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(All parts in English)

Why is a signifier a ‘grasp of’ the slightest thing; how can it grasp the slightest thing? That is the question.

A question of which it is perhaps not excessive to say that it has not yet been posed, because of the form classical logic has taken. Indeed, the principle of predication, which is the universal proposition, implies only one thing: that what one grasps are nullifiable beings, dictum de omni et nullo.

For those to whom these terms are not familiar and who consequently do not understand very well, I am recalling what I have been in the process of explaining to you for several sessions now, namely, taking the support of EULER’s circle— all the more legitimately in that what was at issue was to substitute something else— EULER’s circle, like every circle, if I may say so, ‘naive’, a circle with respect to which the question does not arise of whether it encloses a piece, a fragment, the property of the circle— does it detach a fragment from this hypothetical surface implied?— is that it can progressively be reduced to nothing. The possibility of the universal is nullity.

All professors— I told you one day, because I chose this example so as not always to fall back into the same problems— all professors are lettered. Well then, if by chance somewhere no professor deserves to be qualified as lettered, never mind, we shall have null professors. Observe carefully that this is not equivalent to saying that there is no professor. The proof is that null professors, well then, we do have them on occasion! When I say ‘have’, take this ‘have’ in the strong sense, in the sense at issue when we speak of ‘being and having’. It is not some slippery word meant to let the soap slip away.

When I say ‘we have them’, that means that we are accustomed to having them, just as we have lots of things like that, we have the Republic… As a peasant with whom I was talking not very long ago said: ‘This year we had hail, and then afterward, the boy scouts.’ Whatever the definitional precariousness of these meteors may be for the peasant, the verb ‘to have’ therefore indeed has its meaning here.

For example, we also have psychoanalysts… And that is obviously much more complicated, because psychoanalysts begin to make us enter into the order of existential definition. One enters it by way of condition. One says for example: ‘There is no… No one will be able to call himself a psychoanalyst if he has not been psychoanalyzed.’

Well then, there is a great danger in believing that this relation is homogeneous with what we evoked previously, in the sense that, to make use of EULER’s circles, there would be: the circle of the psychoanalyzed, but since, as everyone knows, psychoanalysts must be psychoanalyzed, the circle of psychoanalysts could therefore be drawn as included within the circle of the psychoanalyzed.

I do not need to tell you that if our experience with psychoanalysts causes us so many difficulties, it is probably because things are not so simple, namely that after all, if it is not obvious, at the level of the professor, that the very fact of functioning as professor could suck up within the professor, like a siphon, something that empties him of all contact with the effects of the letter, it is on the contrary entirely obvious for the psychoanalyst that everything is there.

It is not enough to refer the question back to: ‘What is it to be psychoanalyzed?’ For of course what one believes one is doing there, and of course naturally, would only be to divert no one from putting in the foreground the question of what it is to be psychoanalyzed. But in relation to the psychoanalyst, that is not what is at issue to grasp if we want to catch the conception of the psychoanalyst; it is to know: what does being psychoanalyzed do to him, to the psychoanalyst, this insofar as psychoanalyst, and not insofar as part of the psychoanalyzed.

I do not know whether I am making myself clearly heard, but I am going to bring you back once more to the b-a, ba, to the elementary.

And yet, upon hearing the oldest example in logic, the first step taken to push SOCRATES into the hole, namely: ‘All men are mortal…’. How long we have had our ears drummed with this formula! I know very well that you have had time to toughen yourselves, but for any somewhat fresh being, the very promotion of this example to the heart of logic cannot but be the source of some discomfort, some feeling of swindle. For in what does such a formula interest us, if it is man that is at issue to grasp?

Unless what is at issue— and this is precisely what the concentric circles of Eulerian inclusion spirit away— is not knowing that there is a circle of mortals and inside it the circle of man, which has strictly no interest, but rather knowing: what does it do to man to be mortal, to catch the whirlpool produced at the center, somewhere, of the notion of man, by virtue of its conjunction with the predicate ‘mortal’, and that it is indeed for that that we are running after something. When we speak of man, it is precisely this whirlpool, this hole that is made there, in the middle somewhere, that we touch.

I was recently opening an excellent book, by an American author of whom one can say that his work increases the patrimony of thought and logical elucidation. I will not tell you his name, because you are going to look for who it is. And why do I not do so? Because I was surprised to find, in the pages where he works so well, such a lively sense of the actuality of the progress of logic, where precisely my inner figure-eight intervenes. He does not make the same use of it as I do at all; nevertheless, I came to the thought that some mandarin among my listeners would come one day to tell me that that is where I fished it out.

As for the originality of the move from Mr. JAKOBSON, I count indeed on the strongest reference. It must be said that in this case— I believe I began to push forward metaphor and metonymy in our theory somewhere on the side of the Discourse of Rome, which was published— it was while speaking with JAKOBSON that he said to me:

‘Of course, this story of metaphor and metonymy, we twisted that together, remember, on 14 July 1950.’

As for the logician in question, he has long been dead, and his little inner figure-eight unquestionably precedes its promotion here. But when he strides briskly into his examination of the universal affirmative, he uses an example that has the merit of not dragging around everywhere. He says:

‘All saints are men, all men are passionate, therefore all saints are passionate.’

He gathers this up because you must surely feel, in such an example, that the problem is indeed to know where this predicative passion is, the outermost one of this universal syllogism, to know what sort of passion returns to the heart in order to make sainthood.

All this, I thought about this morning, I mean to say it to you just like this, in order to make you feel what is at issue regarding what I called ‘A certain whirlpool movement’.

What are we trying to get hold of, with our apparatus concerning surfaces, surfaces in the sense that we mean to give them a use which here— to reassure my listeners anxious about my excursions— is perhaps not very classical, but is all the same something that is nothing other than renewing, re-questioning the Kantian function of the schema.

I think that the radical illogism— in experience— of ‘belonging’, of ‘inclusion’, the relation of ‘extension’ to ‘comprehension’, to ‘Euler’s circles’, this whole direction into which logic has over time engaged itself— is it not, in its very going astray, a reminder of what, at its departure, was forgotten? What, at its departure, was forgotten is that the object at issue, be it the purest, is, has been, will be— whatever one does about it— the object of desire, – and that if it is a matter of enclosing it in order to catch it logically, that is to say with language, – it is because first it is a matter of grasping it as object of our desire, – having grasped it, of keeping it, which means enclosing it, – and that this return of inclusion to the foreground of logical formalization finds its root there in this need to possess, on which our relation to the object as such of desire is founded.

The Begriff evokes grasp, because it is from running after the grasp of an object of our desire that we forged the Begriff. And everyone knows that everything we want to possess that is an object of desire, what we want to possess for desire, and not for the satisfaction of a need, flees us and slips away. Who does not evoke it in the moralist sermon:

– ‘In the end we possess nothing, we shall have to leave all this’ says the famous cardinal, how sad it is!

– ‘We possess nothing, says the moralist preacher, because there is death.’

Another sleight of hand: what is promoted to us here, at the level of the fact of real death, is not what is in question. It is not for nothing that for a long year I had you walk through that space which my listeners called ‘between-two-deaths’. The suppression of real death would settle nothing in this matter of the slipping-away of the object of desire, because what is at issue is the other death, the one that means that even if we were not mortal, if we had the promise of eternal life, the question always remains open whether this eternal life— I mean one from which every promise of an end would be removed— is not conceivable as a form of dying eternally. It assuredly is, since it is our daily condition, and we must take it into account in our logic of analysts because it is thus— if psychoanalysis has a meaning, and if FREUD was not mad— for that is what this point called the death instinct designates.

Already the physiologist, the most brilliant— one may say— of all those who have a sense of this slant of biological approach: BICHAT: ‘Life,’ he says, ‘is the ensemble of forces that resist death.’ If something of our experience can be reflected, can one day take on a meaning anchored on this plane so difficult, it is this precession produced by FREUD of this form of ‘whirlpool of death’ on whose flanks life clings so as not to go under in it.

For the only thing to add, in order to make this function entirely clear to anyone, is that it suffices not to confuse the dead with the inanimate, when in inanimate nature it suffices that, bending down, we pick up the trace of what a dead form is, the fossil, for us to grasp that the presence of the dead in nature is something other than the inanimate. Is it really certain that there, shells and waste, it is a function of life? That resolves the problem somewhat too easily when it is a matter of knowing why life wriggles around like that!

At the moment of taking up again the question of the signifier already approached by the way of the trace, the ironic idea suddenly came to me, coming out of the Platonic dialogues, of thinking that this somewhat scandalous imprint, to which PLATO refers, thinking of the mark left in the stadium sand by the bare buttocks of the beloved, expressions toward which, no doubt, the adoration of lovers rushed and whose propriety consisted in erasing it, they would have done better to leave it in place.

If lovers had been less obsessed by the object of their desire, they would have been able to make use of it and see in it the outline of this curious line that I am proposing to you today. Such is the image of the blindness carried along by every desire that is too intense. Let us therefore start again from our line, which must indeed be taken in the form in which it is given to us, closed and nullifiable, the line of the original zero of the effective history of logic. If we learn there, returning to it already now, that ‘none…’ is the root of ‘all…’, at least the experience will not have been made in vain. This line, for us, we call the cut, a line— this is our starting point— that we must hold a priori to be closed.

There lies the essence of its signifying nature: nothing will ever be able to prove to us— since it is of the nature of each of these turns to found itself as different— nothing in experience can allow us to ground it as being the same line. It is precisely that which allows us to apprehend the real: it is in this that its return, being structurally different, always another time, if it resembles itself, then there is suggestion, probability, that the resemblance comes from the real.

No other means of correctly introducing the function of the similar. But this is only an indication that I am giving you, to be pushed further. It seems to me that I have repeated it many times, if not— so as not to have to come back to it— then all the same, recalling it, I refer you to that work by a precocious genius, and like all precocious geniuses, disappeared too precociously: Jean NICOD[1893-1924], La géométrie du monde sensible, where the passage concerning the axiomatic line, at the center of the work— perhaps some of you who are genuinely interested in our progress can refer to it— shows well how the smuggling-away of the function of the signifying circle, in this analysis of sensible experience, is chimerical and leads the author, despite the incontestable interest of what he promotes, to the paralogism that you will not fail to find there.

At the outset we take this closed line whose existence in the function of topologically defined surfaces served first to overturn for you the deceptive evidence that the inside of the line was something univocal, since it suffices for the said line to be drawn on a surface defined in a certain way— the torus for example— for it to be apparent that, while remaining there in its function as cut, it could in no way fulfill there the same function as on the surface which you will allow me, without more, to call here ‘fundamental’, that of the sphere, namely to define a nullifiable fragment, for example.

For those who are coming here for the first time, this means a closed line, here drawn [d] or again this one [D], which could in no way be reduced to zero, that is to say that the function of the cut they introduce into the surface is something that each time poses a problem.

I think that what is at issue, concerning the signifier, is this reciprocal linkage such that:

if on the one hand, as I made sensible to you last time with regard to the Möbius surface— that pretty little twisted ear of which I gave you some examples— the median cut relative to its field transforms it into another surface, which is no longer this Möbius surface— if indeed the Möbius surface, and on this point I make more than one reservation, can be said to have only one face— assuredly that which resulted from the cut had, without ambiguity, two faces.

What is at issue for us, taking this slant of questioning the effects of desire by approaching from the signifier, is to perceive how the field of the cut, the gaping of the cut, by organizing itself into a surface, brings forth for us the different forms in which the times of our experience of desire can be ordered. There is the torus…

When I tell you that it is from the cut that the forms of the surface at issue are organized, forms with which, in our experience, we must be capable of bringing into the world the effect of the signifier, I illustrate it— I do not illustrate it for the first time:

Here is the sphere, here is our central cut, taken by the inverse route of EULER’s circle. What interests us is not the piece that is necessarily detached, by the closed line on the sphere; it is the cut thus produced and, if you like, already now the hole. It is quite clear that everything must be given from what we will find at the end, in other words that a hole already has there all its meaning, a meaning made particularly evident by our recourse to the sphere.

A hole here puts the inside in communication with the outside. There is only one misfortune, namely that as soon as the hole is made, there is neither inside nor outside any longer, as this is all too evident: this pierced sphere turns inside out with the greatest ease in the world. It is the universal, primordial creature, that of the eternal potter. There is nothing easier to turn inside out than a bowl, that is to say a cap.

The hole would therefore not have much meaning for us if there were not something else to support this fundamental intuition— I think this is familiar to you today— namely that a hole, a cut, undergoes avatars [transformations/alterations], and the first possible one is that two points on the edge stick together: one of the first possibilities concerning the hole is to become two holes.
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Some have said to me, ‘Why do you not refer your images to embryology?’ Believe me, they are never very far from it. That is what I explain before you, but that would only be an alibi, because here to refer myself to embryology is to leave myself to the mysterious power of life, of which one does not know, of course, why it believes it must enter into the world only by way of, through the intermediary of this globule, of this sphere which multiplies, depresses itself, invaginates, swallows itself, then singularly— at least up to the level of the batrachian— the blastopore, namely that something which is not a hole in the sphere but a piece of the sphere that has gone into the other.

There are enough physicians here who have done a tiny bit of elementary embryology to remember that something which begins dividing in two in order to initiate that curious organ called the neurenteric canal, completely unjustifiable by any manifest function in the organism, this communication of the inside of the neural tube with the digestive tube being rather to be considered as a baroque singularity of evolution, moreover promptly resorbed: in subsequent development nothing more is said of it.

But perhaps things would take on a new turn if they were taken as a metabolism, a metamorphosis guided by structural elements whose presence and homogeneity with the plane in which we move, in the maintenance of the signifier, are the term of an isolation, in some sense pre-vital, of the trace of something that could perhaps lead us to formalizations which, even on the plane of the organization of biological experience, could prove fruitful.

Be that as it may, these two holes isolated on the surface of the sphere are the ones which, joined to one another, stretched, prolonged, then conjoined, gave us the torus.

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This is not new; I would simply like to articulate the result for you. The result first is that if there is something that, for us, supports the intuition of the torus, it is this: a macaroni that rejoins itself, that bites its own tail. It is what is most exemplary in the function of the hole: there is one in the middle of the macaroni and there is one ‘draft-hole’ [air-current hole], which means that by passing through the hoop it forms…

There is a hole that puts the inside in communication with the inside, and then there is another, even more formidable one, which puts a hole at the heart of the surface, which is there a hole while being fully outside. The image of boring/drilling is introduced, for what we call a hole is this, this corridor that would sink into a thickness [a], a fundamental image which, as regards the geometry of the sensible world, has never been sufficiently distinguished, and then the other hole [b], which is the central hole of the surface, namely the hole that I shall call ‘the draft-hole’.

What I claim to advance in order to pose our problems is that this irreducible ‘draft-hole’, if we encircle it with a cut, is precisely where, in the effects of the signifying function, the object (a) as such stands. Which means that the object is missed, since there can in no case be anything there but the contour of the object, in all the senses you can give to the word ‘contour’. Another possibility still opens up, one which for us enlivens, gives its interest to the structuring and structural comparison of these surfaces, namely that the cut can, on a surface, be articulated otherwise.

On the hole here drawn on the surface of the sphere, we can state, formulate, wish, that each point be conjoined to its antipodal point, that without any division of the gaping, the gaping organizes itself into a surface in this way that exhausts it completely without the medium of this intermediate division.

I showed it to you last time, and I will show it to you again: this gives us the surface qualified as bonnet or cross-cap.

(1) (2) (3)

Namely something about which it is fitting that you not forget that the image I gave you is only an image, properly speaking twisted, since what seems so to everyone who for the first time has to reflect on it, what creates an obstacle there, is the question of that famous line of apparent penetration of the surface through itself, which is necessary in order to represent it in our space. This, which I designate here in a trembling manner [line of penetration], is done to indicate that it must be considered as wavering, not fixed.

In other words, we never have to take account of everything that moves about here on one side, outside the surface, which could not pass outside what is on the other side— since there is no real meeting of the faces— but on the contrary could only pass to the other side, therefore inside the other face, I say the other, relative to the observer placed here [blue arrow].

Thus, representing things in this way, concerning this form of surface, is due only to a certain incapacity of the intuitive forms of 3-dimensional space to provide the support of an image that really accounts for the continuity obtained under the name of this new surface called cross-cap, the bonnet in question.

In other words, what does this surface support? We shall call it— since these are the theses I am advancing first, and which will then allow us to give its meaning to the use I shall propose that you make of these various forms— we shall call this surface not the hole— for as you see there is at least one that it spirits away, that disappears completely in its form— but ‘the place of the hole’.

This surface thus structured is particularly apt to make function before us that most elusive element called desire as such, in other words lack. It remains nevertheless that for this surface which fills the gaping, despite the appearance that makes all these points— which we shall call, if you like, antipodal— equivalent points, they can nevertheless function in this antipodal equivalence only if there are two privileged points. These are represented here:

by this very small circle [a], about which the perspicacity of one of my listeners has already questioned me:

‘What indeed do you want to represent in this way by this very small circle?’

Of course, this is in no way something equivalent to the central hole of the torus, since everything that, at whatever level you place yourself from this very privileged point, everything exchanged from one side to the other of the figure, will pass here through this false decussation [b], this chiasm or crossing, which in fact makes its structure.

Nevertheless, what is thus indicated by this thus-circled form is nothing other than the possibility below, if one may so express it, this point, of passing from one outer surface to the other. It is also the necessity of indicating that a non-privileged circle on this surface [a]:

[a] [b] [c]

a reducible circle, if you make it slide, if you extract it from its appearance of semi-occultation, beyond the line apparently, here, of recrossing and penetration, in order to bring it to extend, to develop itself thus toward the lower half of the figure [b] and thus to isolate itself here in a form outside the figure, will always here have to go around something that does not allow it, in any way, to transform itself into what would be its other form, the privileged form of a circle insofar as it goes around the privileged point and as it must be figured thus [c] on the surface in question.

This one, indeed, can in no way be equivalent to it, since this form is something that passes around the privileged point, the structural point around which the whole structure of the surface thus defined is supported. This double point and simple point at once, around which the very possibility of the intercrossed structure of the bonnet or cross-cap is supported, this point, it is by it that we symbolize what can introduce any object (a) into the place of the hole.

This privileged point, we know its functions and its nature: it is the phallus, the phallus insofar as it is through it, as operator, that an object (a) can be put in the very place where we grasp in another structure [the torus] only its contour. That is the exemplary value of the structure of the cross-cap that I am trying to articulate before you: the place of the hole is in principle this point of a special structure, insofar as it is a matter of distinguishing it from other forms of points, this one for example, defined by the recutting of a cut upon itself, the first possible form to give to my inner figure-eight.

We cut something in a sheet of paper, for example, and a point will be defined by the fact that the cut passes again over the place already cut. We know well that this is in no way necessary for the cut to have on the surface a completely definable action and to introduce there this change of which it is a matter that we take the support to image certain effects of the signifier. If we take a torus and cut it thus:

that makes this form that we have drawn here. Passing to the other side of the torus, you can see clearly that at no moment does this cut rejoin itself. Try it on some old inner tube, you will see what it gives: it will give a continuous surface, organized in such a way that it turns over on itself twice before rejoining itself. If it had turned over only once, it would be a Möbius surface. Since it turns over twice, that makes a two-faced surface, which is not identical to the one I showed you the other day after section of the Möbius surface, since that one turns over three times and once differently again.

But the interest is to see what exactly this privileged point is insofar as, as such, it intervenes, it specifies the fragment of surface on which it remains, where it remains irreducibly, giving it the particular accent that allows it, for us, at once to designate the function according to which an object there from always is, even before the introduction of the reflections, of the appearances that we have had of it in the form of images, the object of desire. This object is to be taken only in the effects for us of the function of the signifier, and yet in it one only rediscovers its destination of always.

As object, it is the only absolutely autonomous object, primordial with respect to the subject, decisive with respect to it, to the point that my relation to this object is in some way to be inverted:

– that if, in fantasy, the subject— by a mirage in every respect parallel to that of the imagination of the mirror stage, though of another order— imagines itself, by the effect of what constitutes it as subject, that is to say the effect of the signifier, to support the object that comes for it to fill the lack, the hole of the Other— and that is fantasy,

– inversely, can one say that the whole cut of the subject, what in the world constitutes it as separate, as rejected, is imposed on it by a determination no longer subjective, going from the subject toward the object, but objective, from the object toward the subject, is imposed on it by the object (a), but insofar as at the heart of this object (a) there is this central point— this whirlpool-point through which the object emerges from a beyond of the imaginary, idealist, subject-object knot that has up to now always constituted the deadlock of thought— this central point which, from that beyond, promotes the object as object of desire.

That is what we shall pursue next time.