Numbers or Things?-- Yes, Please!

In reflecting on the debate between Deleuze and Badiou, my thoughts have turned to Hegel's dialectic of quantum in The Science of Logic (cf. pgs. 202-313). In these passages it seems to me that Hegel all but anticipates the split between the respective ontologies of Deleuze and Badiou. This dialectic can be situated in terms of Hallward's discussion of Badiou's fundamental ontological decision in Badiou: A Subject to Truth. In a subsection to chapter 3, Hallward argues that the fundamental decision of any ontology begins with a decision as to what is first or more fundamental: numbers or things? The first orientation is, of course, that of Plato, where eidos constitutes true reality and the world of objects about us is composed of illusory and fleeting appearances from which thought must depart if it is to find truth, whereas the second is that of Aristotle where all thought begins with "primary substances" or individuals, and "things" such as numbers are merely constructions or abstractions of mind, not truly existing beings. In this second orientation, there is truth only insofar as it refers back to the buzzing world of objects.

Badiou's own neoplatonic option, then, implies (at various stages of the argument) the destitution of the old categories 'substance,' 'thing,' 'object,' and 'relation'; the ontological primacy of mathematical over physical reality; the distinction of mathematics from logic and the clear priority of the former over the latter. In this Platonic tradition, that mathematics is a form of thought means, first of all, that it 'breaks with sensory immediacy,' so as to move entirely within the pure sufficiency of the Ideal. Badiou refuses any cosmological-anthropological reconciliation, any comforting delusion that there is some deep connection (such as that proposed by Jung and his followers) between our ideas or images and the material world we inhabit... His ontology everywhere presumes the radical cut of symbolic representation from the nebulous cosmos of things and experiences that was first proposed by Descartes and subsequently given a particularly strident formulation by Lacan, who insists again and again that we 'can only think of language as a network, a net over the entirety of things, over the totality of the real' (S1, 399/262)... 'Reality is at the outset marked by symbolic neantisation,' and as Badiou confirms, every 'truth is the undoing, or defection, of the object of which it is a truth. .' In particular, 'all scientific progress consists in making the object as such fade away,' and replacing it with symbolic-mathematical constructions. (53)

Badiou's point is very simple. The first step in science, according to this orientation, consists in no longer trusting in appearances. It is by turning away from appearances that scientific knowledge becomes possible. The most striking example of this is seen with regard to the shift from Ptolemy to Copernicus, where the appearance of how the sun and stars moves certain supports the geocentric hypothesis of Ptolemy, but where Copernicus' break with appearances is what first sets us on the path of getting things right with regard to planetary motion. While I'm more than sympathetic to the jab taken at Jung here, I wonder if this is truly an accurate characterization of science. It is certainly true that there is no science without mathematization, but can science do away with the object in the way that Hallward and Badiou here propose? Can science proceed on the basis of mathematization alone? As I tried to argue in a previous post, Badiou's ontology runs into problems when dealing with questions of the relationship between the pure order of being (multiplicity qua multiplicity) and actual situations. What Badiou lacks is any sort of explanatory principle that would allow us to understand why one situation arises rather than another situation from the infinite field of pure multiplicty belonging to ontology. The mathematical enthusiasts of Badiou have sternly lectured me for asking such an impudent question when mathematicians are only interested in studying the formal structure of numbers (is this use of mathematics a new rhetorical strategy?), but when Badiou argues that number is being and that it is of the essence of being to appear as a situation, then he has shifted from a discussion of purely formal, deductive, and possible fields explored with the sovereignity of thought by mathematicians, to a field of actuality that cannot be deduced. The question now becomes unavoidable, and it becomes clear that there is a difference between the materiality of a situation and the "materiality" of mathematics.

What is interesting here is that the debate between Badiou and Deleuze concerning quantitative and qualitative multiplicities closely mirrors Hegel's dialectic of quantum. In treating the dialectic of quantum, Hegel distinguishes between extensive magnitudes and intensive magnitudes. In an articulation of this distinction that is almost identical to Deleuze-Bergson's distinction between quantitative and qualitative multiplicities, the difference between extensive and intensive magnitudes lies in the fact that the former can be divided without changing in kind, whereas the latter cannot be divided without changing in kind. Thus, for instance, no matter how much I divide a bit of space (say my study), I still get more spatial units of measure such that this space is composed of those units of space; however, when dealing with intensive magnitudes such as boiling water, I cannot meaningfully suggest that this magnitude is composed of all the lower temperatures below it. Rather, an intensive magnitude such as boiling is a sort of indivisible and irreducible threshold at which things occur. Hegel goes on to argue that this distinction is, in fact, artificial and that we cannot properly think these two sorts of magnitudes apart:

...it is quite correct that there are no merely intensive and merely extensive magnitudes, any more than there are merely continuous and merely discrete ones; and hence, these two determinations of quantity are not independent species that confront one another. Any intensive magnitude is also extensive, and conversely. So, a certain degree of temperature, for instance, is an intensive magnitude, to which, as such, there corresponds a wholly simple sensation; and if we then go to the thermometer we find a certain expansion of the column of mercury corresponds to this degree of temperature, and this extensive magnitude changes together with the temperature taken as an intensive magnitude. It is the same in the domain of spirit, too; a more intense character exerts influence over a wider range than a less intense one. (The Encyclopaedia Logic, Geraets trans., 164-5)

It is not difficult to see that Badiou has sought to comprehend being purely in terms of extensive magnitudes, whereas Deleuze has sought to understand the world purely in terms of intensive magnitudes. For Deleuze, for instance, we must understand the manner in which a soap bubble individuates itself as resulting from an equalization of intensive magnitudes or surface tensions among the molecules composing the bubble. I understand an entity when I understand the intensities to which the actualized entity is a response or solution. These intensities are the dynamic factor driving system organization. Badiou, by contrast, is able to arrive at the idea that being is pure multiplicity without one because, when we focus on extensive magnitude alone we can endlessly divide any set (take the subsets of any set) without ever reaching a final set or ultimately primitive set. As such, being becomes pure dissemination without one. In Badiou's case, it is interesting to note that in his most recent work, Logiques des mondes, he has been forced to extensively discuss intensity to account for the structure of worlds or situations. Unfortunately, Badiou's concept of intensity pertains to the degree to which elements appear in a situation, rather than serving an energetic function presiding over the actualization of a new organization. For instance, in the situation comprising the United States it could be said that those elements called "leftist" are an intensity of a very low degree as there is very little in the way of a genuine leftist discourse or movement in the United States (even those who say they are on the left in the Democratic party are generally concervatives or supporters of the state... which have a very high degree of intensity by being more predominantly present in the U.S. situation). It's clear that this concept of intensity is radically different than that advocated by Deleuze or Hegel.

Yet it is necessary to wonder whether Badiou's characterization of science is accurate. Can science proceed as pure mathematization, making no reference to intensive magnitudes? Perhaps this holds for astronomy, where we describe planetary motion without referring to intensive factors, but certainly disciplines such as quantum mechanics or chemistry would be all but impossible if we didn't take into account intensive magnitudes. Moreover, I am not at all clear that this is an accurate picture of Lacanian psychoanalysis. Lacan certainly advocated the mathematization of psychoanalytic concepts, but it must never be forgotten that these concepts take their sense and orientation from the clinic. If it were not for variations in intensity, I could never locate, for instance, the objet a or a particularly significant signifier... These instances are accompanied by affect presented in the speech and behavior of the analysand (think of the Rat Man's violent gestures and his famous facial expression portraying a "horror at an enjoyment of which he was unaware" that came to function as the ciphered key to his entire analysis), that cannot be strictly mathematized.

What I find interesting in reading the Deleuze-Badiou debate revolving around extensive and intensive multiplicities as mirroring Hegel's dialectic of quantum is that the dialectic of quantum is the final moment of the broader dialectic of being in the "Logic of Being" portion of the Science of Logic. Those familiar with the Science of Logic will recall that this final moment resolves itself in the dialectic of measure, which then passes over into the the doctrine of essence, or relation and appearing. That is, by following the paths of Hegel (and I'm certainly not suggesting becoming Hegelian or that there is some inherent telos at work here in these debates), it might prove possible to productively think Deleuze and Badiou together, formulating an ontology of essence or appearing that goes beyond both of their positions. What this would be, I'm not exactly sure. At the very least, when offered alternatives such as that proposed by Hallward or Badiou between number and thing, I'm led to wonder why I should have to choose exclusively for one at all.