1. a a Reflexivity
2. (a b) and (b c) --> (a c) Transitivity
3. (a b) --> (b a) Symmetry

It is the axiom of symmetry which preoccupies us, inasmuch as it proposes that the rapport is such that the terms which correlate to it do so identically: the term a has to b the same rapport as b to a. In other words, in regard to the same rapport, the terms a and b have no singularity of position. Moreover, their difference does not enter into the rapport conceived as fixing places.

Consequently: a rapport faithful to the singularity of terms which it joins cannot accept the indifferent symmetry of the places it prescribes. It should admit only one case in which this indifference subsists: that in which, in reality, the two terms joined are the same. This is written, if refers to such a rapport:

(a b) and (b a) --> a = b

The axiom is that of anti-symmetry. Essentially, a rapport between singularities is always anti-symmetrical, which, in conformance with an intuition of Nietzsche, means also that it is always a rapport of power. Thus, if the term a enters a place given in rapport with the term b, supposedly different, it will not in any way cede its place to this term.

And, in effect, love is such that there is never a question of its ceding its place. This is why it is appropriate and provisionally convenient to represent the matrix of rapport between singularities as a relation of order.

The relation of order is, like the relation of equivalence, reflexive and transitive. But it is anti-symmetrical and constitutes by this fact the first relation appropriate to unsubstitutable singularities. Its axioms are:

1. a a Reflexivity
2. a b and b c --> a c Transitivity
3. a b and b a --> (a = a) Antisymmetry

Such rapports are continually rife in the general space constituted by human animals. There's no place, moreover, to interpret them as rapports of subjection or of hierarchy. They indicate only: for reflexivity, that all living singularity is related to itself; for transitivity, that all rapport goes through what Sartre named phenomena of seriality; for anti-symmetry, that there is never indifference of a singularity to its place in a rapport.

The crucial thesis is, then, that the sexed distribution of human animals "is not inscribed in a rapport," the rapport being precisely conceived as a relation of order. If we name M and W the generic positions of "man" and "woman," we can say that there is not, in regard to sex, either M W, nor W M.

Technically, this means that "man" and "woman" are, in regard to rapport, supposedly articulated as sexual, without rapport or incompatible. Which we note:

M W

However, there are various ways to conceive incompatibility. And, in my opinion, it is a crucial point to think of sexual non-rapport with a precise appreciation of its disjunctive character. Lacking which, love becomes unintelligible, or it collapses without ado in the imaginary. Which is the thesis, in my opinion, false, of the pessimistic French moralists, who see in love only an empty parade whose sexual desire is the only real.

One can think of the non-rapport as an integral disjunction, in the sense of the curse of Sodom and Gomorrah: the two sexes die, each on their own side. That means that there is no common term in the position M and W. Or, that what is in common rapport to these two terms is empty. Thus:

[(t M) and (t W)] --> t = 0

Let's name this thesis the segregative thesis. It is thus in its empirical consequences, step by step, that it affects each of the sexed positions in regard to the functions and the places which inscribe pure disjunction. But I do not believe that one can, without resorting to some imaginary point about feminine mystery, assert as real that the non-rapport is pure disjunction. We thus pose that there is at least one non-null term that enters in its place in rapport with the two sexed positions. We will inscribe this term, supposed local mediator of global non-rapport, under the letter u, which connotes rather well the ubiquity that is attested to everywhere by blind usage. Must we also declare uniqueness? Or is there, besides the minimal u, a mass of common predicators, capable of activation in every encounter of M and W?

So I advance what I would call the humanistic thesis, which is that the two positions M and W share a multitude of predicatives allowing for detailing almost to infinity their common membership in Humanity. This thesis in reality hearkens back to the non-rapport, to support a detailed description: that what the two terms have in common makes a sort of acceptable approximation of a rapport.

It is clear that if there is certainly an element which ties up the two non-related terms in the space of non-rapport, it is certain that this element is absolutely indeterminate, indescribable, uncomposable. It is, in fact, atomic, in the sense in which nothing singularizable enters into its composition. The thesis is that there is something which is simultaneously in rapport with the two positions, but that this something-in which one recognizes the phantom of an object-is not composed of nothing, and cannot make the object from any analytical description. In other words: it is true that a non-nul term which enters into rapport with the two sexed positions and which makes a local hole, or a relational support, in their non-rapport, exists. But one would say that nothing enters in rapport with this term, which empties it out. This would be written:

(u M and u W) --> [(t u) --> t = 0]

This fundamental axiom indicates that if the sexed positions are such that u intersects one and the other of them, then with regard to u itself, it remains unformulated, or indeterminate, in such a way that only the void enters in rapport with it.

A third way of conceiving non-rapport, in accordance with an arrangement which goes back at least to the myth of Aristophanes in Plato's The Banquet, is that the sexed positions are, at the same time, totally disjointed and complementary. Namely, that their joining convergence reconstitutes all of humanity. By which, of course, they are not directly in rapport, but they have the mediating rapport such that their supposed conjunction equals a whole, or a completed One. This would be written:

( t) t M or t W (not a third space)

With the result that the union of M and W composes the One (or the Whole):

M W = 1

We no longer believe that there is some real to look for in this thesis of complementarities, which is latent subversion of non-rapport through conjunctive totalization. We thus pose that no totality results from the supposition of a pairing of the two positions, which is written: there exists at least one non-total term which escapes the distribution of the positions as:

( t) [not (t W) and not (t M)]

Or again:

M W 1

One can then recapitulate what I called the elementary axiom of non-rapport.

First M and W are incompatible, or not related to each other, as M W, or also:

not (W M) and not (M W)

Secondly, they have one term in common, namely u:

( u) [u W and u M]

Thirdly, this term is incomplete, or atomic:

(t u) --> (t = 0)

Finally, M and W do not compose a whole:

M W 1

One must in passing be interested in the derivation that leads from the real thesis to the segregative thesis. It includes in fact a double process. On the one hand, the common atom u is excised from its feminine inclusion, which states: a woman has nothing indeterminate that destines her to public space. Or again: a woman is essentially a private creature. On the other hand, since nothing atomic joins a woman to a man, there is no masculine knowledge of the space occupied by a woman. Thus the supposition of a potentially infinite expansion of the feminine, which could well equal the whole, having nul commonality. I inscribe it directly:

W --> (W - u) --> (W = 1?)
excision dilation

That there is a double operation of excision and dilation of the feminine makes it clear that the thesis of a radical exteriority, being the basis of the strictly private and segregated character of the feminine, and the thesis of a mysterious infinity of the same feminine, should be homogeneous, such that, apart from the public man, it perhaps occupies the totality of space. A woman is then at the same time a destitute being (in regard to what has public value) and an emphatically overvalued being (in regard to the infinity of the situation).

On the other hand, the four axioms of non-rapport include a knotting, but atomic and thus unanalyzable, or formally unwritable. And there is an exteriority which is not complementarity.

It is on this identification that it is possible to produce a formal understanding of love. Let us give the first definition of it.

An amorous encounter is what allocates descriptively a double function to the atomic and unanalyzable intersection of the two sexed positions: that of the object, where a desire finds its cause, and that of a point from which the Two are counted, thus initiating an investigation of the sharing of the universe.

It all is based on the fact that u ¾ M and u ¾ W can be read in a double manner: thus one can summon here the inaugural non-rapport of M and W, inasmuch as it affects the unanalyzable u of non-being that circulates in the non-rapport. The two positions M and W are then only in the misunderstanding about the atom u, cause of their common desire, a misunderstanding that nothing sayable can arise, since u is unanalyzable. This is the first reading.

Or one can read in another sense: after u is constructed, by internal linked excision, or by pairing the two external "halves" side by side through u, (W - u) and (M - u), the atom u supports the Two of the positions while being subtracted.

Let us say that in one case, the misunderstanding of the object supports the lack of rapport. And in the other case the excess of the object asserts that the conjunction depends not only on it , but also on the being as such of the sexed positions.

In the first case, u is the One from which the Two slips away, or is undetermined.

In the second case, u is the separated common One from which the Two is positioned in the universe.

The amorous event is no more than the hazardous authorization given to the double reading, that is to say, to the double function of u. I must immediately make two remarks:

a) We search in vain in u itself for the secret of its double function, since it is atomic and thus uncomposed. This is why one cannot have there a pure and simple reduction of love to the localized ambiguity of the object.

b) One cannot inversely reduce love to the second function. Because one would read its separation in vain. It remains assigned to the atomic knotting of non-rapport as it is to the only real of what makes an encounter.

Thus love is the continual exercise of the double function. This is its whole difficulty, as we know. Because what it brings as scenario of the Two does not go harmoniously with the sexual as such. The double function is unsound, and the amorous march is continually enacted to invent this limping. And it is true that it marches along poorly, that is to say, limping. But it is also how it marches on.

The limping is the supplement as such. One can say it is a march as well as an interdiction to march. And this without a resolving dialectic.

One can also say that, conceived as a process of supplement, or as a limping march, love is placed between two boundaries. On one side that which, reduced to the scheme of misunderstanding of the object, can be named sexual adventure. That one can march very well, without limping, but it does not construct any scene of Two, and it basically is only an activation of the structure. And, on the other extremity, that which, only assuming the Two, without sharing of the object, can be named sublime love, or platonic love, which has, if I may say so, no marching orders, but proposes imaginarily that the segregation itself, or the sexual mystery, be singularized as encounter. It is with this theme that the rest of the artistic scheme of sublime renunciation flirts.

The essence of love is to be neither trivial nor sublime. This is why, as everyone knows, it is on the order of hard labor, which is the limping march of the double function of an indeterminacy, the atom u.

Of what order is this labor? Love cannot avoid the return of sexual non-rapport in the modality of a misunderstanding of the object which assures the equivocal triumph of the One and erases the contours of the Two. It can only construct the Two toward the exterior, and not, as such, toward the interior object. Love is thus the alternating movement of an external expansion of the Two, which constructs the scene, and of a return of the atomic object, which erases the two as such, but not exactly the virtual outline of its expansion.

What should be understood by external expansion of the Two? Empirically, it is a matter of those innumerable common practices, or shared inquiries about the world, without which love has no scene of its own except as a sexual adventure.

Logically, it's a matter of this: since in accordance with its aspect of internal conjoined excision the Two comes into existence from the indeterminacy which co-belongs to it, it is possible to identify what surrounds this Two as such, precisely the exception made of the object which entered into the composition of the one and the other. Thus the terms of the situation which surround the subtraction of u on the "male" side as well as on the "female." Let t be one such term. We have:

(W - u) t and (M - u) t and not (u t)

Thus: t has predicative value for M excised from u as it does for W excised from u.

The ensemble of these terms designs the virtual scene of the Two, and they are constructed in the duration of a love, as its non-sexual material, although the identification of this scene built by subtraction of u may be sexually animated throughout.

The limping rhythm of love can be described as the diastole of its expansion around the conjoined excision of u, and the systole of what, irresistibly, leads to the central atomicity of what was subtracted.

One can pose then, that, in the systole which ineluctably leads a love toward centering on its sexual indeterminacy, something of the scene constructed of the Two "sticks" to the M and W positions, in such a way that it is not exactly in the same configuration that the misunderstanding inscribes. Let us say that on traversing-not without the greatest care, because the absence of u governs this labor-the valid terms of (W - u) and of (M - u), love prescribes the "aura" which its atomicity lacks.

The result is that sexual non-rapport is topologically situated in another configuration than that in which it was originally deployed. Or, if you wish, it is saturated by the construction of the scene of Two.

One could not in any way assert that this saturation is itself a sexual facilitation or an adequate mise-en-scene of desire. It could certainly be quite the contrary. But there is no longer any necessity that it should operate in this way. Love is played as truth precisely there, in the radical knowledge of those who capture its effect.

One must also see that what comes to stick to the avatars of desire from the elements of the scene of Two is itself absolutely undetermined. There is an external expansion of the Two, correlative to the conjoined excision of the object. But since the object is again in the efficacy of non-rapport, what there is of this "there is" displacing the sexual topology enters into a radical indeterminacy.

So that the amorous limping, taken, so to speak, to bed, or to the snares of the sexual, can also be defined as the repression of the sexed positions which form a couple between two indeterminacies: the included indeterminacy which is the object, and the exterior indeterminacy which is such-and-such unpredictable fragment of the scene of Two. Between u and what, from a term t subsuming W - u and M - u, returns on u, the difference of two indeterminations is opened.

It is then easy to say that a love is the descriptive establishment of a difference which sets up the lack of its rapport, and the indeterminacy which sets up an excess on top of its non-rapport. Finally, such is the Two it institutes. And such is the formal principle of a supplement. The intelligence that love delivers is that the Two, as such, thought of as process, is neither stuck to the One which makes the difference opaque, nor detached from it to the point that one could count, like a third term, the interval which separates the components of it. Nor is it the Two which counts Two for One; nor is it the Two counted as One by the Three. It is an immanent construction of an indeterminate disjunction, which does not pre-exist it.

And of course, one must at this point come to some transcendental deduction of the sexes. This can wait for another time. Because "woman" and "man" do not enter into the subject-Two in the same way.

It remains that love is the only available experience of a Two counted from itself, of an immanent Two. Each singular love has this of the universal-that, were it ignored by everyone, it contributed on its part, while limping along for as long as it could, to establishing that the Two can be thought in its place, a place supported partially by the hegemony of the One as well as by the inclusion in the Three.

Neither absolute transcendence, nor the Trinitarian doctrine. It is from this point of view tat one can see to what degree love is atheistic. Because atheism is, in the end, nothing other than the immanence of the Two. Love is atheistic in the sense that the Two never pre-exists its process.

It is thus, as an experience in which God fails to guarantee the pre-existence of the separation, that I understand for my part the enigmatic sentence that the poet Pessoa pronounced to his heteronym Caeiro, and with which I'm pleased to conclude: "Love is a thought."