🦋🤖 Robo-Spun by IBF 🦋🤖

(All parts in English)

I had announced that I would continue today on the phallus; well then, I will not speak to you about it! Or rather, I will speak to you about it only under this form of the inverted eight, which is not especially reassuring. It is not a new signifier that is at issue, as you will see; it is always the same one I have been speaking about, in sum, since the beginning of this year. Only, why do I bring it back as essential? It is in order properly to renew contact with the topological basis at issue, namely, what it means, the introduction made this year of the torus.

It is not at all certain that what I said about anxiety was so well understood. Someone very sympathetic, and who reads, because he is someone from a milieu where people work, very opportunely pointed out to me — I must say that I choose this example because it is rather encouraging — that what I said about anxiety as the desire of the Other overlapped with what one finds in KIERKEGAARD.

On first reading, for it is entirely true — you can well imagine that I remembered it — that KIERKEGAARD, in order to speak of anxiety, evoked the young girl at the moment when, for the first time, she notices that she is desired. Only, if KIERKEGAARD said it, the difference from what I say is, if I may use a Kierkegaardian term, that I repeat it. If there is someone who pointed out that it is never for nothing that one says, ‘I say it and I repeat it,’ it is precisely KIERKEGAARD.

If one feels the need to stress that one repeats it after having said it, it is because probably it is not at all the same thing to repeat it as to say it, and it is absolutely certain that, if what I said last time has a meaning, it is precisely in this: that the case raised by KIERKEGAARD is something altogether particular and which, as such, obscures — far from illuminating — the true meaning of the formula that anxiety is the desire of the Other, with a capital O. It may be that this Other becomes embodied for the young girl at a moment in her existence in some hack. That has nothing to do with the question I raised last time, and with the introduction of the desire of the Other as such in order to say that this is anxiety, more exactly, that anxiety is the sensation of this desire.

Today, then, I am going to return to my path of this year, and all the more rigorously because last time I had had to make an excursion. And that is why, more rigorously than ever, we are going to do topology. And it is necessary to do it, because you cannot do anything but do it at every moment, I mean: whether you are logicians or not, whether you even know the meaning of the word topology or not, you use, for example, the conjunction ‘or.’ Now it is rather remarkable, but surely true, that the use of this conjunction was only well articulated, well specified, well brought out — in the field of technical logic, the logic of logicians — at a rather recent period, much too recent for its effects, in sum, truly to have reached you.

And that is why it is enough to read the slightest current analytic text, for example, to see that thought stumbles at every moment as soon as it is a matter, not only of the term ‘identification,’ but even of the practice of identifying anything whatsoever in the field of our experience. One must start again from schemas which, let us say it, remain unshaken in your thought, unshaken for two reasons:

– first, because they pertain to what I shall call a certain incapacity properly speaking, proper to intuitive thought, or more simply to intuition, which means to the very bases of an experience marked by the organization of what is called the visual sense. You notice this intuitive powerlessness very easily — if I have the good fortune that after this little talk you set yourselves to posing simple problems of representation concerning what I am going to show you can happen on the surface of a torus — you will see the trouble you will have not to get mixed up. A torus is nevertheless very simple, a ring. You will get mixed up, and then I get mixed up like you: it took me practice to find my way around it a little and even to notice what it suggested, and what it made it possible to ground practically.

– The other term is linked to what is called ‘instruction,’ namely that this sort of intuitive powerlessness, everything is done to encourage it, to install it, to give it an absolute character, of course with the best intentions. This is what happened, for example, when in 1741, Mr. EULER, a very great name in the history of mathematics, introduced his famous circles which — whether you know it or not — in sum did much to encourage the teaching of classical logic in a certain sense which — far from opening it up — could only tend to make regrettably obvious the idea simple schoolchildren could form of it.

The thing happened because EULER got it into his head — God knows why — to teach a princess: the princess of ANHALT-DESSAU. For an entire period people were very much occupied with princesses; they still are, and it is regrettable. You know that DESCARTES had his own, the famous Christine. She is a historical figure of another relief: he died from it! That is not altogether subjective: there is a very particular kind of stench that emanates from everything surrounding the entity ‘princess’ or ‘Prinzessin.’ We have, during a period of roughly three centuries, something that is dominated by letters addressed to princesses, princesses’ memoirs, and this occupies a certain place in culture.

It is a sort of substitute for that ‘Lady’ whose function I tried to explain to you, so difficult to understand, so difficult to approach in the structure of courtly sublimation, and after all I am not sure I made you perceive what its true scope really is. I could truly give you only sorts of projections of it, as one tries to depict in another space four-dimensional figures that one cannot represent to oneself.

I learned with pleasure that something of it has reached ears near mine, and that people are beginning to take an interest elsewhere than here in what ‘courtly love’ might be. That is already a result. Let us leave the princess and the embarrassments she may have caused EULER. He wrote her 241 letters, not only to make her understand Euler’s circles. Published in 1775 in London, they constitute a sort of corpus of scientific thought at that date.

What has effectively remained afloat of it are only those little circles, those EULER circles, which are circles like all circles; it is simply a matter of seeing the use he made of them. It was to explain the rules of syllogism and in the end ‘exclusion,’ ‘inclusion,’ and then what one may call ‘the overlap’ of two what? of two fields, applicable to what? well, my God, applicable to many things:
– applicable, for example, to the field where a certain proposition is true,
– applicable to the field where a certain relation exists,
– applicable quite simply to the field where an object exists.

You see that the use of the EULER circle — if you are accustomed to the multiplicity of logics, as they have been elaborated in an immense effort, of which the greater part lies in propositional logic, relational logic, and class logic — has been distinguished in the most useful way.

I cannot even think of entering, of course, into the details that would be required to give the distinction among these elaborations. What I simply want to make recognized here is that you surely remember such or such a moment in your existence when there reached you, under this supporting form, some logical demonstration whatsoever of some object as logical object, whether it be proposition, relation, class, or even simply an object of existence. Let us take an example at the level of class logic, and represent, for example, by a small circle inside a large one:
mammals in relation to the class of vertebrates.

This goes by itself, and all the more simply because class logic is certainly what at the outset opened the way most easily to this formal elaboration, and because there one refers to something already embodied in a signifying elaboration, that of zoological classification quite simply, which truly provides its model. Only, the ‘universe of discourse’ — as one rightly puts it — is not a zoological universe, and in wanting to extend the properties of zoological classification to the whole universe of discourse, one easily slips into a certain number of traps that prompt you to errors and rather quickly let one hear the alarm signal of the signifying impasse.

One of these drawbacks is, for example, an inconsiderate use of negation. It is precisely at a recent period that this use was opened up as possible, namely just at the period when it was remarked that, in the use of negation, this outer EULER circle of inclusion had to play an essential role, namely that it is absolutely not the same thing to speak without any precision, for example, of what is non-man, or of what is non-man within animals. In other words, for negation to have a roughly assured meaning, usable in logic, one must know relative to what set something is denied.

In other words, if A’ is ‘non-A,’ one must know in what it is ‘non-A,’ namely here in B.

Negation, you will see — if on this occasion you open ARISTOTLE — leads into all sorts of difficulties. It nevertheless remains incontestable that no one either waited for these remarks, nor moreover made the slightest use of this formal support; I mean that it is not normal to make use of it in order to use negation, namely that the subject in his discourse frequently makes use of negation in cases where there is not in the least in the world any possibility of securing it on this formal basis.

Hence the utility of the remarks I make to you on negation by distinguishing negation at the level of enunciation, or as constitutive of negation at the level of the statement. This means that the laws of negation, precisely at the point where they are not secured by this altogether decisive introduction, and which dates from the recent distinction of the logic of relations from the logic of classes, that it is, in sum for us, altogether elsewhere than where it found its footing that we have to define the status of negation.

This is a reminder, a reminder intended to illuminate for you retrospectively the importance of what, since the beginning of this year’s discourse, I have been suggesting to you concerning the primordial originality, with respect to this distinction, of the function of negation. You see therefore that these Euler circles, it is not EULER who used them for this end: since then, the work of BOOLE, then of DE MORGAN, had to be introduced for this to be fully articulated.

If I return to these Euler circles, it is therefore not because he himself makes such good use of them, but because it is with his material, with the use of his circles, that the subsequent advances could be made, and among these I give you one that is not the least, nor the least famous, in any case particularly striking, immediate to make felt. Between EULER and DE MORGAN, the use of these circles made possible a symbolization that is as useful as it seems to you moreover implicitly fundamental, and which rests on the position of these circles, which are structured thus:

This is what we shall call two circles that overlap, which are especially important for their intuitive value, which will appear incontestable to everyone if I point out to you that it is around these circles that there can first be articulated two relations which should be strongly emphasized, which are that — first — of union.

Whatever it may be that I enumerated earlier, their union is the fact that after the operation of union, what is unified are these two fields. The so-called operation of union, which is ordinarily symbolized thus U — and it is precisely what introduced this symbol — is, as you see, something that is not altogether the same as addition.

It is the advantage of these circles to make this felt. It is not the same thing as adding, for example, two separate circles or uniting them in this position:

There is another relation illustrated by these overlapping circles, that of intersection, symbolized by this sign ∩, whose meaning is altogether different. The field of intersection is included in the field of union.

In what is called BOOLE’s algebra:
– one shows that, up to a certain point at least, this operation of union is sufficiently analogous to addition for one to symbolize it by the sign of addition: +.
– One also shows that intersection is structurally sufficiently analogous to multiplication for one to symbolize it by the sign of multiplication: x.

I assure you that I am making here an ultra-rapid extract intended to lead you where I have to lead you, and for which I of course apologize to those for whom these things present themselves in all their complexity, as for the omissions all this involves, for we must go further.

And on the precise point that I have to introduce, what interests us is something that, until DE MORGAN — and one can only be astonished by such an omission — had not properly speaking been brought out as precisely one of those functions that follow, that should follow, from an altogether rigorous use of logic: it is precisely this field constituted by the extraction, in the relation of these two circles, of the zone of intersection.

And consider what is the product, when two circles overlap, at the level of the field thus defined:

that is, union minus intersection; this is what is called ‘the symmetric difference.’ This symmetric difference is this thing, which will detain us, which for us, you will see why, is of the highest interest.

The term symmetric difference is here a designation that I ask you simply to take in its traditional usage; that is what it has been called, do not try to give a grammatically analyzable meaning to this so-called symmetry. Symmetric difference means that, that means these fields, in the two EULER circles, insofar as they define as such an exclusive ‘or.’

Concerning two different fields, symmetric difference marks the field as it is constructed if you give to ‘or’ not the alternative sense, which implies the possibility of a local identity between the two terms — this is the current use of the term ‘or,’ which means that in fact the term ‘or’ applies here very well to the field of union.

If a thing is ‘A or B,’ that is how the field of its extension can be drawn, namely under the primary form where these two fields are covered.

‘A or B’

If, on the contrary, it is exclusive, ‘either A, or B,’ this is how we can symbolize it, namely such that the field of intersection is excluded.

‘either A, or B’

This must lead us back, to a reflection concerning what the use of the circle supposes intuitively as a base, as a support, of something formalized as a function of a boundary. This is sufficiently defined in the fact that, on a plane of ordinary use — which does not mean a natural plane, but a fabricable plane, a plane that has quite entered into our universe of tools, namely a sheet of paper…

We live much more in the company of sheets of paper than in the company of tori. There must be reasons for that, but reasons that are not obvious. Why after all would man not make more tori? Besides, for centuries, what we now have in the form of sheets were rolls, which must have been more familiar with the notion of volume in other periods than in ours. In short, there is certainly a reason why this flat surface is something that suffices for us, and more exactly, with which we suffice ourselves. These reasons must be somewhere. And — I indicated it earlier — one cannot attach too much importance to the fact that, contrary to all the efforts, by physicists as well as philosophers, to persuade us of the contrary, the visual field, whatever one may say, is essentially two-dimensional

…on a sheet of paper, on a practically simple surface, a drawn circle delimits in the clearest way an inside and an outside. There is the whole secret, the whole mystery, the simple spring of the use made of it in the Eulerian illustration of logic. I put the following question to you: what happens if EULER, instead of drawing this circle, draws my inverted eight, the one I have to speak to you about today?

In appearance it is only a particular case of the circle, with the inner field it defines and the possibility of having another circle inside. Simply, the inner circle touches — this is what some, at first glance, will be able to tell me — the inner circle touches the boundary constituted by the outer circle. Only it is still not quite that, in the sense that it is very clear, from the way I draw it, that the line here of the outer circle continues into the line of the inner circle to come back here.

And then, simply to mark straightaway the interest, the scope of this very simple form, I would suggest to you that the remarks I introduced at a certain point in my seminar, when I introduced the function of the signifier, consisted in this: reminding you of the paradox, or supposedly such, introduced by the classification of sets — remember — that do not include themselves.

I remind you of the difficulty they introduce: should one, these sets that do not include themselves, include them or not in the set of sets that do not include themselves?

You see the difficulty there:

– if yes, then they will include themselves in this set of sets that do not include themselves,

– if no, we find ourselves faced with an analogous impasse.

This is easily resolved, on this simple condition at least that one notices this — it is moreover the solution given by the formalists, the logicians — that one cannot speak, let us say in the same way, of ‘sets that include themselves’ and of ‘sets that do not include themselves.’

In other words, that one excludes them as such from the simple definition of sets, that one ultimately posits that ‘sets that include themselves’ cannot be posited as sets.

I mean that far from this inner zone…
of objects so considerable in the construction of modern logic as sets
…far from an inner zone…
defined by this image of the inverted eight, by the covering, or the doubling in this covering, of a class, a relation, an arbitrary proposition by itself, by its second-power scope [i.e., self-application / iteration to the second degree]
…far from this leaving, in a notable case, the class, the proposition, the relation in a general way, the category within itself, in a way somehow heavier, more accentuated, this has the effect of reducing it to homogeneity with what is outside.

How is this conceivable? For after all one must still indeed say that, if this is how the question presents itself, namely among all sets a set that covers itself, there is no a priori reason not to make it a set like the others.

You define as a set, for example, all works concerning what relates to the humanities, that is to say to the arts, the sciences, ethnography… You make a list. The works that are works made on the question of what should be classified as humanities will be part of the same catalogue, that is to say that what I have just now even defined by articulating the title: ‘works concerning the humanities,’ is part of what there is to catalogue. How can we conceive that something which thus posits itself as doubling itself within the dignity of a certain category can practically lead us to an antinomy, to a logical impasse such that we are, on the contrary, constrained to reject it?

There is something there that is not of as little importance as you might believe, since one has practically seen the best logicians see in it a sort of failure, a sticking point, a point of vacillation of the entire formalist edifice, and not without reason. And yet here is something that makes to intuition a sort of major objection, all by itself inscribed, sensible, visible in the very form of these two circles which present themselves, in the Eulerian perspective, as one included in relation to the other.

It is precisely on this point that we are going to see that the use of the intuition of the representation of the torus is altogether usable. And, given that you clearly sense, I imagine, what is at issue, namely a certain relation of the signifier to itself, I told you: it is insofar as the definition of a set increasingly approached a purely signifying articulation that it led to this impasse.

This is the whole question of the fact that it is for us a matter of putting in the foreground: that a signifier cannot signify itself
— in fact it is an excessively stupid and simple thing —
except by positing itself as different from itself. This very essential point:
that the signifier, insofar as it can serve to signify itself, must posit itself as different from itself. This is what it is a matter of symbolizing first and foremost because it is also this that we are going to find again, up to a certain extension that it is a matter of determining, in the whole subjective structure, including desire.

When one of my obsessionals, very recently again, after having developed all the refinement of the science of his exercises with regard to the feminine objects to which — as is common among other obsessionals, if I may say so — he remains attached through what one may call ‘a constant infidelity’ — at once the impossibility of leaving any one of these objects, and extreme difficulty in maintaining them all together — and when he adds that it is quite evident that in this relation, in this so complicated relation which requires such high technical refinements, so to speak, in maintaining relations which in principle must remain external to one another, impermeable, one might say, to one another, and yet linked, that if all this, he tells me, has no other end than to leave him intact for a satisfaction at which he himself here stumbles, then it must therefore be found elsewhere: not only in an ever-receding future, but manifestly in another space, since from this intactness and its end he is in the final analysis incapable of saying upon what, as satisfaction, this can open out.

We nonetheless have here, tangibly, something that for us poses the question of the structure of desire in the most everyday way. Let us return to our torus and inscribe our Euler circles on it. This will require making — I apologize for it — a very small return that is not, whatever it may appear to someone entering my seminar for the first time right now, a geometric return — perhaps it will be, quite at the very end, but very incidentally — which is properly speaking topological.

There is no need whatsoever for this torus to be a regular torus, nor a torus on which we can make measurements. It is a surface constituted according to certain fundamental relations that I am going to be led to recall to you, but since I do not wish to seem to go too far from what is the field of our interest, I will limit myself to the things I have already begun and which are very simple. I pointed it out to you, on such a surface, we can describe this type of circle [1]

which is the one I connoted for you as reducible, the one which, if it is represented by a little string that passes at the end through a loop, I can, by pulling on the string, reduce to a point, in other words to zero. I pointed out to you that there are two species of other circles or loops, whatever their extent, for it could just as well, for example this one [2], have that form [2’]:

That means, a circle that passes through the hole, whatever its more or less tight, more or less loose form, that is what defines it: it passes through the hole, it goes to the other side of the hole. Here it is represented in dotted lines, whereas there it is represented in a solid line. This is what that symbolizes: this circle is not reducible. Which means that if you suppose it realized by a string always passing through this little arch that would serve us to tighten it, we cannot reduce it to something punctiform; there will always remain, whatever its circumference, at the center, the circumference of what one may call here the ‘thickness of the torus.’

This irreducible circle, from the point of view that interested us a moment ago, namely the definition of an inside and an outside, if it shows on one side a particular resistance, something that with respect to the other circles confers on it an eminent dignity, on this other point all of a sudden it will appear singularly fallen from the properties of the preceding one. For if this circle I am speaking to you about, you materialize it for example by a cut with a pair of scissors, what will you obtain? Absolutely not — as in the other case — a little piece that goes away and then the rest of the torus. The torus will remain entirely intact in the form of a tube, or of a sleeve if you wish.

If on the other hand you take another type of circle [3], the one I have already spoken to you about, the one that is not the one that passes through the hole, but that goes around it: that one is in the same situation as the preceding one as regards irreducibility. It is also in the same situation as the preceding one regarding the fact that it is not enough to define an inside or an outside. In other words, if you follow it, this circle, and if you open the torus with the aid of a pair of scissors, what will you have in the end?

Well, the same thing as in the preceding case: it has the shape of the torus, but it is a shape that presents only an intuitive difference, which is quite essentially the same from the point of view of structure. You still have, after this operation, as in the first case, a sleeve, only it is a very short and very wide sleeve.

You have a belt if you wish, but there is no essential difference between a belt and a sleeve from the topological point of view. Call it still a band if you wish. So here we are in the presence of two types of circles, which from this point of view moreover make only one, which do not define an inside and an outside.

I observe in passing that if you cut the torus successively along one and the other, you still do not thereby manage to do what is at issue, and yet you obtain immediately with the other type of circle, the first one I drew for you [1], namely two pieces. On the contrary, the torus not only remains quite whole, but this was, the first time I spoke to you about it, a flattening-out that results from it and that allows you to:
– symbolize the torus eventually in a particularly convenient way as a rectangle that you can, by pulling it a little, spread out like a skin pinned at the four corners,
– define the properties of correspondence of its edges to one another, and also the correspondence of its vertices: the four vertices coming together in one point, and thus have — in a way much more accessible to your ordinary intuitive faculties — a means of studying what happens geometrically on the torus.

That is to say, one of these types of circles will be represented by a line like this [2], another type of circles by lines like this [3] representing two opposite points [x-x’, y-y’], defined beforehand as being equivalent on what are called the edges of the spread-out surface, laid flat, the laying-flat as such being impossible, since this is not a surface that is metrically identifiable with a plane surface, I repeat, purely metrically, not topologically. Where does this lead us?

The fact that two sections of this kind are possible, with moreover the necessity of intersecting one another without in any way fragmenting the surface, while leaving it whole, leaving it in a single strip, so to speak, this suffices to define a certain genus of surface. All surfaces are far from having this genus. If in particular you make such a section on a sphere, you will always have only two pieces, whatever the circle. This to lead us to what? Let us no longer make a single section, but two sections on the surface of the torus. What do we see appear?

We see appear something that surely will astonish us at once, namely that if the two circles intersect, the field called ‘the symmetric difference’ does indeed exist. Can we say that, for all that, the field of the intersection exists? I think that this figure as it is constructed is sufficiently accessible to your intuition for you to understand well, straightaway and immediately, that this is not so.

That is to say that this something that would be intersection, but is not, and which — I say: for the eye, for of course there is not even for a single instant any question that this intersection exists — but which, for the eye, and as I have presented it to you thus on this figure as it is drawn, might perhaps be found somewhere, around here [1] in this perfectly continuous field of a single block, of a single strip, with that field there [2] which could analogically, in the crudest way for an intuition precisely accustomed to grounding itself on things that happen only on the plane, correspond to that external field where we could define, with respect to two intersecting EULER circles, the field of their negation:

Namely:
– if here we have circle A,
– and here circle B,
– here we have A¹: negation of A [ ],
– and here we have B¹: negation of B. [ ]

And there is something to formulate concerning their intersection in these possible outer fields.

Here then we see, illustrated in the simplest way by the structure of the torus, this: that something is possible, something that can be articulated thus: two fields intersect, being able as such to define their difference insofar as symmetric difference, but which are nonetheless two fields of which one can say that they cannot be united and that they also cannot overlap. In other words, that they can serve neither for a function of ‘either-or,’ nor for a function of multiplication by themselves:
– they literally cannot be taken up again to the second power,
– they cannot be reflected through one another and in one another,
– they have no intersection: their intersection is the exclusion of themselves.

The field where one expected the intersection is the field where one exits what concerns them, where one is in the non-field. This is all the more interesting in that for the representation of these two circles we can substitute our inverted eight from earlier. We then find ourselves before a form that for us is even more suggestive.

For let us try to remember what I thought to compare them to straightaway, these circles that go around the hole of the torus: to something — I told you — that has to do with the metonymic object, with the object of desire as such.

What is this inverted eight, this circle that takes itself up again within itself? What is it, if not a circle that at the limit doubles and recaptures itself, which makes it possible to symbolize — since it is a matter of intuitive evidence and Eulerian circles seem to us particularly suitable for a certain symbolization of the boundary — which makes it possible to symbolize this boundary insofar as it takes itself up again, identifies itself with itself. Reduce more and more the distance separating the first loop, let us say, from the second, and you have the circle insofar as it grasps itself.

Are there for us objects that have this nature, namely that subsist only in this grasping of their self-difference? For one of two things: either they grasp it, or they do not grasp it. But there is one thing, in any case, that everything that happens at this level of grasping implies and necessitates, and that is that this something excludes any reflection of this object upon itself.

I mean that, suppose it is (a) that is at issue — as I already indicated to you, that that was what these circles were going to serve us for — this means that a2, the field thus defined, is the same field as what is there, that is to say non(a) or -a. Suppose for the moment, I have not said that this was demonstrated, I am telling you that today I am providing you with a model, an intuitive support for something that is precisely what we need regarding the constitution of desire.

Perhaps it will seem to you more accessible, more immediately within your reach, to make of it the symbol of desire’s self-difference with respect to itself, and the fact that it is precisely in its doubling back upon itself that we see appear that what it encloses slips away and flees toward what surrounds it.

You will say, stop there, suspend yourselves here, for it is not really desire that I intend to symbolize by the double loop of this inner eight, but something that suits much better the conjunction of (a) — of the object of desire as such — with itself. For desire to be effectively, intelligibly supported in this intuitive reference to the surface of the torus, it is appropriate to introduce into it, as is only to be expected, the dimension of demand.

This dimension of demand, I told you moreover that the circles enclosing the thickness of the torus, as such, could serve very intelligibly to represent it, and that something — moreover partly contingent, I mean linked to an entirely external, visual apprehension, itself too marked by common intuition not to be refutable, you will see, but in short such that you are forced to represent the torus to yourselves, namely as something like this ring — you easily see how readily what happens in the succession of these circles capable of following one another, as it were helically and according to a repetition that is that of thread around the bobbin, how readily demand, in its repetition, its necessary identity and distinction, its unwinding and its return upon itself, is something that easily finds support in the structure of the torus.

That is not what I mean today to repeat once more. Besides, if I merely repeated it here, that would be altogether insufficient. It is on the contrary something to which I would like to draw your attention, namely this privileged circle constituted by this: that it is not only a circle that goes around the central hole, but that it is also a circle that passes through it. In other words, that it is constituted by a topological property that conflates, that adds together the loop constituted around the thickness of the torus with the one that would be made by a turn made, for example, around the inner hole

This sort of loop is for us of altogether privileged interest, for it is what will allow us to support, to image the relations as structural of demand and desire. Let us see, indeed, what can occur concerning such loops: observe that there can be some constituted in this way, such that another one neighboring it completes itself, returns upon itself, without at all cutting the first.

You see it, given what I have here tried to articulate well, to draw well, namely the way it passes to the other side of this object — which we suppose massive, because that is how you so easily intuitionize it, and which obviously is not — the line of circle [1] passes here, the other line [2] passes a little farther on, there is no sort of intersection between these two circles.

Here are two demands which, while implying the central circle with what it symbolizes on this occasion: the object, and to what extent it is effectively integrated into demand, these two demands involve no sort of overlap, no sort of intersection, and even no sort of articulable difference between them, although they have the same object included within their perimeter. On the contrary, there is another type of circuit, the one that here indeed passes to the other side of the torus, but far from rejoining itself at the point from which it started, initiates here another curve so as to come a second time to pass here and return to its starting point.

I think you have grasped what is at issue: it is nothing less than something absolutely equivalent to the famous inverted-eight curve I spoke to you about earlier. Here the two loops represent the reiteration, the reduplication of demand, and then involve this field of difference from itself, of self-difference, the one on which we placed emphasis earlier, that is to say that here we find the means of symbolizing in a tangible way, at the level of demand itself, a condition for it to suggest, in all its ambiguity, and in a way strictly analogous to the way it is suggested in the earlier reduplication of the object of desire itself, the central dimension constituted by the void of desire.

All this, I bring it to you only as a sort of proposal for exercises, mental exercises, exercises with which you have to familiarize yourselves, if you want to be able, in the torus, to find for what follows the metaphorical value I will give it when I have, in each case, whether it concerns the obsessive, the hysteric, the pervert, or even the schizophrenic, to articulate the relation of desire and demand.

That is why it is under other forms, under the form of the unfolded torus, laid flat from earlier, that I am going to try to mark out clearly for you what the various cases I have so far evoked correspond to. Namely the first two circles, for example, which were circles that went around the central hole, and which intersected by constituting properly speaking the same figure of symmetric difference as that of EULER’s circles.

Here is what that gives on the spread-out torus:

Certainly, figured in this way more satisfactorily than what you saw earlier, in this respect that you can touch with your finger this fact that there is no symmetry, let us say between the four fields two by two [1,2,3,4], as they are defined by the overlap of the two circles.

Earlier you might have said to yourselves, and certainly not in a way that would have been the sign of little attention, that in drawing things thus, and in giving a privileged value to what I call here symmetric difference, I am only doing something rather arbitrary there, since the two other fields [3,4], which I pointed out to you merge together, perhaps occupied with respect to these two [1,2] a symmetrical place. You see here that this is not so, namely that the fields defined by these two sectors, however you reconnect them — and you could do so — are in no way identifiable with the first field. The other figure, namely that of the inverted eight, presents itself thus:

The non-symmetry of the two fields is even more evident. The two circles that I then drew successively on the perimeter of the torus as defining two circles of demand insofar as they do not overlap, here they are symbolized thus:

There is one [A] that we can identify purely — I am speaking of the two circles of demand as I have just defined them insofar as they also included the central hole — one can very easily be defined, situated on the spread-out torus as an oblique line connecting diagonally one vertex to the same point that it really is on the opposite edge, at the opposite vertex of its position: AB.

The second loop [A’] that I had drawn earlier would be symbolized thus: beginning at some point here, we have here A’, here C — a point C that is the same as this point C’ — and ending here at B’: A’C C’B’. There is here no possibility of distinguishing the field that is in AA’; it has no privilege with respect to this field here [BB’]. It is not the same, if on the contrary it is the inner eight that we symbolize, for it presents itself thus:

Here is one of its fields: it is defined by the shaded parts here. It is manifestly not symmetrical with what remains of the other field, however you strive to compose it.

It is perfectly obvious that you can recompose it in the following way, with that element there, let us call it x, coming here, that y coming there, and that z coming here, you have the form defined by the self-difference drawn by the inner eight:

This, whose use we shall see later, may seem to you somewhat tedious, even superfluous, at the very moment when I am trying to articulate it for you. Nevertheless I would like to point out to you what it is for. You can clearly see it: all the emphasis I place on the definition of these fields is intended to mark for you in what they are usable, these fields of symmetric difference and of what I call self-difference, in what they are usable for a certain end, and in what they are supported as existing with respect to another field that they exclude. In other words, if I take so much trouble to establish their dissymmetric function, it is because there is a reason.

The reason is this, that the torus, as it is structured purely and simply as a surface: it is very difficult to symbolize in a valid way what I will call its dissymmetry. In other words, when you see it spread out, namely in the form of this rectangle of which, to reconstitute the torus, it will be a matter that you conceive: first, that I fold it back and make a tube, second, that I bring one end of the tube back onto the other and make a closed tube.

It remains nonetheless that what I did in one direction I could have done in the other. Since it is a matter of topology, and not of metric properties, the question of the greater length of one side with respect to the other has no meaning, that this is not what interests us, since it is the reciprocal function of these circles that is to be used. Now, precisely in this reciprocity they appear able to have strictly equivalent functions.

This possibility, moreover, is at the basis of what I had first let emerge, appear, from the very beginning — for you — in the use of this function of the torus as a possibility of a tangible image concerning it. It is that in certain subjects, certain neurotics for example, we see in some sense in a tangible way the projection, if one may so express oneself, of the very circles of desire to the extent that for them it is a matter, if I may say so, of getting out of them into demands exacted from the Other. And this is what I symbolized by showing you this:

It is that if you draw a torus, you can simply imagine another one that encloses, so to speak, the first in this way. One must clearly see that each of the circles that are circles around the hole can have, by simple rolling, their correspondence in circles that pass through the hole of the other torus, that one torus in some sense is always transformable at all its points into an opposite torus.

What is at issue, then, is seeing what makes one of the circular functions original, that of the full circles for example, in relation to what we called at another moment the empty circles. This difference exists very obviously. One could, for example, symbolize it, formalize it by indicating, with a little sign on the surface of the torus spread out as a rectangle:

if you wish, the anteriority according to which the overlap would occur, and if we call this side a, and this side b, note for example a < b, or inversely.
That would be a notation no one has ever thought of in topology, and which would have something altogether artificial about it, for one does not see why a torus would in any way be an object that has a temporal dimension. From that moment on, it is quite difficult to symbolize it otherwise, even though one clearly sees that there is something irreducible there and that properly speaking makes up the entire exemplary virtue of the toric object.

There would be another way of trying to approach it. It is quite clear that it is insofar as we consider the torus only as a surface, and take its coordinates only from its own structure, that we are placed before this impasse, heavy with consequences for us, since, if obviously the circles, which you see I am going to tend to make serve in order to fix demand there, of course in its relations with other circles that have to do with desire, if they are strictly reversible, is that something we desire to have for our model? Assuredly not!

On the contrary, it is the essential privilege of the central hole that is at issue, and consequently the topological status we are seeking as usable in our model is going to flee from us and escape us. It is precisely because it flees from us and escapes us that it is going to prove fruitful for us.

Let us try another method, to mark what mathematicians, topologists, do perfectly well without in the definition, the use they make of this structure of the torus in topology: they themselves, in the general theory of surfaces, have brought out the function of the torus as an irreducible element of every reduction of surfaces to what is called a normal form. When I say that it is an irreducible element, I mean that one cannot reduce the torus to anything else. One can imagine surface forms as complex as one likes, but one will always have to take account of the torus function in every planning, if I may express myself thus, in every triangulation in the theory of surfaces.

The torus is not sufficient; other terms are needed: the sphere is needed, expressly, that to which I could not even today make allusion, the possibility of what is called a cross-cap must be introduced, and the possibility of holes. When you have the sphere, the torus, the cross-cap, and the hole, you can represent any surface called compact, in other words a surface that is decomposable into strips.

There are other surfaces that are not decomposable into strips, but we leave them aside. Let us come to our torus and to the possibility of its orientation. Are we going to be able to do it in relation to the ideal sphere on which it is attached? We can — this sphere — always introduce it, namely that with a sufficient power of blowing, any torus can come to present itself as a simple handle on the surface of a sphere that is a part of itself sufficiently inflated.

Through the intermediary of the sphere, are we going to be able, if I may say so, to plunge the torus back into what — you can feel it well — we are seeking for the moment, namely this third term that allows us to introduce the dissymmetry we need between the two types of circles?

This dissymmetry, however, so obvious, so intuitively tangible, so irreducible even, and yet such that it manifests itself aptly as that something we always observe in every mathematical development: the necessity, for it to work, of forgetting something at the outset. You find this again in every kind of formal progress: this something forgotten and which literally slips away from us, flees us in formalism. Are we going to be able to grasp it, for example in reference to something called a tube relative to the sphere? Indeed, look carefully at what happens, and what we are told every formalizable surface can give us, in the reduction, the normal form. We are told: this will always be reduced to a sphere — with what? — with tori inserted on it, and which we can validly symbolize thus:

I am skipping the theory. Experience proves that it is strictly exact. That in addition we will have what are called cross-caps. As for these cross-caps, I am giving up speaking to you about them today; I will have to speak to you about them because they will render us the greatest service. Let us content ourselves with considering the torus. It might occur to you that a handle like this one, which would be not outside the sphere, but inside with a hole to enter it, is something irreducible, ineliminable, and that one would have somehow to distinguish exterior tori and interior tori.

In what does this interest us? Very precisely with regard to a mental form that is necessary to all our intuition of our object. Indeed, in the Platonic, Aristotelian, Eulerian perspective of an Umwelt and an Innenwelt, of a dominance placed from the outset on the division of inside and outside, are we not going to place everything we experience, and notably in analysis, in the dimension of what I called the other day ‘the underground,’ namely the corridor that goes off into the depths, in other words, at the maximum, I mean in its most developed form according to this form?

It is extremely exemplary to make felt on this point the non-absolute independence of this form, for I repeat to you, insofar as one arrives at reduced forms, which are the inscribed forms, vaguely sketched on the board in the drawing, to provide support for what I am saying, it is absolutely impossible to maintain, even for an instant, in the difference, the possible originality of the inner handle in relation to the outer handle, to use the technical terms.
It is enough, I think, to have a little imagination to see that if it is a matter of something we materialize in rubber, it is enough to introduce the finger here [at X]

and to hook from the inside the central ring of this handle as it is thus constituted, in order to extract it to the outside according exactly to a form that will be this one, that is to say a torus, exactly the same, without any sort of tearing, nor even properly speaking inversion. There is no inversion; what was interior, namely x, the path thus of the inside of the corridor, becomes exterior because it always was.

If this surprises you, I can still illustrate it in a simpler way that is exactly the same because there is no difference between this and what I am now going to show you, and that I had shown you from the first day, hoping to make you feel what was at issue. Suppose that it is in the middle of its course — which is exactly the same thing from the topological point of view — that the torus is caught in the sphere. You have here a little corridor that runs from one hole to another hole.

There I think it is sufficiently tangible to you that it is not difficult, simply by making bulge a little what you can grasp through the corridor with the finger, to make appear a figure that will be approximately this one, of something that is here a handle and whose two holes communicating with the interior are here in dotted lines.

We therefore arrive at one more failure, I mean at the impossibility, by a reference to a third dimension, here represented by the sphere, of symbolizing that something which sets the torus, if one may say so, in its seat with respect to its own dissymmetry. What we see once again being manifested is that something introduced by this very simple signifier that I first brought you, of the inner eight, namely the possibility of an inner field as always homogeneous with the outer field. This is a category so essential, so essential to mark, to imprint in your mind, that I believed I had to insist on it today, at the risk of boring you, even tiring you, during one single one of our lessons. You will see, I hope, its use in what follows.