FUNCTION OF APPEARING AND FORMAL DEFINITION OF THE TRANSCENDENTAL

Ontologically, a multiple cannot differ ‘more or less’ from another. A multiple is only identical to itself, and it is a law of being-qua-being (the axiom of extension) that the slightest local difference—for example one which concerns a single element amid an infinity of others—entails an absolute global difference.

The axiom of extension declares that two multiple-beings are equal if and only if they have exactly the same multiple composition, and therefore the same elements. In formal terms:

(α = β) ↔ ∀x[(x ∈ α) ↔ (x ∈ β)]

A contrario, the existence of a single element that belongs to the one but not the other entails that the two beings are absolutely distinct:

∃x [(x ∈ α) and ¬ (x ∈ β)] → ¬ (α ∈ β)

It also follows that if two beings are globally different, there certainly exists at least one element of the one that is not an element of the other (this will turn out to be of crucial importance). Therefore there exists a local difference, or difference ‘in a point’, which can serve to test the global or absolute difference between the two beings.

This means that the ontological theory of difference circulates univocally between the local and the global. Every difference in a point entails the absolute difference of two beings. And every absolute difference implies that there exists at least one difference in one point. In particular, there can

be no purely global difference, meaning that in being as such there is no purely intensive or qualitative differentiation.

But the same cannot be said in terms of appearing. It is clear that multiples in situation can differ more or less, resemble one another, be nearer or farther and so on. It is thus necessary to admit that what governs appearing is not the ontological composition of a particular being (a multiple) but rather the relational evaluations which are determined by the situation and which localize that being within it. Unlike the legislation of the pure multiple, these evaluations do not always identify local dif-ference with global difdif-ference. They are not ontological. That is why we will give the name ‘logic’ to the laws of the relational network which determine the worldly appearing of multiple-being. Every world possesses its own logic, which is the legislation of appearing, or of the ‘there’ of being-there.

The minimal requirement for every localization is being able to deter-mine a degree of identity (or non-identity) between an element α and an element β, when both are deemed to belong to the same world. We there-fore have good reason to think that in every world there exists what we will call a function of appearing, which, given two elements of that world, measures their degree of identity. We will write the function of appearing as Id (α, β). It designates the extent to which, in terms of the logic of the world, we can say that the multiples α and β appear identical.

The function of appearing is the first transcendental indexing: it is a question of knowing what is the degree of identity between two beings of the same world, that is the degree according to which these two beings appear identical.

But what are the values of the function of appearing? What measures the degree of identity between two appearances of multiplicities? Here too there is no general or totalizing answer. The scale of evaluation of appearing, and thus the logic of a world, depends on the singularity of that world itself. What we can say is that in every world such a scale exists, and it is this scale that we call the transcendental.

Like everything that is, the transcendental is a multiple, which obviously belongs to the situation of being of which it is the transcendental. But this multiple is endowed with a structure which authorizes the fact that on its basis one can set out the values (the degrees) of identity between the multiples that belong to the situation, that one can establish the value of the function of appearing Id (α, β), whatever α and β may be.

This structure has properties that vary depending on the worlds. But it also possesses general invariant properties, without which it could not

operate. There is a general mathematics of the transcendental, which we will develop below.

The idea—a very simple one—is that in every world, given two beings α and β which are there, there exists a value p of Id (α, β). To say that Id (α, β) = p means that, with regard to their appearing in that world, the beings α and β—which remain perfectly and univocally determined in their multiple composition—are identical ‘to the p degree’, or are p-identical.

The essential requirement then is that the degrees p are held in an order-structure, so that for instance it can make sense to say that in a fixed referential world, α is more identical to β than to γ. In formal terms, if Id (α, β) = p and Id (α, γ) = q, this means that p > q.

Note that saying ‘it makes sense to say that α is more identical to β than to γ’ does not indicate an obligation: the relation of identity from α to β may also not be comparable to the relation of identity from α to γ: order is not necessarily total. For the time being, it suffices to keep in mind that the logic of appearing presents itself as an order and that transcendental operations present themselves as indexings of beings on the algebraic and topological resources harboured by this order.

3. EQUIVALENCE-STRUCTURE AND ORDER-STRUCTURE It is useful to recall here what an order is. After all, the transcendental makes possible evaluations and comparisons, composing a scale of measurement for the more or less, and the simplest abstract form of such a power is the order-relation—in other words, that which allows us to say that a given element α is ‘greater than’ (or placed ‘higher’ on the scale of comparison, or of superior intensity, etc.) than another element β.

To get a good grasp of the essence of the order-relation, it is helpful to compare it to another primitive relation, the one that establishes the strict (or rigid) identity between two elements. This is called the equivalence-relation. It axiomatically decrees the identity (equivalence) between two elements α and β in the following fashion:

a. An element α is always identical with itself (reflexivity).

b. If α is identical to β, and β to γ, then α is identical to γ (transitivity).

c. If α is identical to β, then β is identical to α (symmetry).

Note that the relationship of equivalence decrees a rigorous symmetry, which is formalized by the appropriately-named ‘axiom of symmetry’. In

this relationship, the relation between an element and another is the same as the relation between this element and the first. That is why the equivalence-relation is used most frequently in order to sanction the sub-stitution of β for α in a formula, once we know that β is equivalent to α.

We could even say that our three axioms (reflexivity, transitivity and symmetry) are axioms of substitutability.

But this entails that the very essence of relation, the essence of every differentiated evaluation or of every comparison, is not yet captured by the

‘relation’ of equivalence. For a comparative evaluation always presumes that we are able to contrast really distinct elements, which is to say non-substitutable elements. For this to be the case, we must reject the third axiom, the one that declares the symmetry of the linked elements.

Basically, the order-relation is the result of this rejection. It is ‘like’ the equivalence-relation, with the proviso that it replaces symmetry with antisymmetry.

The order-relation assumes that difference is axiomatically perceivable.

Of course, the fact that a term is linked to itself can constitute a primitive given. Likewise, the ability for the relation to transit (in the sense that if α = β and β = γ then α = γ) is a useful property of expansion. We will therefore retain reflexivity and transitivity. But in the end it is there where two terms cannot be substituted in terms of what links them that the relationship between relation and singularity is affirmed and that differen-tiated evaluations become possible. We will thus explicitly reject symmetry.

A relation between elements of a set A is an order-relation, written ≤, if it obeys the following three axiomatic dispositions:

a. Reflexivity: x ≤ x.

b. Transitivity: [(x ≤ y) and (y ≤ z)] → (x ≤ z)].

c. Antisymmetry: [(x ≤ y) and (y ≤ x)] → (x = y)].

Antisymmetry is what distinguishes order from equivalence, and what allows us truly to enter the domain of the relation between non-substitutable singularities. In the (order-)relation that x entertains with y, the element x cannot trade places with y unless these two elements are rigorously ‘the same’. The order-relation is really the very first inscription of a demand of the Other, inasmuch as the places that it establishes (before

≤ or after ≤) are in general not exchangeable. Reflexivity and transitivity are Cartesian dispositions: self-relation and order of reasons. But they are properties that also pertain to identity or equivalence. The order-relation

retains these Cartesian properties. But with antisymmetry it formalizes a certain type of non-substitutability.

For the sake of intuitive ease, and since we have comparisons in mind, x ≤ y may be read in one of the two following ways: ‘x is lesser than or equal to y’ or ‘y is greater than or equal to x’. We could even say ‘smaller than’ or

‘bigger than’. But we should bear in mind that the dialectic of the great and the small in no way subsumes the entire, axiomatically established field of the order-relation. These are merely ways of reading the symbolism; the essence of the order-relation is comparison ‘in itself’.

The concept of an order-relation does not contain the fact that it links together all the elements of the base set A. That is why, in the general case, a set endowed with an order-relation is called a partially-ordered set, or POS.

In a POS, there are three possible cases for any two elements x and y: either x ≤ y, or y ≤ x, or neither x ≤ y nor y ≤ x. In the third case, we say that x and y are incomparable (or not linked).

Relation is said to be total in the particular case in which two elements taken at random are always comparable: we have either x ≤ y or y ≤ x.

Every transcendental T of a given world is a partially-ordered set (a POS), with the proviso that in certain worlds, which are in point of fact very special, the transcendental is the bearer of a total order-relation.

What we could call the ontology of the transcendental is summed up by the existence, in every world, of a POS.

4. FIRST TRANSCENDENTAL OPERATION: THE MINIMUM OR ZERO