Lacan Jacques. Seminar XXII, R.S.I.

In Euclidean geometry - it is indeed strange - the point has no dimension at all, contrary to the line, the surface, the solid, which respectively have one, two, three. You know the Euclidean definition of the point as the intersection between two straight lines. Isn't there, I will permit myself to say, something that sins (pèche) here? For what, finally, prevents (empêche) these two

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lines from slipping over one another? Me, I would like, for defining a point, three straight lines disposed like this (fig. 4).

These straight lines are not here simple edges, strokes of a saw, shadows; they are effectively consistent straight lines, which realize the essence of the Borromean knot, which is to say, determining, gripping, a point as such. But it no longer suffices to speak here of a dimension which might be zero, which, in other words, does not dimension. We must invent something else.

Let us make the effort to say that these are not simply strokes, but three surfaces. You will obtain the said effect of a point in a fashion as valuable as with three cords.

With two infinite straight lines, take note of it, we can, by knotting a single round of thread, maintain the property of the Borromean knot (fig. 5). On this condition alone, that the two straight lines would not be able to overlap themselves between this knot and infinity except in a single fashion, that is, taking the straight line R, which must be pulled, if I can say so, forward, while line S can only be pulled behind. They must not be linked two-by-two. This is excluded by figure 6, where we see the blue infinite straight line pass beneath the one below and above the one above, to put it simply. On this condition, the Borromean knot functions. Labeling as I do the blue round as real, the black as symbolic, and the red as imaginary is situated by a flattening out, in other words, by a reduction of the imaginary. The imaginary always tends to reduce itself to a flattening out. All figuration is founded on this. Of course, it is not because you will have wadded up these rounds of thread that they would be less Borromeanly knotted. In the real - which is to say, in regard to the fact that one of them unknotted liberates the other two - that changes nothing.

But how is it that we need this flattening out to be able to figure any topology whatsoever? This is very certainly a question that touches on that of the debility I have qualified as mental, inasmuch as this debility is rooted in the body itself.

I have written a here, at the central point. In the imaginary, but also in the symbolic, I inscribe the said function of sense. The two other functions to be defined in regard to the central point are two jouissances.
One of these two jouissances - but which? - we could define as enjoyment of life (jouir de la vie). If the real is life - but is this so certain? - since this jouissance participates also in the imaginary of sense - we must situate it here. This is no less a point than the central point, the point called the object "a" since it conjoins on the occasion three surfaces which are also wedged together.

What then is the other mode of jouissance?

These are the points we will have to elaborate on, since they are also those that we interrogate.
Moreover, Freud, to return to him, has stated something triadic: inhibition, symptom, anxiety. Can we situate these three terms?

Inhibition, as he articulates it himself, is always an affair of the body, that is, of function.

And to indicate it already on this schema, I will say that it is what somewhere stops interfering in a figuration of the hole of the symbolic. Is what an animal encounters, where there is a central inhibitor in the nervous system, of the same order as this imaginary arrest of functioning for the speakingbeing? How would it be conceivable that the putting in function, for the being presumed non-speaking, in the nevrax, in the central nervous system, of a positive inhibitory activity would be of the same order as what we know to be exterior to sense, exterior to the body? How, in other words, can we topologize this surface in way that, as I have said to you, it

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is assuredly only on two surfaces that it is figured? How can inhibition have any business with this effect of arrest which results from its intrusion into the field of the symbolic? We will have to discuss this.

It is anxiety, insofar as it takes its departure from the real, that is going to give sense to the nature of the jouissance which is produced by the overlapping, in the Eulerian sense, of the real and of the symbolic.

Finally, to define the third term, it is as symptom that we identify what is produced in the field of the real. The notion of the symptom was introduced well before Freud by Marx, as the sign of what does not work in the real. If we are capable of operating on the symptom, it is because it is from the effect of the symbolic in the real.

Inasmuch as this knot, although only reflected in the imaginary, is indeed real, and encounters a certain number of inscriptions by which some surfaces respond, I can advance that the unconscious is what answers from (répond de) the symptom. Thenceforth, we shall see, it can be responsible for its reduction.

Marginal Note for Figure 3

It is obvious (!) that this kind of Borromean chain has an "end" - without which it is unknottable one by one (one-by-one round). For traction does not make a knot: dissociation of force and ex-sistence.

Thus there are two fashions to buckle it (in the "sense" of making it hold in a knot).

One is to close it in a circle. Which is true of all other Borromean chains. But this must be put aside for the moment.

The true Borromean chain remains open: cf. the three-linked chain.

There is nothing easier than to reproduce this three-linked chain with what we sketch here. Here is the flattening out that demonstrates it (fig. 7).

As soon as this chain is longer, if only by a single round, the round (F) that closes it here must double itself at the other end of the open Borromean "chain." It can also be filled in for in its function of One by what follows it: 1=2. Whence the privilege of the three-linked chain, which, as we shall see, distinguishes it from the four-linked chain, where the order begins to be no longer any whatsoever. We shall dot the i's concerning this.



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