Theory of Multiplicities in Bergson
Cours du 30/11/1969

... I wanted to propose to you an investigation [recherche] into the history of a word, a still very partial, very localized history. That word is “multiplicity.” There is a very current use of multiplicity: for example, I say: a multiplicity of numbers, a multiplicity of acts, a multiplicity of states of consciousness, a multiplicity of shocks [ébranlements]. Here “multiplicity” is employed as a barely nominalized adjective. And it's true that Bergson often expressed himself thus. But at other times, the word “multiplicity” is employed in the strong sense, as a true substantive, thus, from the second chapter of Time and Free Will onward, the number is a multiplicity, which does not mean the same thing at all as a multiplicity of numbers.

Why do we feel that this use of multiplicity, as a substantive, is at once unusual and important? (The concept of multiplicity, Time and Free Will 224-26) It's because, so long as we employ the adjective multiple, we only think a predicate that we necessarily place in a relation of opposition and complementarity with the predicate ONE: the one and the multiple, the thing is one or multiple, and it's even one and multiple. On the contrary, when we employ the substantive multiplicity, we already indicate thereby that we have surpassed [dépassé] the opposition of predicates one/multiple, that we are already set up on a completely different terrain, and on this terrain we are necessarily led to distinguish types of multiplicity. In other words, the very notion of multiplicity taken as a substantive implies a displacement of all of thought: for the dialectical opposition of the one and the multiple, we substitute the typological difference between multiplicities. And this is exactly what Bergson does: throughout all his work he continually denounces the dialectic as an abstract thought, as a false movement that goes from one opposite to the other, from the one to the multiple and from the same to the one, but which thus always lets the essence of the thing escape, that is the how many, the poson [Greek term for “how much”]. That's why in chapter three of Creative Evolution he will reject the question: is élan vital one or multiple? For élan vital is like duration, it's neither one nor multiple, it's a type of multiplicity. Even further: the predicates one and multiple depend upon the notion of multiplicity, and only agree precisely with the other type of multiplicity, that is to say with the multiplicity that is distinguished from that of duration or élan vital: “Abstract unity and abstract multiplicity are determinations of space or categories of the understanding” (Creative Evolution 280-81).

Therefore there are two types of multiplicity: one is called multiplicity of juxtaposition, numerical multiplicity, distinct multiplicity, actual multiplicity, material multiplicity, and for predicates it has, we will see, the following: the one and the multiple at once. The other: multiplicity of penetration, qualitative multiplicity, confused multiplicity, virtual multiplicity, organized multiplicity, and it rejects the predicate of the one as well as that of the same. Obviously it's easy to recognize behind this distinction between two multiplicities the distinction between space and duration; but what's important is the fact that, in the second chapter of Time and Free Will, the space/duration theme is only introduced as a function of the prior and more profound theme of the two multiplicities: “there are two quite different kinds of multiplicity,” the numerical multiplicity that implies space as one of its conditions, and the qualitative multiplicity that implies duration as one of its conditions. Note: Numerical multiplicities have two dimensions: space and time; the others: duration and pre-spatial extension.

Now Bergson begins with a study of numerical multiplicities. And his study, I believe, includes a very original principle: not that there was a multiplicity of numbers, but each number is a multiplicity, even unity [unité] is a multiplicity. And from this three theses flow [découle], theses that I will only summarize:
1. The reduction of number to exclusively cardinal notions: the number as collection of units [unités], and the ordinal definition of the number of a collection is purely extrinsic or nominal, counting having no other goal than finding the name of the number that was already thought.
2. Space as condition of number, even if only an ideal space, the time that arises in the ordinal series arising only secondarily, and as spatialized time, that is to say as space of succession.
3. The divisibility of the unit; for a number is a unity only by virtue of the cardinal colligation, that is to say the simple act of the intelligence that considers the collection as a whole; but not only does the colligation bear on a plurality of units, each of these units is one only by virtue of the simple act that grasps it, and on the contrary is multiple in itself by virtue of its subdivisions upon which the colligation bears. It's in this sense that every number is a distinct multiplicity. And two essential consequences arise from this: at once that the one and the multiple belong to numerical multiplicities, and also the discontinuous and the continuous. The one or discontinuous qualifies the indivisible act by which one conceives one number, then another, the multiple or continuous qualifying on the contrary the (infinitely divisible) matter colligated by this act.

There we are, how numerical multiplicities are defined, and in a certain way these are the ones that engender space: Time and Free Will, page 91-92.
But there is something quite odd. Time and Free Will appears in 1889. In 1891 Husserl's Philosophie der Arithmetik appears. There Husserl also proposes a theory of number: he there explicitly affirms the exclusively cardinal character of number, the colligation as synthesis of number and the divisible character of the unit. If he differs from Bergson, it's only on the relation of the colligation to space, Husserl thinking that the colligation is independent of spatial intuition; but even this difference is seriously mitigated if one considers the notion of ideal space in Bergson, space being in no way a property of things but a scheme of action, that is to say an original and irreducible intellectual synthesis (cf. Matter & Memory 210-11). So there is an astonishing parallelism. Furthermore, Husserl in turn considers number as a type of multiplicity.

Furthermore, Husserl opposes this type of multiplicity that is number to another type: when I enter a room and see that there are “lots of people,” when I look at the sky and see “lots of stars, or lots of trees in the forest,” or a line of columns in a temple. There, actually, there is no numerical multiplicity: it's in its very looming up [surgissement] that a sensorial aggregate presents a mark that makes it recognizable as a multiplicity, and as a multiplicity of a totally different type than the numerical multiplicity, without any explicit colligation: this is an “implied” multiplicity, a qualitative multiplicity. Husserl speaks of “quasi-qualitative characteristics,” or of an organized multiplicity, or of “figural factors.”

It's a property of the Whole, which, as it's too easy to say, is in no way independent of its elements, but which has complex relations with its elements that are completely different than those a numerical collection has with its elements. And Husserl doesn't fail to cite the example of melody. It's quite evident that Husserl here agrees with the work of his contemporary Ehrenfels who, in 1890, spoke of Gestalt qualities, distinct from the qualities proper to the elements, of another order than those qualities, and above all and explicitly the work of Stumpf who, in 1885, invoked the notion of Verschmelzung to designate a sort of passive (non-intellectual) synthesis, the apprehension of qualities of an order superior to that of the elements.

Thus there we have what the non-numerical multiplicity is. Now this seems quite far from Bergson. And yet it's not so: the strokes of the clock, in chapter two of Time and Free Will, can enter into a numerical multiplicity, but when I am distracted, what happens? They are based in a non-numerical qualitative multiplicity. Multiplicity of fusion, of interpenetration. It's true that in Bergson it involves a fusion, but there's nothing of the kind in Husserl or Stumpf, who observe that the more clearly the elements, the notes of a melody are perceived, the more forcefully the quality of the set [ensemble] affirms itself.