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(All parts in English)

Art, “the art of producing a necessity of discourse”—that is, for the last time, the formula I slipped in, rather than proposed, as to what logic is. I left you amid the hubbub of everyone rising, to point out that it is not enough that Freud noted as a characteristic of the unconscious that it neglects, that it disregards the principle of contradiction, for—contrary to what some psychoanalysts imagine—logic to have no role in its elucidation.

If there is discourse, discourse that deserves to be pinned to the new analytic institution, it is more than likely that, as with any other discourse, its logic must be brought to light.

Let me remind you in passing that discourse is something about which the least one can say is that meaning remains veiled. In truth, what constitutes it is precisely made of the absence of meaning. No discourse exists that does not receive its meaning from another.

And if it is true that the emergence of a new structure of discourse gains meaning, it is not only in receiving it—it is also in the appearance of this analytic discourse, such as I situated it for you last year, that it represents the latest slippage over a tetrahedral, quadripodal structure… as I called it in a text published elsewhere… through the latest slippage of what is articulated in the name of signifiance, it becomes apparent that something original arises from this closing circle.

“The art of producing—a necessity of discourse,” I said—is something other than that necessity itself. Logical necessity—think on this, there can be no other kind—is the fruit of that production. Necessity, ἀνάγκη [ananké], begins only with the speaking being, and just as well everything that could appear from it, emerge from it, is always the fact of a discourse.

If this is indeed what tragedy is about, it is so precisely insofar as tragedy materializes as the fruit of a necessity that is none other… it is obvious, since it concerns only speaking beings… than a necessity, I say, that is logical.

Nothing, it seems to me, appears elsewhere than in the speaking being that properly pertains to ἀνάγκη [ananké]. This is also why Descartes regarded animals as nothing more than automatons. Which is surely a case of illusion—an illusion whose impact we shall demonstrate in passing, concerning what we are about to, concerning this art of producing a necessity of discourse… “what we are about to”: I shall attempt it… attempt to clear a path.

“To produce,” in the double sense:

– to demonstrate what was already there: it is precisely in this that it is not certain that something does not reflect, does not contain the germ of the necessity at issue in what precedes, in the precedence of animal existence. But lacking demonstration, what is to be produced must indeed be considered as previously nonexistent.

– another sense of produce, the one upon which a whole field of research derived from the elaboration of an already constituted discourse, called the discourse of the Master, has already advanced under the term: to realize through labor.

This is indeed what is done insofar as I myself am the logician in question, the product of the emergence of this new discourse, such that production in the sense of demonstration can be here announced before you. What must be supposed to have already been there, by the necessity of the demonstration, is the product of the supposition of the necessity of always, but just as much it testified to the—not lesser—necessity of labor, of actualizing it.

But in this moment of emergence, this necessity at the same time provides proof that it can only be initially supposed under the status of the nonexistent.

What is necessity, then? No! What must be said is not “what, then,” but directly “what is,” this “what, then” already containing too much being in itself.

It is directly: “What is necessity?” such that, by the very fact of producing it, it can, before being produced, only be supposed nonexistent. Which means posited as such within discourse. There is an answer to this question as to any question, for the reason that one only asks it, like any question, in already having the answer. You have it, then, even if you do not know it.

What answers this question “What is necessity?” is what you enact logically, even if you do not know it, in your everyday tinkering, that tinkering which a certain number here… being in analysis with me—there are some, of course not all… come to entrust to me without being able, before a certain step is taken, to feel that in doing so, in coming to see me, they suppose that I myself am—this tinkering—in enacting it, that is to say, all of them, even those who do not confide it to me, are already answering.

How? By simply repeating it, this tinkering, tirelessly. This is what is called:
– at one level, the symptom,
– at another: automatism, an ill-fitting term, but one whose history can be accounted for.

At every moment—you enact it, insofar as the unconscious exists—the demonstration on which the inexistence is based as the prerequisite of the necessary: it is the inexistence of what lies at the principle of the symptom, it is its very consistency in the so-called symptom, since the term, having emerged with Marx, acquired its value. What lies at the principle of the symptom is, namely, the inexistence of the truth it supposes, even though it marks the place of it.

That is for the symptom, inasmuch as it is connected to a truth that no longer holds. In that sense, one can say that, like anyone who persists in the modern age, none of you is foreign to this mode of answering.

In the second case, that of so-called automatism, it is the inexistence of jouissance that the automatism called “of repetition” brings to light, of the insistence of this tramping at the threshold, which presents itself as an exit toward existence.

Only, beyond that, it is not exactly what one calls an existence that awaits you, it is jouissance as it operates as necessity of discourse, and it operates, as you see, only as inexistence.

Only, here you are, recalling these ritornellos, these refrains I repeat indeed with the intention of reassuring you, of giving you the feeling that all I will be doing here is delivering speeches about that in which… in the name of this that would have a certain substance—jouissance, the truth on this occasion as it would be advocated in Freud—nonetheless, if you were to remain at that point, it would not be to the bone of the structure that you could refer.

“What is necessity,” I said, “which is established from a supposition of inexistence?” In this question, it is not what is nonexistent that matters, it is precisely the supposition of inexistence, which is nothing but the consequence of the production of necessity. Inexistence is only a question insofar as it already has a response—double, to be sure—of jouissance and of truth, but it already inexist.

It is not through jouissance nor through truth that inexistence gains status, that it may inexist, that is to say, come to the symbol that designates it as inexistence—not in the sense of not having existence, but of being existence only of the symbol that would render it nonexistent and that itself exists: it is a number, as you generally know, designated as zero. Which shows clearly that inexistence is not what one might believe: the void.

For what could come out of it, except belief, belief in itself? There are not 36 beliefs! God made the world from the void—it is no surprise that this is a dogma.

It is belief in itself, it is this rejection of logic which is expressed… one of my students once found this on his own… and which is expressed according to the formula he gave it—I thank him: “Surely not, but still” [Octave Manoni?].

That can in no way suffice for us. Inexistence is not the void. It is, as I have just told you, a number that is part of the series of whole numbers. There is no theory of whole numbers if you do not account for what zero is.

This is what was realized in an effort which, not by chance, is precisely contemporary—slightly earlier, admittedly—than Freud’s research: it is the one inaugurated by someone named Frege, born 8 years before him and who died some 14 years prior, in logically questioning what is at stake in the status of number.

This is greatly intended [for]…
in our questioning of what constitutes the logical necessity of the discourse of analysis…
this is very precisely what I was indicating, that which risked escaping your attention, concerning the reference I was just illustrating as an application—in other words, a functional usage—of inexistence, that is, that it only occurs in the after-effect from which necessity first emerges, namely, from a discourse in which it manifests before the logician—I’ve told you—arrives there himself as a secondary consequence, that is, at the same time as inexistence itself.

It is its end to be reduced where it manifests before him, this necessity—I repeat it—demonstrating it this time even as I state it.

This necessity is repetition itself: in itself, by itself, for itself, that is, that through which life demonstrates itself to be nothing but necessity of discourse, since it finds, to resist death—that is, its share of jouissance—nothing other than a trick, namely, recourse to that same thing produced by an opaque programming…
which is quite something else, I have emphasized, than “the power of life,” “love,” or other nonsense…
which is this radical programming that only begins to be somewhat elucidated for us through what biologists are doing at the bacterial level, and whose consequence is precisely only the reproduction of life.

What discourse does…
in demonstrating this level at which nothing of a logical necessity manifests except in repetition…
appears here to join, as a semblance, what is carried out at the level of a message that is by no means easy to reduce to what we know of that term, which belongs to the order of something situated at the level of a short combinatorics, whose modulations are those that pass from deoxyribonucleic acid to what is transmitted from it at the level of proteins, with the good will of a few intermediaries known notably as enzymatic, or catalysts. That this is what allows us to refer to what repetition is, this can only be done by elaborating precisely what the fiction is, through which something seems suddenly to reverberate from the very depths of what once made the living being capable of speaking.

There is indeed one among all who does not escape a particularly senseless jouissance, which I will call local in the sense of accidental, and which is the organic form that sexual jouissance has taken for him.

He colors with jouissance all his elementary needs, which in other living beings are only patchworks with respect to jouissance. If the animal eats regularly, it is quite clear that it is in order not to experience the jouissance of hunger.

So he colors with it, the one who speaks…
and it is striking, it is Freud’s discovery…
all his needs, that is, those by which he defends himself against death.

One must not believe at all, however, that sexual jouissance is life.
As I told you earlier, it is a local, accidental, organic production, and very precisely linked to, centered on, what concerns the male organ.
Which is obviously particularly grotesque.

Detumescence in the male has engendered that special kind of call which is articulated language, thanks to which the necessity to speak is introduced in its dimensions. It is from there that logical necessity reemerges as the grammar of discourse.
You see how thin it is! It took nothing less than the emergence of analytic discourse to notice it.

“La signification du phallus,” in my Écrits somewhere, I took care to house this enunciation that I had made, very precisely in Munich, sometime before 1960—a long time ago! I wrote below it “die Bedeutung des Phallus.”
It wasn’t for the pleasure of making you believe I know German—still, still, it was in German, since it was in Munich, that I thought I should articulate what I gave there as a retranslated text.

It had seemed appropriate to me to introduce under the term Bedeutung what, in French—given the level of culture we had reached at the time—I could only decently translate as signification.
Die Bedeutung des Phallus was already there, but the Germans themselves, given that they were analysts…
I note the gap with a little note, reproduced at the beginning of that text…
the Germans hadn’t…
of course I speak of the analysts; we were just coming out of the war, and one can’t say that analysis had made much progress during that time…
the Germans hadn’t grasped a thing.

All that seemed to them, as I emphasize at the very end of that note, quite literally unheard of. It is curious, moreover, that things have changed to the point where what I am recounting today may have already, and justifiably, become commonplace for a number of you.

Bedeutung, however, was indeed referred to the usage Frege makes of this word to oppose it to the term Sinn, which corresponds exactly to what I thought necessary to remind you of in today’s enunciation, namely, the meaning—the meaning of a proposition.

One could express differently…
and you will see that it is not incompatible…
what is at stake in the necessity that leads to this art of producing it as a necessity of discourse.

One could express it differently: what is needed for a word to denote something?
Such is the meaning…
pay attention, the small exchanges are beginning…
such is the meaning that Frege gives to Bedeutung: denotation.

It will appear clear to you, if you are willing to open this book called The Foundations of Arithmetic…
which a certain Claude Imbert—who, if memory serves, once attended my seminar—translated, which leaves it there for you, within your reach, entirely accessible…
it will appear clear to you—as was to be expected—that in order for there to be denotation with certainty, it is not a bad idea to turn first, timidly, to the field of arithmetic as defined by the whole numbers. There was a certain Kronecker who could not help saying—so strong is the need for belief—that “whole numbers were created by God.” Whereupon, he added, man has to do everything else. And since he was a mathematician, “everything else” for him was everything that concerns the remainder of number.

It is precisely to the extent that nothing of that kind is certain, that is to say, that a logical effort can at least attempt to account for whole numbers, that I bring Frege’s work into the field of your consideration.

Nevertheless, I would like to pause for a moment—if only to encourage you to reread it—on this:
that this enunciation I produced under the heading “The Signification of the Phallus”…
of which you will see that at the point I’ve reached—and this is a small merit I claim for myself—there is nothing to revise, even though at that time no one really understood anything about it: I was able to see that firsthand…
what does The Signification of the Phallus mean?

This deserves attention, for after all, in a determinative linkage like this, one must always ask whether it is an “objective” or “subjective” genitive, such as I illustrate the difference by the juxtaposition of the two meanings, marked here by two small arrows:

— a desire → of a child, it is a child that is desired: [objective genitive]
— a desire ← of a child, it is a child who desires: [subjective genitive]

You may practice this—it is always very useful.
The law of retaliation that I write below without adding commentary may have two meanings:
– the law that is retaliation, I institute it as law,
– or what retaliation articulates as law, that is, “an eye for an eye, a tooth for a tooth.” These are not the same thing.

What I would like to point out to you is that The Signification of the Phallus…
and what I will go on to develop will be made to lead you to discover it
in the sense I have just clarified of the word “sense,” that is, the little arrow…
is neutral. The Signification of the Phallus has this cleverness: that what the phallus denotes is the power of signification.

It is therefore not—this Φx—a function of the ordinary type, it is that which makes it so that, provided one uses—for placing it as argument—something that does not need to have any meaning at all beforehand, under the sole condition of articulating it with a quantifier: “there exists” or “for all,”
under that condition, according only to the quantifier…
itself the product of the search for logical necessity and nothing else…
that which will be pinned from this quantifier will take on signification of man or woman, according to the quantifier chosen, that is:

– either the “there exists” [:], or the “there does not exist” [/],
– either the “for all” [;], or the “not all” [.].

Nevertheless, it is clear that we cannot fail to take into account what has emerged from a logical necessity, in confronting it with the whole numbers, for the reason from which I started: that this after-effect necessity implies the supposition of what inexist as such.

Now it is remarkable that in his attempt to question the whole number, in having tried to establish its logical genesis, Frege was led to nothing other than founding the number 1 on the concept of inexistence. It must be said that, to have been led there, one must believe that what had circulated up until then regarding the foundation of 1 did not satisfy him—not the satisfaction of just anyone, but of a logician.

It is certain that for quite some time people made do with very little. One believed it wasn’t difficult: there are several, there are many… well, we count them. That of course raises insoluble problems for the emergence of the whole number.

Because if it’s merely a matter of what is customarily done, of a sign to count them, such a thing exists—someone just brought me a little booklet showing me how… there’s an Arabic poem on this, a poem that explains in verse what should be done with the little finger, then with the index finger, and with the ring finger, and a few others, to pass on the sign of number.

But precisely because a sign must be made, number must have another kind of existence than merely to designate—even if each time with a bark—each, for example, of the persons present here: in order for them to have the value of 1, as has always been noted, they must be stripped of all their qualities without exception. So what is left?

Of course, there were some philosophers called “empiricists” who tried to articulate this by using small objects like little balls: a rosary, of course, is the best example. But that does not at all resolve the question of the emergence of 1 as such.

This had already been seen clearly by someone named Leibniz, who believed he had to begin—as was necessary—with identity, namely by first positing:
2 = 1 + 1
3 = 2 + 1
4 = 3 + 1
and believing he had solved the problem by showing that by reducing each of these definitions to the previous one, one could demonstrate that 2 and 2 make 4.

Unfortunately, there is a small obstacle that the logicians of the 19ᵗʰ century quickly noticed: his demonstration is valid only on the condition of neglecting the completely necessary parentheses to be placed on 2 = 1 + 1, namely the parentheses enclosing (1 + 1), and that it is necessary—which he neglects—that it is necessary to posit the axiom that: (a + b) + c = a + (b + c).

It is certain that such negligence on the part of a logician as truly a logician as Leibniz surely deserves to be explained, and that in some way something justifies it. Be that as it may, its omission is enough, from the logician’s point of view, to reject the Leibnizian genesis, not to mention that it neglects all foundation of what 0 is.

I am here only pointing out to you from which notion of the concept, of the concept supposed to denote something, one must choose them so that it fits. But after all, one cannot say that the concepts…
those he chose: satellites of Mars or even Jupiter…
do not have this scope of denotation sufficient to allow one to say that a number can be associated with each of them.

Nevertheless, the subsistence of number can only be assured starting from the equinumerosity of the objects subsumed by a concept. The order of numbers can thus only be given by this trick that consists in proceeding in exactly the opposite direction of what Leibniz did, to subtract 1 from each number, to say that the predecessor is the one…
the concept of number, derived from the concept…
the predecessor number is the one which…
aside from a given object that served as support in the concept of a certain number…
is the concept which—excepting that object—is found to be identical to a number that is very precisely characterized by not being identical to the preceding one, let’s say, off by 1.

Thus Frege regresses to the conception of the concept as void, which contains no object, which is not that of the void since it is a concept, but of the nonexistent—and it is precisely in considering what he believes to be the void, namely the concept whose number would be equal to 0, that he believes he can define it by the formulation of the argument:
x different from x, x ≠ x, that is, different from itself.

That is to say, what is a denotation that is undoubtedly extremely problematic, for what do we reach? If it is true that the symbolic is what I claim it to be, namely entirely within speech, that there is no metalanguage, then from where can one designate, within language, an object of which it is certain that it is not different from itself?

Nevertheless, it is on this hypothesis that Frege establishes the notion that the concept “equal to 0” gives a number different…
according to the formula he first gave for that of the predecessor number…
gives a number different from what 0 is defined as, considered—and truly so—as the void, that is, from that to which not the equality to 0, but the number 0 itself, applies.

Hence, it is in reference to the following:
— that the concept to which the number 0 applies is based on the fact that it is identical to 0, but not the same as 0,
— that the one which is simply the same as 0 is regarded as its successor and as such equated to 1.
The thing is founded, founded on what is called the starting point of equinumerosity; it is clear that the equinumerosity of the concept under which no object falls, by virtue of inexistence, is always equal to itself. Between 0 and 0, no difference. It is this “no difference” that, by this path, Frege intends to use to found 1.

And in any case, this conquest remains valuable to us insofar as it gives us 1 as being essentially…
pay close attention to what I’m saying…
the signifier of inexistence.

Nevertheless, is it certain that 1 can be founded upon it?
Certainly the discussion could be pursued through purely Fregean avenues.

Nevertheless, for your clarification, I believed I should reproduce something which may be said to have no direct relation with the whole number, namely the arithmetic triangle. The arithmetic triangle is organized in the following way: it begins, as data, from the sequence of whole numbers.

Each term to be written is formed without further commentary—it concerns what is below the line—by the addition…
you’ll notice I have not yet spoken of addition, nor has Frege…
by the addition of the two digits: the one immediately to its left, and the one to its left and above.

You will easily verify that this gives us something…
for instance, when we have a whole number of points that we shall call “monads”…
which automatically provides, given a number of these points, the number of subsets that can, within the set that contains all those points, be formed of any number chosen as being below the whole number in question.

Thus, for example, if you take the row corresponding to the “dyad”: 0, 1, 3, 6, 10, 15, 21…
upon encountering a dyad, you immediately obtain that there are in the dyad 2 monads.
A dyad is not hard to imagine: it is a line with two terms, a beginning and an end.

And if you question what there is—let’s take something more amusing—about the “tetrad,”
you obtain a “tetrad”:
– 0, 1, 5, 15, 35…
you obtain something that is 4 possibilities of triads, in other words to picture it for you:
– 4 faces of the tetrahedron: 0, 1, 4, 10, 20…
Then you obtain six dyads, that is to say:
– the six edges of the tetrahedron: 0, 1, 3, 6, 10, 15…
and you obtain:
– the four vertices of a monad: 0, 1, 2, 3, 4, 5…
This is to support what must be expressed solely in terms of subsets.

It is clear that you see that as the whole number increases, the number of subsets that can arise within it far and quickly exceeds the whole number itself: 0, 1, 4, 10, 20… This is not what concerns us. But simply that in order for me to account for the same process, for the series of whole numbers, I had to begin with what is very precisely at the origin of what Frege did.

Frege, who comes to designate that the number—the number of objects that satisfy a concept as concept of number, of number N by name—will itself be what constitutes the successor number.
In other words, if you count starting from 0: 0, 1, 2, 3, 4, 5, 6, it will always yield what is there, namely 7—7 what?—7 of that something I have called nonexistent, being the foundation of repetition.

Still, in order for the rules of this triangle to be satisfied, this 1 that repeats here must emerge from somewhere. And since we have framed this triangle everywhere with 0—0, 1, 1, 1, 1, 1…—there is therefore a point here, a point to be located at the level of the line of 0s, a point which is 1, and which articulates what?

What matters to distinguish in the genesis of 1 is precisely the distinction of the lack of difference among all these 0s, starting from the genesis: 0, 1, 0, 0, 0, 0… of what repeats, but repeats as nonexistent.

Frege does not account for the sequence of whole numbers, but for the possibility of repetition. Repetition is posited first as repetition of the 1, as the 1 of inexistence.

Is there not—
I can only raise the question here—
something that suggests that in this fact, that there is not just one 1 but:

– the 1 that repeats,

– and the 1 that is posited in the sequence of whole numbers,

in this gap we have to find something that belongs to the order of what we questioned when positing, as a necessary correlate of the question of logical necessity, the foundation of inexistence?