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(All parts in English)

I’m starting right now because I was asked… I was asked due to things prevailing in the way this place functions… I was asked to finish earlier, much earlier than usual. There it is!

So, to address what is to come, like this, in a framework whose memory I hope is not too distant for you, I take it up again from the “Yad’lun,” which I have already uttered for those who are here, parachuting in from a distant land, I repeat what it means, because it does not have a very familiar ring to it.

“Yad’lun,” it seems to come from I don’t know where. From the One, from the One, right? We do not usually express ourselves like that. Yet, that is indeed what I am talking about. From the One: L, apostrophe, U, N, “there is.”

It is a way of expressing oneself that, I hope at least for you, will be found to be in accord with something which I hope is not new to everyone here. And thank God, I know I have ears, some of them attuned to the fields that it so happens I must touch on to address what is at stake in psychoanalytic discourse.

It will be shown to be in accord—I will explain how—with what historically took place in set theory. You have heard about that! You have heard about that because that’s how mathematics is now taught starting in the eleventh grade. It is of course not certain that this much improves understanding.

But then, in relation to what a theory is, one whose driving force is writing… not, of course, that set theory implies a univocal writing, but that—as with many things in mathematics—it cannot be stated without writing… the difference then with this formula, this “Yad’lun” that I am trying to convey, is precisely all the difference there is between writing and speech. It is a gap that is not always, always easy to bridge.

That is indeed what I am attempting on this occasion, and you should immediately be able to understand why, if it is true that, as I have rewritten them on the board, the upper two of these four formulas where I am trying to fix what substitutes for what I have called “the impossibility of writing” precisely, what there is about the sexual relation, it is to the extent that, at the higher level, two terms confront each other, one being “there exists” and the other “there does not exist,” that I bring—I try to bring—the contribution that can usefully be made starting from set theory.

It is already remarkable, is it not, it is striking that “there is One” has never been the subject of any surprise. Still, perhaps it is going a bit too fast to formulate it like that, for one can credit to what I call astonishment… what I call upon you to be astonished about… one can credit to it precisely what I have spoken of, what I have most energetically invited you to become acquainted with, namely that famous Parmenides, right, of dear Plato, who is always so poorly read, at least in any case—as for me—I endeavor to read it in a way that is not quite the received one.

As for the Parmenides, it is quite striking to see to what extent, at a certain level, which is properly that of academic discourse, it creates embarrassment. The way all those who utter wise things in the name of the University always do so with prodigious embarrassment. As if it were a kind of challenge, a sort of purely gratuitous exercise, a ballet.

And the unfolding of the eight hypotheses concerning the relations of the One and of Being remains somehow problematic, a source of scandal.

Some, of course, distinguish themselves by showing its coherence, but this coherence as a whole appears gratuitous and the confrontation of the interlocutors themselves seems to confirm the ahistorical character, so to speak, of the whole.

I would say… if indeed I may venture something on this point… I would say that what strikes me is really quite the opposite, and that if anything gave me the idea that there is in the Platonic dialogue some kind of primary basis for a properly analytic discourse, I would say it is indeed that one, the Parmenides, that would confirm it for me.

It is indeed quite clear, if you remember what I gave, what I wrote down as structure… excuse me for being silent while I write, because otherwise it would cause complications [microphone problem…]… what I gave as structure is indeed that something, and it is not by chance that it is inscribed as the signifier indexed 1 [S1], which appears at the level of production in analytic discourse.

And this is already something which, although I admit it cannot appear obvious to you right away, I am not asking you to take it as self-evident, it is an indication of the opportunity to focus very precisely on—not the numeral—but the signifier One, in our subsequent questioning.

It is not self-evident that there is One. It seems self-evident like that, because, for example, there are living beings and you have, all of you here so well arranged, so to speak, quite the appearance of being entirely independent from one another and of each constituting what is nowadays called an organic reality, of holding as an individual. That is indeed where an entire initial philosophy has of course found a certain foundation.

What is striking, for example, is that at the level of Aristotelian logic, the fact of placing in the same column—that is to say, on this occasion I remind you—of placing under the principle of the same specification of X, namely—as I have said, as I have already stated—of man, of the being who qualifies himself among speakers as masculine, if we take “there exists”: there exists at least one for whom ΦX is not admissible as an assertion : : §, well from that point of view, from the point of view of the individual, we find ourselves facing a position that is distinctly contradictory, namely that Aristotelian logic, which is based on this intuition of the individual whom it posits as real: Aristotle tells us that, after all, it is not the idea of the horse that is real, it is the living horse itself, about which we are precisely forced to ask how, how the idea comes, from where we derive it. He overturns, not without peremptory arguments, what Plato was talking about, namely: that it is by participating in the idea of the horse that the horse is sustained, that what is most real is the idea of the horse.

If we position ourselves from the Aristotelian angle, it is clear that there is a contradiction between the statement that: “for all x, x fills the function of argument in ΦX,” and the fact that “there is some X which can fill the place of argument only in the enunciation”: exact negation of the first.

If we are told that “every horse—whatever you want, in the end—is spirited” and if we add that “there is some horse—at least one—which is not”: in Aristotelian logic, that is a contradiction.

What I am putting forward is intended to make you grasp that precisely if I can, if I dare to put forward two terms, those on the right in my group of 4 terms—it is not by chance that there are 4—if I can put forward something that is manifestly lacking in so-called logic, it is certainly to the extent that the term “existence” has changed its meaning in the interval and that it is not the same existence when it is a matter of the existence of a term that is capable of taking in a mathematically articulated function the place of the argument. Nothing here yet bridges this “Yad’lun” as such with this “at least one” which is precisely what is formulated by the notion reversed E x : :, there exists an x, at least one which gives, to what is posed as a function, a value that can be qualified as true. This distance that is posited of “existence,” if one can say so… I will not call it otherwise today for lack of a better word… “natural existence,” which is not limited to living organisms.

These Ones, for example, we can see them in the celestial bodies, which are among the first to have attracted properly scientific attention, and this is very precisely because of their affinity with the One. They appear as if inscribed in the sky as elements all the more easily marked as Ones since they are point-like, and it is certain that they have done much to emphasize—as a form of transition—the accent on the point.

If, between the individual and what I will call “the real One” in the interval, the elements that signify themselves as point-like have played a prominent role in their transition, is it not apparent to you, and certainly has it not caught your ear in passing, that I speak of the One as a Real, a Real that might as well have nothing to do with any reality?

I call “reality” what is reality, namely for example your own existence, a mode of support which is assuredly material, and above all because it is bodily.

But it is a question of knowing what one is talking about when one says Yad’lun, in a certain way in the path that science is taking. I mean from that turning point where it is decidedly to “number” as such that it entrusted itself in what is its great turning point, the Galilean turning point, to name it.

It is clear that from this scientific perspective, the One we can qualify as individual, One and then something that is stated in the register of the logic of number, there is not much reason to question the existence, the logical support that one can give to a unicorn as long as no animal is conceived in a more appropriate way than the unicorn itself.

It is indeed from this perspective that we can say that what we call “reality,” natural reality, we can take at the level of a certain discourse… and I do not hesitate to claim that analytic discourse is that one… reality, we can always take it at the level of fantasy.

This real I am speaking of, and which analytic discourse is meant to recall that its access is the symbolic. The so-called “real” is in and through this impossible that only the symbolic defines, that we access it. I return to it at the level of the natural history of a Pliny.

I do not see what differentiates the unicorn from any other animal, it being perfectly existent in the natural order. The perspective that interrogates the real in a certain direction compels us to state things in this way. I am not at all, for all that, speaking to you about anything that resembles progress.

What we gain on the scientific plane, which is indisputable, does not in any way, for example, increase our critical sense in the field of political life, for instance. I have always emphasized that what we gain on one side is lost on the other, insofar as there is a certain limitation inherent in what can be called “the field of adequacy” in the speaking being.

It is not because we have made, concerning life, biology, progress since Pliny, that it is an absolute progress. If a Roman citizen saw how we live… it is unfortunately out of the question to evoke him in person on this occasion… but still, he would probably be overwhelmed with horror. Since we can only judge from the ruins left by that civilization, the idea we can form of it is to see, or to imagine, what will be the remains of ours in a time, if such a time is conceivable, equivalent.

This, is it not, is so that you do not delude yourselves, so to speak, on the subject of a particular trust I would place in science. The analytic discourse is not a scientific discourse, but a discourse for which science provides the material, which is quite different.

So it is clear that the grasp of the speaking being on the world in which he conceives himself as immersed… a schema already tinged with fantasy, is it not?… that this grasp, all the same, does not keep increasing—that much is certain—this grasp only increases to the extent that something is elaborated, and that is the use of number.

I claim to show you that this number simply reduces to this “Yad’lun.” So, one must see what, historically, allows us to know about this Yad’lun a little more than what Plato makes of it, if I may say so, by laying it flat with what there is of Being.

It is certain that this dialogue is extraordinarily suggestive and fruitful, and that if you look closely you will already find there a prefiguration of what I can… on the basis, on the theme of set theory… articulate as this “Yad’lun.”

Just begin the statement of the first hypothesis: if the One—it is to be taken for its meaning—if the One is One, what can we possibly do with it? The first objection he introduces is this: that this One will be nowhere, because if it were somewhere, it would be within an envelope, within a limit, and that this is indeed contradictory to its existence as One.

What is there? Well, there you go! I speak softly. That’s how it is, too bad, that’s how I speak today, perhaps because I simply can’t do better.

For the One to have been elaborated in its existence as One in the way that founds the “Mengenlehre,” set theory, to translate it as it has been translated—not without merit—into French, but certainly with an accent that does not quite correspond to the sense of the original German term which, from the viewpoint of what is aimed at, is no better.

Well, this only came late, and only emerged in connection with the entire history of mathematics itself, of which, of course, it is out of the question for me to trace even the briefest of summaries, but in which one must take into account this, which has gained its full emphasis, its full import, namely what I could call the extravagances of number.

This, obviously, began very early, since already in Plato’s time the irrational number was a problem and he found himself inheriting… he gives us the account with all its developments in the “Theaetetus,” does he not… the Pythagorean scandal of the irrational character of the diagonal of the square, the fact that it never ends… this can be demonstrated on a figure. And this is what was most fortunate to make them realize at that time the existence of what I call “numerical extravagance” [here: deviation from the order of One], I mean something that steps outside the field of the One.

After that, what? Something we can, in the so-called method of exhaustion of Archimedes, consider as an avoidance of what comes so many centuries later—in the form of the paradoxes of infinitesimal calculus, in the form of the statement of what is called the infinitely small, something that takes a very long time to be developed by positing, by positing some finite quantity of which it is said that in any case, a certain way of proceeding will result in being smaller than the said quantity, that is to say, in the end, using the finite to define a transfinite.

And then the appearance—indeed, one cannot fail to mention it—the appearance of Fourier’s trigonometric series, which is certainly not without posing all sorts of foundational theoretical problems. All this is combined with the reduction to perfectly finitist principles of the so-called infinitesimal calculus, which continues at the same time and of which Cauchy is the great representative.

I make this ultra-rapid evocation only to date what it means for Cantor to take up again, in his writing, what is the status of the One. The status of the One, from the moment one tries to ground it, can only proceed from its ambiguity. Namely, the driving force of set theory depends entirely on the fact that the One that is the One of the set is distinct from the One of the element.

The notion of the set rests on this: that there is a set even with only one element. That is not usually stated like this, but the very nature of speech is to move forward with heavy boots. Indeed, it suffices to open any exposition of set theory to touch with one’s finger what this implies.

Namely, that if the element posited as fundamental to a set is that something which the very notion of the set allows to be posited as an empty set, well then, the element is perfectly acceptable.

Namely, that a set can have the empty set as its constituent element, and it is in this respect absolutely equivalent to what is commonly called a “singleton” precisely so as not to immediately reveal the card of the number 1.

And this in the most justified way, for the very good reason that we can only define the number 1 by taking the class of all sets that have only one element and highlighting their equivalence as being precisely what constitutes the foundation of the One.

Set theory is thus made to restore the status of number. And what proves that it indeed restores it—this from the perspective of what I am stating—is that precisely, by stating as it does the foundation of the One and by making number rest on it as a class of equivalence, it leads to the highlighting of what it calls the uncountable, which is very simple and, as you will see, immediately accessible, but which, translated into my own vocabulary, I call—not “the uncountable,” an object I would not hesitate to qualify as mythical—but “the impossibility to count.”

This is demonstrated by the method… here I apologize for not being able to immediately illustrate its construction on the board, but really, after all, what is stopping those among you who are interested in this discourse from opening any treatise called Naive Set Theory to realize that:… by the so-called diagonal method, one can make it clear that it is possible to state—the sequence of natural numbers in a variety of ways, for in truth one can state it in thirty-six thousand ways, that it will be immediately accessible to show that, whatever way you have arranged it, there will be… by simply taking the diagonal, and in this diagonal, by changing at each time, according to a previously determined rule, the values… yet another way to count them.

It is precisely in this that the real attached to the One consists. And if today I cannot push the demonstration as far as I would like within the time I promised I would limit myself to, I will nonetheless, right now, emphasize what this ambiguity at the foundation of the One as such entails.

It is exactly this:—that contrary to appearances, the One cannot be founded on “sameness,”—but that it is precisely, on the contrary, by set theory, marked as needing to be founded on pure and simple difference.

What determines the foundation of set theory consists in this, that when you note, let’s say to keep it simple, 3 elements, each separated by a comma, thus by two commas, if any of these elements in no way appears to be the same as another, or if it can be joined to it by any sign of equality whatsoever, it is purely and simply identical with this one.

At the first level of construction that constitutes set theory, there is the axiom of extensionality which means very precisely this: that at the outset, there can be no question of sameness.

It is exactly a matter of knowing at what moment in this construction “sameness” arises. “Sameness” not only arises late in the construction, and, so to speak, at one of its edges, but moreover I can add that this “sameness” as such is counted in the number, and so the emergence of the One, insofar as it is qualifiable as “the same,” only arises, so to speak, in an exponential way.

I mean that it is from the moment when the One in question is nothing other than this aleph zero [aleph-zero, with play on Hebrew letter א], where the cardinal of infinity is symbolized, of numerical infinity, this infinity that Cantor calls “improper,” and which is made up of the elements of what constitutes the first proper infinity, namely the aforementioned aleph zero, it is in the course of the construction of this aleph zero that the construction of the same itself appears, and that this same, in the construction, is itself counted as an element.

This is why, let’s say, it is inadequate in the Platonic dialogue to make participation in anything existing a matter of the order of the similar. Without the crossing by which the One is first constituted, the notion of the similar cannot appear in any way.

This, I hope, is what we will see. If we do not see it here today, since I am limited to a quarter of an hour less than usual, I will continue elsewhere. And why not next time, at the Thursday meeting at Sainte-Anne, since a certain number of you know the way there.

Nevertheless, what I want to point out is what results from this very beginning of set theory and from what I would call—why not?—the cantorization, provided we write it c.a.n, of number.

Here is what it is about.

To found in any way “the cardinal” [of a set], there are no other paths than those of what is called “the one-to-one mapping of one set onto another.”

When one wants to illustrate it, one finds nothing better, nothing else, than to evoke alternately some kind of primitive potlatch rite for the prevalence from which at least a provisional chief will be established, or more simply the operation known as that of the maître d’hôtel, the one who matches, one by one, each element of a set of knives with a set of forks.

It is from the moment there is still One left on one side and nothing on the other—whether it is the herds each of the two competitors makes cross a certain threshold as chief, or whether it is the maître d’hôtel who is doing his tally—what will appear?

The One begins at the level where there is One missing. The empty set is therefore properly legitimized by this, that it is, if I may say so, the door whose crossing constitutes the birth of the One, the first One that is designated in an admissible experience, I mean mathematically admissible, in a way that can be taught, for that is what “matheme” means, and not something that appeals to that kind of crude representation which is… it is about the same thing that constitutes the One and very precisely justifies it, which is designated only as distinct, and not by any other qualifying reference, it is that it only begins from its lack.

And this is indeed what appears to us, in the reproduction I have made here of Pascal’s triangle, the necessity to distinguish each of these lines of which you know… I think for a while now, I have emphasized it enough… how they are constituted, each being made of the addition—of what is above, and on the same line of what is noted on the right, each of these lines is thus constituted:

It is important to realize what each of these lines designates. The error, the lack of foundation stated in Euclid’s definition, which is very precisely this:

“Μονάς ἐστι καθ’ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται, Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος”
“The monad is that by which each of the beings is said to be One, and the number, arithmos, is precisely that multiplicity which is made up of monads.” (Euclid, Elements, VII, 1-2)

Pascal’s triangle is not here for nothing. It is there to figure what is called in set theory, not the elements, but the subsets of these sets. At the level of the subsets, the parts stated monadically of any set are of the second line: the monad is second. What shall we call the first, the one that is ultimately constituted by this empty set whose crossing is precisely what constitutes the One? Why not use the echo given to us by the Spanish language and call it the “nade”?

What is at stake in this repeated One of the first line is very properly the nade, namely the entryway that is designated by lack. It is from what is at stake in the place where a hole is made, from that something which, if you want a figure for it, I would represent as being the foundation of the “Yad’lun,” there can only be One in the form of a bag, which is a bag with a hole in it. Nothing is One that does not come out, or which—from the bag, or in the bag—does not enter: this is the original foundation—taken intuitively—of the One.

I cannot, due to my promises, and I regret it, go further here today with what I have brought. Simply know that we will question… as I had already indicated here the figure… that we will question, starting from the triad, the simplest form in which the subsets—the sub-sets made of the subsets of the set—where these subsets can be represented in a way that satisfies us, in order to return to what happens at the level of the dyad and at the level of the monad.

You will see that by questioning not these prime numbers, but these first numbers, a difficulty will be raised, the fact that it is a figurative difficulty, I hope, will not prevent us from understanding what its essence is, and from seeing what the foundation of the One is.