30 Dec 2009

Objective Divisibility §32. Ch.2.Duration as Immediate Datum. Bergsonism. Deleuze

[The following summarizes parts of Deleuze's Bergsonism. My commentary is in brackets. Paragraph subheadings are my own.]

Gilles Deleuze

Le bergsonisme

La Durée comme donné immédiate
Duration as Immediate Datum

[For Bergson, duration changes in kind when we divide it. This is because it is thoroughly heterogeneous. But duration does not extend like space does. So we do not have half a duration when we split it. Rather, we have wholly different durations. There are never any self-same phases throughout a duration. It is continuously heterogeneous. (Consider seeing this entry. It discusses Riemann's influence on Bergson's notion of continuous multiplicity.]

§32 Objective Divisibility

[Duration is subjective. But we are also conscious of objective things in the world around us. And these extend in homogeneous space, unlike duration. Hence,] objective things (objects) are divisible, and when we divide them, they do not change in kind. Hence objectivity is what divides by differences in degree. [Deleuze cites Matter and Memory p.273. (Reference to be added later). Bergson writes here: "As long as we are dealing with space, we may carry the division as far as we please; we change in no way, thereby, the nature of what is divided. This is because space, by definition, is outside us; it is because a part of space appears to us to subsist even when we cease to be concerned with it; so that, even when we leave it undivided, we know that it can wait, and that a new effort of our imagination may decompose it when we choose. As, moreover, it never ceases to be space, it always implies juxtaposition and consequently possible division. Abstract space is, indeed, at bottom, no thing but the mental diagram of infinite divisibility. But with duration it is quite otherwise" (Bergson 273c.d).]

Numbers are the model for those things that can divide without changing their nature. Like numbers, objects are numerically divisible into units. Hence an object is a "numerical multiplicity / multiplicité numérique" (41d/34d). In a sense, such numerically divisible things already have the division implied within them. So Deleuze writes: "number has only differences in degree, or that its differences, whether realized or not, are always actual in it / le nombre n'a que des différences de degré, ou que ses différences, réalisées ou non, sont toujours actuelles en lui" (41d/34-35).

Something that does not extend also does not have parts and cannot be divided into parts. So what about the basic numerical units? If they are unities, then can they be divisible? If not, then they would have different degrees of difference within them. Rather, perhaps they might be like heterogeneities which change kind if they are divided. Consider for example when we add two units together. We regard each of these units as indivisible wholes that are combined with each other. So are such units indivisible? But later, we might divide that doubled unit by four. By doing so, we divide each of the previous wholes into halves. So in this way, the unity of the unit is provisional. Bergson says it obtains its unity in the mental act that regards it as such, for the temporary purpose of addition. But then for the purpose of division, the mind then regards the unit as divisible. Bergson writes in Time and Free Will §54, [underlined are the parts that Deleuze quotes]:

by looking more closely into the matter, we shall see that all unity is the unity of a simple act of the mind, and that, as this is an act of unification, there must be some multiplicity for it to unify. No doubt, at the moment at which I think each of these units separately, I look upon it as indivisible, since I am determined to think of its unity alone. But as soon as I put it aside in order to pass to the next, I objectify it, and by that very deed I make it a thing, that is to say, a multiplicity. To convince oneself of this, it is enough to notice that the units by means of which arithmetic forms numbers are provisional units, which can be subdivided without limit, and that each of them is the sum of fractional quantities as small and as numerous as we like to imagine. How could we divide the unit, if it were here that ultimate unity which characterizes a simple act of the mind? How could we split it up into fractions whilst affirming its unity, if we did not regard it implicitly as an extended object, one in intuition but multiple in space? You will never get out of an idea which you have formed anything which you have not put into it; and if the unity by means of which you make up your number is the unity of an act and not of an object, no effort of analysis will bring out of it anything but unity pure and simple. No doubt, when you equate the number 3 to the sum of 1 +1 + 1, nothing prevents you from regarding the units which compose it as indivisible: but the reason is that you do not choose to make use of the multiplicity which is enclosed within each of these units. Indeed, it is probable that the number 3 first assumes to our mind this simpler shape, because we think rather of the way in which we have obtained it than of the use which we might make of it. But we soon perceive that, while all multiplication implies the possibility of treating any number whatever as a provisional unit which can be added to itself, inversely the units in their turn are true numbers which are as big as we like, but are regarded as provisionally indivisible for the purpose of compounding them with one another. Now, the very admission that it is possible to divide the unit into as many parts as we like, shows that we regard it as extended. (Bergson Time and Free Will 80d.82b)

Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness, Transl. F. L. Pogson, (New York: Dover Publications, Inc., 2001).

Available online at: http://www.archive.org/details/timeandfreewill00pogsgoog

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

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

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