, the lengths of the semi-major and semi-minor axis. The difference in the conditions of equilibrium of an ideal gas which is given by two independent variables, such as pressure and temperature, form a two-dimensional manifold, likewise the points on a sphere, or the system of pure tones (in terms of intensity and pitch). According to the physiological theory which states that the sensation of colour is determined by the combination of three chemical processes taking place on the retina (the black-white, red-green, and the yellow-blue process, each of which can take place in a definite direction with a definite intensity), colours form a three-dimensional manifold with respect to quality and intensity, but colour qualities form only a two-dimensional manifold. This is confirmed by Maxwell’s familiar construction of the colour triangle. The possible positions of a rigid body form a six-dimensional manifold, the possible positions of a mechanical system having n degrees of freedom constitute, in general, an n-dimensional manifold. The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colours, tones) may be specified by the giving of n quantities, the “co-ordinates,” of which are continuous functions within the manifold. (Weyl, Space, Time, Matter, 84b.d)

We will look now at what Riemann himself wrote. I base the following summary on the French translation. At the end of this posting, I will give the text for the French translation, as well as page images from both the German and French versions. [The text is: “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” in Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass. The French translation: Sur les hypothèses qui servent de fondement a la géométrie” in Œuvres Mathématiques de Riemann.]

According to Riemann, we may determine magnitudes in two different and incompatible ways: either as continuous or as discrete values. And so we consider manifolds either as discrete manifolds (discrete Mannigfaltigkeit / variété discrète) or as continuous manifolds (stetige Mannigfaltigkeit / variété continue). A discrete determination is a point, and a continuous determination is a range. We customarily consider values as discrete quantities. But everyday life does not often present us with values that we consider continuous. Yet when we do experience continuous magnitudes, it is usually in objects of sensation, for example, with colors.

We might consider an isolated discrete unit that is distinct from every other one in the manifold. Or we might consider a limit along a continuum. In both cases, we call the manifold’s part a ‘quantum.’ We may also then want to compare two amounts of quanta. To do so in discrete manifolds, we count the quanta. So the discrete manifold has already within it intrinsically the parts which measure or quantify its magnitude. Hence the principle of measurement of discrete manifolds is found within the manifold itself.

But in continuous manifolds, we measure the quantitative values rather than count them. Quantifying continuous manifolds requires that we maintain a standard of comparison, like a ruler, that remains the same as we carry it between the continuous manifolds that we compare. If we do not have a standard such as a ruler, then we can overlay the two continuous magnitudes to see which one fits within the other. But as we see, either way, quantifying continuous manifolds requires an extrinsic comparison. And thus unlike with discrete manifolds, the principle of measurement of continuous manifolds is not found within the manifold itself. [See this entry on Maurice Trask’s similar explanation of continuous and discrete multitudes. He equates them with analog and digital forms of representation and computation.] Yet, when we place one continuous manifold within another, this can only tell us which one is greater or lesser, but it does not determine the numerical difference between them.

If we were to divide a continuous manifold into standardized parts, they would not be units as much as regions of variation. Dealing with quantities as continuous values is useful in certain differential calculus operations, but because it is imprecise, it also can present difficulties. (French 282-283, German 273-274)

Weyl cites the key passage in Philosophy of Mathematics and Natural Science:

“that for a discrete manifold the principle of measurement already contained in the concept of this manifold, but that for a continuous one it must come from elsewhere.” (Weyl, Philosophy of Mathematics and Natural Science, 43)

He further writes:

Riemann contrasts discrete manifolds, i.e. those composed of single isolated elements, with continuous manifolds. The measure of every part of such a discrete manifold is determined by the number of elements belonging to it. Hence, as Riemann expresses it, a discrete manifold has the principle of its metrical relations in itself, a priori, as a consequence of the concept of number. In Riemann’s own words:

“The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metrical relations of space. In this question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete manifold, the principle or character of its metric relations is already given in the notion of the manifold, whereas in a continuous manifold this ground has to be found elsewhere, i.e. has to come from outside.” (Weyl, Space, Time, Matter, 97)

So in Deleuze’s Bergsonism, he explains that multiplicity for Bergson is a matter of distinguishing continuous from discrete multiplicities, rather then differentiating the Multiple from the One. Hence Bergson builds his distinction from Riemann’s. According to Deleuze, Riemann defines multiplicities as “things that could be determined in terms of their dimensions or their independent variables.” (Deleuze, 39) And he reiterates the distinction: discrete multiplicities “contain the principle of their own metrics (the measure of one of their parts being given by the number of elements they contain); and continuous multiplicities find their “metrical principle in something else, even if only in phenomena unfolding in them or in the forces acting in them.” (39b)

But Bergson breaks with Riemann by defining continuous multiplicities in a different way. This is because Bergson considers duration to be a continuous multiplicity. I suggest we take-up Riemann’s terminology, and call it a continuous manifold. We never experience instants of duration. However, we might ideally divide up duration. Doing so will never produce discrete units. It will always yield more continuous ranges of qualitative variation. But if we could divide duration down to an instant, we would find that each instant is qualitatively different than the rest. So in a sense, duration is qualitatively discontinuous. It varies at every point.

Husserl’s model is very similar. He also writes of a non-extending now limit. And phenomena are continually altering. But also for Husserl there is unity to the manifold, and he writes that discontinuity is not possible each instant. [For more on Husserl and time synthesis, see this entry and this one.] This would suggest that while time might have an inextensive now point, the phenomenal flow passing through it is always extensive and qualitatively continuous. So the Husserlian manifold involves modification, but nonetheless there is a qualitative continuity of change as well as a durational continuity of temporal flow. Likewise, the Bergsonist manifold has a continuous and extending temporal flow, but it is a flow of qualitative discontinuity. The instants do not contract because they associatively assimilate. They contract because that is how we perceive things: our whole past is continually contracted with the present, and each moment of that contraction forces new qualitative differences into the whole.

So the Bergsonist durational manifold might be more like Deleuze’s characterization of Spinoza’s continuous variations of affection [in fact, Deleuze specifically says that Bergson’s duration coincides with Spinoza’s. Also see Ch. 12 and especially 13 of Deleuze’s Expressionism. And as well this entry]. No instant has duration. But when we go from one instant to the next, there is a qualitative change. That change can be more-or-less. But what is important is the speed changes. The more unpredictable, the more intense. Because they are unpredictable, one does not imply the next. So in that sense they are discontinuous and inextensive. And also, because at each instant there is a different instantaneous tendency toward change, they also are in this way continuously qualitative discontinuities. Bergson’s duration, then, might be the experience of this continual-fluctuation of changing tendencies.

Riemann text of the French translation :

Les concepts de grandeur ne sont possibles que là où il existe un concept général qui permette différents modes de détermination. Suivant qu'il est, ou non, possible de passer de l'un de ces modes de détermination à un autre, d'une manière continue, ils forment une variété [ft.1] continue ou une variété discrète; chacun en particulier de ces modes de détermination s'appelle, dans le premier cas, un point, dans le second un élément de cette variété. Les concepts dont les modes de détermination forment une variété discrète sont si fréquents que, étant donnés des objets quelconques, il se trouve toujours, du moins dans les langues cultivées, un concept qui les comprend (et les mathématiciens étaient par conséquent en droit, dans la théorie des grandeurs, discrètes, de prendre pour point de départ la condition que les objets donnés soient considérés comme de même espèce). Au contraire, les occasions qui peuvent faire naître les concepts dont les modes de détermination forment une variété continue sont si rares dans la vie ordinaire, que les lieux des objets sensibles et les couleurs sont à peu près les seuls concepts simples dont les modes de détermination forment une variété de plusieurs dimensions. C'est seulement dans les hautes Mathématiques que les occasions pour la formation et le développement de ces concepts deviennent plus fréquentes. [ft.1 : Varietas, Mannigfaltigkeit. Voir Gauss, Theoria res. biquadr., t. II, et Anzeige zu derselben (Werke, t. II, p. 110, 116 et 118). — (J. Houel.)]

Une partie d'une variété, séparée du reste par une marque ou par une limite, s'appelle un quantum. La comparaison des quanta au point de vue de la quantité, s'effectue, pour les grandeurs discrètes, au moyen du dénombrement; pour les grandeurs continues, au moyen de la mesure, La mesure consiste dans une superposition de grandeurs à comparer; il faut donc, pour mesurer, avoir un moyen de transporter la grandeur qui sert d'étalon de mesure pour les autres. [282-283] Si ce moyen manque, on ne pourra alors comparer entre elles deux grandeurs, que si l'une d'elles est une partie de l'autre, et encore, dans ce cas, ne pourra-t-on décider que la question du plus grand ou du plus petit, et non celle du rapport numérique. Les recherches auxquelles un tel cas peut donner lieu forment une branche générale de la théorie des grandeurs, indépendante des déterminations métriques, et dans laquelle elles ne sont pas considérées comme existant indépendamment de la position, ni comme exprimables au moyen d'une unité, mais comme des régions dans une variété. De telles recherches sont devenues nécessaires dans plusieurs parties des Mathématiques, notamment pour l'étude des fonctions analytiques à plusieurs valeurs, et c'est surtout à cause de leur imperfection que le célèbre théorème d'Abel, ainsi que les travaux de Lagrange, de Pfaff, de Jacobi sur la théorie générale des équations différentielles, sont restés si longtemps stériles. Dans cette branche générale de la théorie des grandeurs étendues, où l'on ne suppose rien de plus que ce qui est déjà renfermé dans le concept de ces grandeurs, il nous suffira, pour notre objet actuel, de porter notre étude sur deux points, relatifs : le premier, à la génération du concept d'une variété de plusieurs dimensions; le second, au moyen de ramener les déterminations de lieu dans une variété donnée à des déterminations de quantité, et c'est ce dernier point qui doit faire clairement ressortir le caractère essentiel d'une étude de n dimensions. (283)

Images of the French translation [click to enlarge]:

Images of the Original German [click to enlarge]:

Deleuze, Gilles. Bergsonism. Transl. Hugh Tomlinson and Barbara Habberjam. New York: Zone Books, 1991.

Deleuze, Gilles. Le bergsonisme. Paris : Presses Universitaires de France, 1966.

Riemann, Bernhard. Sur les hypothèses qui servent de fondement a la géométrie.in Œuvres Mathématiques de Riemann. Transl. L. Laugel. Paris: Gauthier-Villars et Fills, Imprimeurs-Libraires, 1898. Available online at: http://www.archive.org/details/uvresmathmatiqu00kleigoog

The original German can be found in

Riemann, Bernhard. “Ueber die Hypothesen, welche der Geometrie or its Grunde liegen.” in Bernhard Riemann’s Gesammelte Mathematische Werke united Wissenschaftlicher Nachlass. Leipzig: Druck und Verlag B. G. Teubner, 1892. Available online at:

Weyl, Hermann. Philosophy of Mathematics and Natural Science. Transl. Olaf Helmer. Princeton: Princeton University Press, 1949.

Weyl. Hermann. Space, Time, Matter. Transl. Henry L. Brose. New York: Dover Publications, 1952.