12 Dec 2008

Deleuze Cours Vincennes 17/02/1981, summarized

Deleuze will discuss the 17th century's infinitist conception of the individual, and he initially makes three points about the individual:
1) it is relation
2) it is power
3) it is a mode, namely, an intrinsic mode

1) The individual as relation refers to the plane of composition, because individuals are composed of relations.

2) The individual is power, and its intrinsic mode or degree is gradus. Hence the individual is not substance.

So as relation, the individual is not substance, because substance is a term, it is substantial, not relational. As degree of power, the individual is not substance, because the degree is a grade of a quality, and although substance is determined by qualities, it is not grades of them.

The individual as relation:

Deleuze considers the individual as a relation in its pure state, that is, independently of its terms. In Nicolas de Cuses relation is a measure that plunges into the infinite. Cusa dealt with weighing, and thought that "the relative measure of two weights refers to an absolute measure" which "brings the infinite into play."

This is the theme that there is an immanence of pure relation and the infinite.

C’est le thème qu’il y a une immanence du rapport pur et de l’infini.

The reason it is so difficult to think of relation independent of its terms, is because relation is the countermeasuring that itself countermeasures to an absolute and infinite measure; it is difficult to conceive because it brings together the mutual immanence of relation and the infinite.

The intellect is thought to pose relations, and this intellectual activity implies an infinite. (Précisément, dans l’activité intellectuelle, il y a une espèce d’infini qui est impliqué). Philosophers through the Renaissance to the 17th century sought a "first statute of relation independent of its terms," even by mathematical means.

In fact, what provides this statute is infinitesimal calculus, which poses the differential relation dy/dx. dy is an infinitesimally small quantity, called a vanishing quantity (quantité évanouissante): it is smaller than any given or givable quantity.

Thus I can say that dy as a vanishing quantity is strictly equal to zero in relation to y. In the same way dx is strictly equal to zero in relation to x. dx is the vanishing quantity of x. Thus I can write, and mathematicians do write dy/dx = 0/0. This is the differential relation. If I call y a quantity of the abscissa and x a quantity of the ordinate, I would say that dy=0 in relation to the abscissa, dx=0 in relation to the ordinate. Is dy/dx equal to zero? Obviously not. dy is nothing in relation to y, dx is nothing in relation to x, but dy over dx does not cancel out. The relation subsists and the differential relation will present itself as the subsistence of the relation when the terms vanish. They have found the mathematical convention that allows them to treat relations independently of their terms. (see Hegel's description of this phenomenon)

Donc je peux dire que dy, en tant que quantité évanouissante, est strictement égal à zéro par rapport à y. De la même manière, dx est strictement égal à zéro par rapport à x. dx est la quantité évanouissante de x. Donc, je peux écrire, et les mathématiciens écrivent, dy = 0. Évidemment non. dy n’est rien par rapport à y, dx n’est rien par rapport à x, mais dy sur dx ne s’annule pas. Le rapport subsiste et le rapport différentiel se présentera comme la subsistance du rapport quand les termes s’évanouissent. Ils ont trouvé la convention mathématique qui leur permet de traiter des rapports indépendamment de leurs termes.

This mathematical convention is the infinitely small, hence the pure relation "necessarily implies the infinite under the form of the infinitely small."

It's at the level of the differential relation that the reciprocal immanence of the infinite and relation is expressed in the pure state.

C’est au niveau du rapport différentiel qu’est exprimée à l’état pur l’immanence réciproque du l’infini et du rapport.

So even though in one sense dy/dx = 0/0, it does not thereby as a ratio equal zero. It equals something, we will call it z (see § 598 of Hegel's Science of Logic, where he names it P).

One comprehends that dy/dx = z, that is to say the relation that is independent of its terms will designate a third term and will serve in the measurement and in the determination of a third term: the trigonometric tangent. In this sense I can say that the infinite relation, that is to say the relation between the infinitely small, refers to something finite.

On comprend que dy = z, c’est-à-dire que le rapport qui est indépendant de ses termes, va désigner un troisième terme et va servir à la mesure et à la détermination d’un troisième terme: la tangente trigonométrique. Je peux dire en ce sens que le rapport infini, c’est-à-dire le rapport entre infiniment petit, renvoie à quelque chose de fini.

So the pure relation independent of its terms is a countermeasurement countermeasured with infinity, and hence is a mutual immanence of finite and infinite. This mutual immanence is conceived mathematically in the calculus differential, in which a finite value, the derivative, equals infinitely small values that are put in relation to each other. The relation of infinitesimal values is a pure relation independent of its terms, because its terms, quantities x and y, have vanished, leaving a determinate pure relation in the form of a ratio value.

Deleuze brings these notions together by saying that dy/dx tends towards the limit, z, which determines the tangent.

By means of these calculus concepts, 17th century thinkers were able to fuse three key concepts: relations, infinity, and limit.

Although the individual involves a relation to the infinite, it as well involves a relation to a limit marking the individual's finitude.

We are able to grasp the relation independent of its terms, on account of our understanding the limit towards which the terms are directed, because at the limit we have the differential relation made up of zero valued terms. The cause of a relation's existence, then, is the infinite or infinitesimal, because at the infinitesimal level there can be pure relation.

As Descartes says, we may conceive the infinite but just not comprehend it. But because we can know it, there is a reason for knowing that is distinct from its reason for being; for in order to grasp its reason for being, we need also to comprehend it. But only God has the capacity to comprehend the infinite; our understanding is always finite.

To account for the immanence of the infinite in relations, we turn to the logic of relations, which is developed through two stages in Anglo-American philosophy.

The first stage occurs with the English thinker Russell, who conceived of the pure relation in terms of the finite. Prior to this stage was the classical period of the logic of relations when judgments of relation (Pierre is smaller than Paul) and judgments of attribution (Pierre is yellow or white) were confused. It is not that they failed to understand pure relation; for in the 17th century they understood relation as being itself in relation to the infinite.

The individual as power:

The individual is power and not form, because the individual's conatus is a power tending toward a limit.

But this limit is an infinity: the polygon whose sides increasingly tend toward the limit, a circle, is tending toward having infinite sides.

The limit is precisely the moment when the angular line, by dint of multiplying its sides, tends towards infinity.

La limite, c’est précisément le moment où la ligne angulaire, à force de multiplier ses côtés, à l’infini.

In Greek thought, limit is "peras:" contour or time-limit [French for time limit: "le terme;" think "term" used for time, as in "first term."]

The limit is a term, a volume has surfaces for its limits. For example, a cube is limited by six squares. A line segment is limited by two endpoints. Plato has a theory of the limit in the Timeus: the figures and their outlines.

La limite, c’est un terme ; un volume a pour limite des surfaces. Par exemple, un cube est limité par six carrés. Un segment de droite est limité par deux points. Platon a une théorie de la limite dans le Timée: les figures et leurs contours.

Contour finds its importance in idealism, because something is defined by its limits, where it terminates, and what lies between those limiting determinations can be conceived as the idea of the term.

Henceforth the individual will be the form related to its outline

Dès lors, l’individu, ce sera la forme rapportée à son contour.

When we look to the idea's concrete thing, it is defined by a tactile-optical outline. Although the Ancient Greek notion of eidos suggests the optical world, it also refers to the tactile, for even Aristotle's conception of form and matter involves the tactile outline defining the form.

The pure soul grasps the eidos, and this could be a pure spiritual sight that grasps it, but it could also be a pure spiritual touch as well.

Centuries later the Stoics conceive the limit differently. Their critique is that conceiving something in terms of its limit is to only consider it superficially, as though Nature creates with molds. It matters whether the square is made of wood, marble or whatever else, because when Platonics conceive something only in terms of its outline, they neglect what is inside.

The Stoics devise a new image of limit, they wonder not where does the form stop, but where the action stops.

Bateson, a genius, wrote the article "Why does everything have an outline?" If we think there is something outside the subject, then the subject must have a boundary.

In wondering about the limits of action, the Stoics evoke the example of a sunflower seed stuck in the crack of a wall.

Their favorite example is: how far does the action of a seed go? A sunflower seed lost in a wall is capable of blowing out that wall. A thing with so small an outline. How far does the sunflower seed go, does that mean how far does its surface go? No, the surface is where the seed ends.

Leur exemple favori c’est: jusqu’où va l’action d’une graine? Une graine de tournesol perdue dans un mur est capable de faire sauter ce mur. Une chose qui avait un si petit contour. Jusqu’où va la graine de tournesol, est-ce que ça veut dire jusqu’où va sa surface? Non, la surface, c’est là où se termine la graine.

Stoics criticize Plato for only telling us what things are not, because he defines things according to their terminations. The Stoics argue that things are not ideas, things are bodies, in the sense that things are the actions of bodies.

The limit of something is the limit of its action and not the outline of its figure.

La limite de quelque chose, c’est la limite de son action et non pas le contour de sa figure.

Deleuze offers a simpler example: We are walking in a dense forest, slowly working our way out towards its outer border. Slowly the forest thins-out. What is the limit of the forest? It is the limit of an action because it is the limit of its power; for, as it thins, it looses its power to overcome the surrounding terrain.

We know that the boundary of the forest is not an outline, because we cannot specify the precise moment in its gradual thinning that it ceases to be a forest.

There was a tendency, and this time the limit is not separable, a kind of tension towards the limit. It's a dynamic limit that is opposed to an outline limit. The thing has no other limit than the limit of its power [puissance] or its action. The thing is thus power and not form. The forest is not defined by a form, it is defined by a power: power to make the trees continue up to the moment at which it can no longer do so.

Il y avait tendance, et cette fois la limite n’est pas séparable, une espèce de tension vers la limite. C’est une limite dynamique qui s’oppose à la limite contour. La chose n’a pas d’autre limite que la limite de sa puissance ou de son action. La chose est donc puissance et non pas forme. La forêt ne se définit pas par une forme, elle se définit par une puissance: puissance de faire pousser des arbres jusqu’au moment où elle ne peut plus.

Everything is body for the Stoics, because unlike the implications of Platonic idealism, a circle made of wood extends in space differently than one made of marble, and likewise for a red or a blue circle.

Thus it's tension. When they say that all things are bodies, they mean that all things are defined by tonos, the contracted effort that defines the thing. The kind of contraction, the embryonic force that is in the thing, if you don't find it, you don't know [connaissez] the thing. What Spinoza takes up again with the expression "what can a body do?"

Donc, c’est la tension. Quand ils disent que toutes les choses sont des corps, ils veulent dire que toutes les choses se définissent par tonos, l’effort contracté qui définit la chose. L’espèce de contraction, la force embryonnée qui est dans la chose, si vous ne la trouvez pas, vous ne connaissez pas la chose. Ce que Spinoza reprendra avec l’expression, «qu’est-ce que peut un corps?»

Consider the Neo-Platonists, who ask, "where does light begin," and "where does it end?"

Light has no tactile limit, and nevertheless there is certainly a limit. But this is not a limit such that I could say it begins there and it ends there. I couldn't say that. In other words, light goes as far as its power goes

La lumière n’a pas de limite tactile, et pourtant il y a bien une limite. Mais ce n’est pas une limite telle que je pourrais dire ça commence là et ça finit là. Je ne pourrais dire ça. En d’autres termes, la lumière va jusqu’où va sa puissance.

Although hostile to the Stoics, Plotinus has an insight that will lead to a reversal of Platonism. Idealities will be more than optical, they will be luminous. But shadow

forms a part of light and you will have a light-shadow gradation that will develop space. They are in the process of finding that deeper than space there is spatialization.

elle fait partie de la lumière et vous aurez une gradation lumière-ombre qui développera l’espace. Ils sont en train de trouver que, plus profond que l’espace, il y a la spatialisation.

Space is the result of expansion. This is not an Ancient Greek conception, rather it is an Oriental one, emerging through Byzantian art.

According to Riegl, Greek art is tactile-optical, because it differentiates the foreground from the background (unlike Egyptian art which places everything on the same plane); and hence Greek art emphasizes the outline of things.

But in Byzantine art, the depth lies between the spectator and the mosaic. Byzantine artists privilege the background, out of which the figure emerges. And what defines the outline in Byzantine art is light and color relations.

the figure goes on as far as the light that it captures or emits goes, and as far as the color of which it's composed goes.

que la figure se poursuit jusqu’où va la lumière qu’elle capte ou qu’elle émet, et jusqu’où va la couleur dont elle est composée.

Thus the black of an eye extends as far as the "black shines;" hence the figure has no outline, but rather an expansion of light-color.

Thus art must not be an art of space, it must be an art of the spatialization of space
There is an outline-limit and there is a tension-limit. There is a space-limit and there is a spatialization-limit.

Donc l’art ne doit pas être un art de l’espace, ce doit être un art de la spatialisation de l’espace.
Il y a une limite-contour et il y a une limite-tension. Il y a une limite-espace et il y a une limite-spatialisation.


Deleuze, Gilles. "Cours Vincennes: 17/02/1981". webdeleuze.com

English and French versions available here. With profound gratitude I thank Richard Pinhas for providing these texts.

image from
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.287a.

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