Reading notes: After Finitude, Number and Numbers
May 10th, 2008i) To think Number “in its being” is to nominate an ontological schema in which numbers (whole, real, infinitesimal…) are given. There is no way to “think Number” without such a gesture of nomination: attempts to derive the being of Number from some other given (the extensionality of concepts; the totality of the thinkable; the permutations of a syntax) either founder in contradiction or yield only a castrated, procedural numericality. A novel axiomatic is required.
ii) Ontological thought is not divination, but nomination. To think something in its being is not to intuit, via some extra-rational means, its eternal essence, perceiving it as it might appear in divine ideation, but to determine what there is within the ontological situation (the count-as-one of the resources of ontology in general) that delivers it to thought.
iii) Through the inscription of a new schema within the ontological situation, we are able to separate Number, thought in its being, from those numbers - numerical beings - that are accessible to our thought. We are able to think a great profusion of numbers, but our thinking of Number tells us that the numbers we are presently able to think are an immeasurably tiny subset of the numbers that there are.
iv) Not even a subset, because the numbers that there are do not form a set. The being of Number is an inconsistent multiplicity: no consistent presentation of all of Number is possible. Our access to Number is through this or that numerical situation, this or that consistent presentation of numericality. No such presentation can exhaust the being of Number.
v) What is presented within the ontological situation, wherein Number is thought in its being through the inscription of an ontological schema, is not the being of Number (which cannot be consistently presented) but the thinking of Number in its being. The ontological schema of Number is a being, a consistent multiplicity (in this case, the doctrine of “surreal numbers”). That of which it is the schema is not a being, but the inconsistent multiplicity of the being of Number itself.
vi) There is no mystery about this. In mathematics there are sets, which must be well-founded and consistent, and proper classes, which are “larger” than sets in that no set can “contain” them. A proper class may be regarded as supplying a “law of the count” for all of the sets that can be drawn from the class; but this “descent” from inconsistent multiplicity to consistent presentation is not the predicative separation of a subset from a set. Neither is it necessarily the determination of a finite section of the infinite. The “largeness” of a proper class is not the vastness of infinity, but the non-containability of the class within any set, even an infinitely large one. A proper class is unpresentable, but the ontological schema of the class (which determines the property that every set within the class must have) is a presentation: what it presents is that there is something which is nevertheless itself unpresentable.
vii) Thus, the question of what is or is not accessible to us of Number is not primarily a question of our limited mental powers - the poverty of our mathematics, its irrecusible subservience to our “species being”, or any such pragmatic or empirical limitation. Such limitations no doubt exist, but the separation we have been speaking of between Number, thought in its being, and the numbers to which any particular numerical presentation may provide access, is not an empirical separation. No being that thinks - that organises its thoughts into consistent multiplicities - can think all of Number at once. But a being that thinks can indeed think that this is so, and for what reasons.
viii) The “ancestral” time prior to the emergence of any animal consciousness is (by definition) inaccessible to thought as something present to it. Thought and the ancestral have never been present to each other, face-to-face. The claim of science to be able to identify the “arche-fossil” (the object residing within the depths of ancestrality) and define its properties does not rest on any powers of mystical divination: the scientist does not presume to be an intimate of the Creator, privileged to share His innermost thoughts. Neither is mathematics conceived of, in neo-Platonist fashion, as a bridge between the human and divine, a means of deciphering the inscription within nature of its maker’s mark. In any case, the unpresentable can no more be present within the Creator’s thoughts than it can within ours.
ix) However, as we have seen, mathematics is able to think that something is that is not thinkable in its totality under the sign of presence. It is this thought that breaks the correlationist circle; for correlationism insists on thinking even the absences in its thought under the sign of presence, as if marking off a register. Its “lacunae” are so many rogues, truants, inscrutable oriental gentlemen; they are not thought in their being, but as components of the being of a thought. The arche-fossil cannot be thought in this way.
x) Neither is it claimed that the arche-fossil is somehow seized by mathematics, which succeeds in apprehending it through the achievement of a higher level of technical refinement than other kinds of thought - smarter detectives, better surveillance equipment. This is not a sales pitch: we are not declaring that mathematics reaches the parts other disciplines cannot reach. What is significant about mathematics is that we may recognise here and there in the work of mathematicians the gesture of inscribing a new form in the ontological situation. Only a gesture of this kind can determine that the arche-fossil is, quite separately from how it is for us.
xi) This act of recognition Badiou calls an evental nomination: the decision that such and such an axiomatisation shall be determined as fixing the being of that which it axiomatises. Meillassoux emphasises the groundlessness of any such decision: it is certainly possible to think the being of Number otherwise than through the schema of the “surreal numbers”, or being in general otherwise than through the Z-F axiomatisation of sets. This emphasis on the contingency of nomination is detectable in the name that Meillassoux gives to the venture of ontological thought: speculative realism.
xii) Speculative realism is a realism because it concerns what is, and not merely what our thought is able to grasp in its totality under the sign of presence. It is speculative because there is no possible ground for the act of evental nomination on which its procedure depends: it must set out anew the terms of its investigation, and proceed on the basis of a fidelity to those terms. We are still in a sense in the situation Lyotard described as that of postmodern science: that of legitimation by paralogy, oriented towards a thought of the sublime.

