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(All parts in English)

MILLER

LACAN

Today you are going to hear a communication from Jacques-Alain MILLER. This… which I warned you about last time, perhaps a bit too late, as part of the audience had already dispersed by the time I made the announcement… marks my desire for this curious name of seminar, which has been tied to my teaching since Sainte-Anne where it was held for ten years, as you know, to remain established.

Speaking only of the last two years here, some of you are not unaware – to your great displeasure – that I wanted this seminar to be held in an effective manner, believing that this effectiveness should be linked to a certain reduction of that audience, so numerous and so sympathetic, which you offer me through your assiduity and your attention.

And – my God – so much assiduity and attention truly deserve many considerations, which made very difficult for me the sorting required by the reduction of the audience. So that, in the end, your reduced number was not so reduced that, from the point of view of quantity, which plays such an important role in communication, things had properly speaking changed scale. Therefore, I will leave unresolved this year the solution to this difficult problem. Until further notice and without committing myself in any way, I will not close any of these Wednesdays, whether final, semi-final or otherwise.

I would merely like this name of seminar to be maintained in a more marked way than we saw at Sainte-Anne, where even in the very last years there were meetings during which I delegated speech to one or another of those who were following me at the time.

Nevertheless, some ambiguity remains that suspends this designation of seminar between the proper usage of a category – a place where something is to be exchanged, where the transmission, the dissemination of a doctrine is to manifest itself as such, that is to say, in the process of being conveyed – and I don’t know what other “usage,” not of the proper name – for the whole discussion of the proper name could begin there – but let’s say of an “eminent naming,” which “eminent naming” would become a “naming by irony.”

Hence, I believe that, in order to clearly indicate that this is not the state of affairs in which I intend the usage of this appellation to stabilize, you will periodically see a certain number of people take part, who will be inclined to do so.

Certainly Jacques-Alain MILLER, to inaugurate the continuation, has some claim to it this year: he has provided you in my book with that reasoned index of concepts which, from what I hear, has been very well received by many, who find great advantage in that Ariadne’s thread which allows them to navigate through this succession of articles, in which such and such a notion or concept – as the term is more judiciously employed – can be found at different levels.

A very small detail: I note, in response to a question I was asked, that in this index, the italic numbers indicate the essential passages, and the straight or Roman numerals indicate passages where the concept is involved in a more “in passing” way. It happens that on the page referred to, what is thus referenced may amount to a note in a single line, which says a lot about the care with which this very usable little device was constructed.

I am told that the book is – as one says in that Franglais that I, for one, do not repudiate – out of print, which means “épuisé.” I find “out of print” kinder, “épuisé” [Laughter]: one wonders what happened to it. I hope this out of print will not last too long. That’s what is called a success, but a sales success! Let us not prejudge the other kind of success, of which everything remains to be expected and which keeps the question open.

It may have been noticed that it is a book I did not rush to put into circulation. If I took so long to do it, one may ask the question “Why now? What is he expecting from it?” It is quite clear that the answer “May it serve you!” was no less valid a year or two ago, and even long before. The question is therefore not simple; it concerns everything regarding my relationship with something that here plays the role of a foundation… namely psychoanalysis in its incarnated form, we would quickly say, or else subjugated… in other words: with psychoanalysts themselves.

Several elements seemed to me to justify that what I was trying to construct should remain within a reserved field, allowing for the selection – which indeed occurred! – of those who would be willing to decide to recognize the consequences implied by the study of FREUD in their practice.

Ultimately, things never quite unfold in the calculated manner, in these difficult matters where resistance is not localized at what must be designated—in the narrow sense of the term—in analytic praxis; it takes another form where the social context is not without impact. This makes it delicate for me to explain myself before such a large audience.

This is precisely why, everything that concerns what I would call the external relations of my teaching… I do not consider any differently all that may manifest as hubbub and commotion around terms with which I do not view myself very favorably associated: such as “structuralism”… I am in no way disposed, unless forced to by some consequence of what I just now called “the success of the book,” to bite into a timed interval.

You see or feel, through your experience over recent years, that I have no time to waste if I am to state things before you at the level of the construction that I inaugurated in its style through my last seminar and at the point where I intended to establish the beginning of this logic that I must develop before you this year.

As this book does exist, nonetheless, along with the initial movements it provokes—which will be followed by others—and as the two or three points I just brought forth as principal—but there are others—risk remaining unresolved, I believe I must inform you that you will find the explanation, at least a sufficient one that may allow you to respond at least in part to the questions that for you might remain unanswered, in two interviews that will appear this week, if my information is correct, in places that are anything but a marketplace: Le Figaro Littéraire [1966-12-01: Interview in Le Figaro Littéraire] and Les Lettres françaises [1966-11-26: Interview with Pierre Daix] [Laughter] You may then know a little more.

Moreover, being unable to stop myself, every time I engage in one of these modes of external relation, from inserting a little of what is currently underway, it is possible that you will find here and there something that relates to our discourse this year.

I feel some scruple—as I said last time—about speaking to you of the repetition of the unary trait as instituting itself fundamentally, of this repetition of which one can say it occurs only once, which means that it is double, otherwise there would be no repetition. This, from the outset, for anyone who wants to dwell on it a bit, establishes at its most radical foundation, the division of the subject.

If I enunciated this notion before you last time, almost in passing, while at that conference at Johns Hopkins in October, I chewed it over for about three-quarters of an hour, perhaps it’s because I credit you more than I did my audience then, as certain echoes received since showed me that the structuralist ear, whoever may be its bearers, is capable of showing itself a bit hard of hearing! [Laughter] In even more unexpected places, where you might perhaps…

X in the audience: “We can’t hear!”

LACAN

What? Who can’t hear? How long have you not been hearing anything? [Laughter]

…In even more unexpected places, you may perhaps find on these various themes, including these little indication-beginnings that never come too early, on certain themes that I will later have to develop. And for example on the function of the preconscious, which—curiously enough—it seems no one has been attending to for quite some time… since everything has been jumbled together, in the belief that it remains distinguished… from the functions FREUD had reserved for it. The preconscious slipped in during one of those interviews, I can’t remember which, to which two others must be added, likely unexpected for you I think: they will be held on the O.R.T.F.

One, next Friday at 10:45 AM, during what I have been assured is “prime time” [Laughter]. I’m willing to believe it, but I think you’ll all be at the hospital. Anyway… you’ll arrange yourselves as best you can and I hope to be able to share the text, if Radio is willing to give me permission. The second interview will take place on Monday. There’s urgency, as you can see. The first is with Georges CHARBONNIER, who kindly gave me the spot, and the second is with Mr. Pierre DAIX, thanks to whom you will perhaps get something livelier than the first, since it will be a dialogue with the person most qualified to sustain it, namely François WAHL—who is here—and who kindly agreed to engage in this exercise with me.

X in the audience: at what time?

That… I promise nothing! Apparently it starts at 6:15, only, the discussion is not only about my book and I can’t tell you at what point this will appear between a quarter past six and seven o’clock, as everyone has their fifteen minutes… What is it, dear Irène?

Irène PERRIER-ROUBLEF: Is it at six in the morning?

It’s a prime time that is usually accompanied by calisthenics. [Laughter] Well, we’ll see how all this unfolds…

Before giving the floor to Jacques-Alain MILLER, I want to share something very amusing, which was brought to me by a faithful follower: a communication from a specialized journal, which mentions both IBM machines and what is being done with them on an experimental level at the Massachusetts Institute of Technology—M.I.T. as it is commonly called—and tells us about the use of one of these high-level machines as it is now being employed, to which has been given—not without reason—the name Elisa, or at least it is called Elisa in the context of the usage I am about to describe.

Elisa is, in a well-known play, “Pygmalion”, the person who is taught “proper speech”… while she is a little flower girl selling bouquets in the busiest streets of London… and is being trained to express herself in high society, without it being noticeable that she does not belong to it.

Something of that order emerges with this said machine. In truth, that is not exactly the point. That a machine is capable of giving articulated responses, simply when spoken to—I do not say when interrogated—now appears to be a game, which raises the question of what may happen: of obtaining these responses in the one who speaks to it.

The matter is not articulated in a way that completely satisfies a situation so usable for us, offering such an interesting reference within the ongoing discourse. It is not stated in a way that accounts for the framework in which we might insert it. Nevertheless, it is very interesting because, in the end, something is suggested which could be considered a therapeutic function of the machine. To say it plainly, it is nothing less than the analogue of a transference that could arise in this relation, a question which is indeed raised. This, which did not displease me, is not unrelated to everything I leave open concerning the way I must handle the dissemination of what you call my teaching.

And I would like you to find there the handling of a first symbolic chain, the notion of which psychoanalysts had to conceive. A notion to which their minds needed to accommodate themselves, in order to properly center upon what FREUD called recollection, and which would provide them the subjective model of the construction of this symbolic chain, and of its own kind of memory.

A memory that is indisputably consistent and even insistent, articulated in what now appears in this book, in the second chapter, in the second moment: in the inverted position where the “Introduction to The Purloined Letter”, which precedes it, is fixed—that is to say, just after The Purloined Letter. I remind those who were listening to me at the time that this construction, like all the others, was made before them and for them, step by step, and that I began with an examination, starting from a text by POE, namely the way the mind works on this theme: “Can one win at the game of even or odd?”

My second step had been to imagine a machine of this nature. What is actually produced today differs in no way from what I had articulated then, except that the machine is supposed by the subject to be equipped with programming such that it takes into account gains and losses.

Starting from this [Cf. The Purloined Letter, Écrits]…
— that the subject would interrogate it—the said machine—by playing the game of even or odd with it,
— and from this sole supposition, that it retains at least for a certain number of moves, the memory of its gains and losses,
…one can construct this sequence of +, +, –, +, –, … which, encompassed and grouped within a parenthesis of a typical length, and which shifts one rank each time, allows us to establish that trajectory I constructed, upon which I founded this most elementary type of model: “that we do not need to consider memory under the register of physiological impression, but only of symbolic memorial.”

And this, starting from a hypothetical game with what perhaps was not yet operational at that level at the time, but which nonetheless existed as such, as an electronic machine—that is to say, just as well as something that can be written on paper—that is the modern definition of the machine.

It is from there… thus well before this came fully to the forefront of the concerns of engineers, who devote themselves to these devices, as you know, always progressing, since nothing less is expected from them than automatic translation… it is from there, that fifteen years ago, I constructed a first model for the specific use of psychoanalysts, with the aim of producing in their mind this sort of necessary detachment from the idea that the functioning of the signifier is necessarily the flower of consciousness, which was, at the time, to be introduced in a step that was absolutely without precedent.

The floor is now given to Jacques-Alain MILLER.

Jacques-Alain MILLER: The Equations of Thought.

For KANT, what is unthinkable in SPINOZA’s system is summed up in this proposition:

“Spinozism speaks of thoughts that think themselves.”

That there are “thoughts that think themselves”—let us say that it is in accepting and in hearing this that FREUD’s discovery has summoned us.

That there are “thoughts that think themselves” receives from FICHTE the name of “postulate of unreason.” This is undoubtedly an expression that should hold our attention, as it marks, unambiguously, the limit of the philosophy of subjectivity, in its inability to conceive of any thought that would not be the act of a subject.

On the contrary, to articulate “the laws of the thought that thinks itself” requires of us that we constitute categories radically incompatible with those of thought conceived by the subject. That is why we will here make use of what has been developed in a domain of science where it has been a matter, from the beginning, of thoughts that think themselves: that articulate themselves in the absence of a subject who animates them.

This domain of science is mathematical logic. Let us say that we must regard mathematical logic as pure logic, for the theoretical game in which are reflected “the laws of the thought that thinks itself” outside the subjectivity of the subject.

Now, it should be noted that the constitution of the domain of mathematical logic was carried out by the progressive exclusion of the psychological dimension, in which it had previously seemed possible to derive the genesis of the elements of specifically logical categories. Let us recall that, from our perspective, the exclusion of psychology leaves us free to follow, in this field, the traces where is marked what must be called “the passage of the subject,” in a definition that owes nothing more to the philosophy of the cogito, insofar as it relates the concept of the subject not to its subjectivity but to its subjection.

In what way does mathematical logic prove to be suited to our reading? Well, in this: that the autonomy and self-sufficiency it strives to ensure for its symbolism make all the more manifest the articulations where the mark of its functioning falters. It is thus quite simply insofar as they articulate, unknowingly, the suggestion of the subjectivity of the subject, that the laws of mathematical logic can hold our attention here.

This is what authorizes me to bring forth, from the origin of mathematical logic, an expression which it has long since abandoned. To propose this expression to you as my subject, I will try to speak a little, partially, about the “equations of thought.”

To retrieve this expression, we must push our reading beyond the formalized apparatus of modern logic. To find it exactly in the first founder of mathematical logic—of whom FREUD is only the second—let us go back to the discovery of George BOOLE [1815–1864]: that algebra can formulate logical relations. The discovery is properly theoretical.

Because algebraic formalization frees itself from the field of numbers, which becomes then only one of its specifications, it liberates mathematical formalization to state that symbolization properly speaking is not dependent on the interpretation of symbols, but only on the laws of their combination.

In doing so, BOOLE strives to establish that the laws of thought are subject to a mathematics, in the same way as the quantitative conceptions of space and time, of number and magnitude. However, although logic indeed recognizes BOOLE’s first book: Mathematical Analysis of Logic as the inaugural event of its history, BOOLE’s second book: An Investigation of the Laws of Thought holds no longer any place in the memory of logical science.

BOOLE, in returning to what logic neglects in its history, will make us aware of what it fails to recognize regarding the conditions of its practice, thereby revealing to us certain of the laws of logic which operate in these places. Logic which, as you know, rises above logician’s logic. This logic, the logic of the signifier, Jean-Claude MILNER and I have had the opportunity to present, in relation to Plato’s Sophist and the Grundlagen, certain elements.

If I am continuing its presentation today, it is undoubtedly because the subject of this year’s lectures by Dr. LACAN lends itself to it, and also because our formal construction has proven, for the psychoanalyst, to be sufficiently manageable to be interpreted freely within the Freudian field. That such an interpretation is possible eminently justifies the constitution of our symbolism and the presentation we have made of it, as a calculus of the subject.

Let us move on to BOOLE’s doctrine, to say right away that he does not innovate, since he conceives language as the product and the instrument of thought, and since he gives the sign as an arbitrary mark. That is to say, meaning is produced from the linking of a word and an idea, or of a word and a thing. You know that these two possibilities are not at all equivalent. For BOOLE, they are equivalent.

Which means that communication is then ensured only by the permanence of an association:
— nothing there but what is entirely classical,
— nothing there that exceeds the Lockean doctrine of language.

Only, let us come to the proposition that founds BOOLE’s enterprise. All operations of language as an instrument of reasoning can be carried out in a system of signs. Of course, all languages—the languages we speak—are systems of signs. But what specifies the sign employed by the algebra of logic is that it can be nothing more than a letter or a simple mark. And this is permitted by the theory of the arbitrariness of the sign. But this is the first time that one properly uses a sign.

It is now necessary to learn—and this can be done fairly quickly in elementary fashion—BOOLE’s symbolism. Let us say that there are three categories of signs to be established:
— firstly: the symbolic letters, which have the function of representing things as objects of our conceptions, which mark things as objects of representation.
— secondly: there are the operational signs: “+”, “–”, “×”, which have the function of representing the operations of the understanding by which our representations are combined and reshaped into new representations.
— thirdly, and this is not the least important: the sign of identity.

1. The symbolic letters.

Let us say that the sign X or the sign Y represents a class of things to which a particular name, or a property, can be attributed. So, let us picture a circle with a certain number of objects, of a certain name or a certain property. This class will be named X.

We will say that the combination X×Y—one may write X.Y—represents the class of objects to which the names and properties of X and Y are simultaneously applicable: the intersection of X.Y.

One may first note that the order of the symbols is indifferent. One may write X.Y = Y.X, that is to say, symbolic letters are commutative. But BOOLE insists that this concerns a law of thought here, and not of nature, and not simply a law of arithmetic either.

2. The operational signs.

Next, one may obtain from BOOLE a certain number of other laws, which moreover are not far from the laws of arithmetic, but which take them up again within the arc of logic:
— one may introduce the sign (+): this will be the sign of the class that unites, for example, the classes X and Y.
— one may introduce the sign (–), which will indicate that one removes from a class a part of its elements.

[Lacan illustrates on the board]

A – B (in gray)

Let us suppose that X and Y have the same meaning. Since the combination of the two symbols expresses the entirety of the class of objects to which the names or properties represented by X and Y are jointly applicable, this combination expresses nothing more than one of the two symbols.

This seems very simple. You will see with what ingenuity BOOLE draws from it a law, which he calls fundamental for thought.

LACAN – Simply, to complete the difference, which is not exactly what you have in mind. [Lacan explains the illustration]

Jacques-Alain MILLER

If the two symbols say nothing more than one of the two: X.Y = X, as Y has the same meaning as X, one may state: X.X = X

This is particularly simple. One can still write this by applying a rule that will translate a symbolism. One can write this entirely innocuous law: X² = X

Since all of this is extraordinarily simple, it is necessary—each time—to emphasize that it is important. This formula X² = X is, in the algebra of logic, given as a major law of thought. What we must say about it is that it governs, in some way, everything that can be defined as belonging to the dimension of meaning.

We must first recall that all the symbols which are to function, in the algebra of logic, as representations of the laws of thought are subjected to this law. If there is not a subject common to logic and arithmetic, there is a community of formal laws. That is the basis upon which BOOLE’s algebra proceeds.

That is why, once we have this formula, we must seek to interpret it through numbers. Now, it becomes immediately apparent that only two numbers are capable of interpreting this formula in a way that satisfies arithmetic. It is quite evident that the only two numbers which can interpret this formula are 0 and 1.

However, one must not believe that all the Xs found in logic, in this logic of thought, must be interpreted by 0 and 1. But it must be said that only 0 and 1 respond, in numeration, to the Boolean law of thought, which we have called the “law of meaning.” From now on, let us say that it is arithmetic that will guide logic.

Let us examine the arithmetic properties of 0.

The simplest: 0 × Y = 0, whatever Y may represent.
That means that class 0 multiplied by Y is identical to the class represented by 0. In other words, there is only one possible interpretation of 0: 0 represents nothing, but this 0 which represents nothing is a class.

Let us now examine the arithmetic property of 1: 1 × Y = Y.

The symbol 1 represents and can only represent a class such that all individuals—whatever the class X may be—are also its members. Result: this class can only be the universe, defined as the class in which are included all individuals of any class whatsoever.

Here you see emerging the category of the universe of discourse about which Dr. LACAN was speaking to you last time. You see it here, through BOOLE, deduced from the most elementary symbolism.

Let us continue in BOOLE’s elaboration.

Let X now be any class. If 1 represents the universe, it is clear that 1 – X is the complement of X: it is the class containing the objects that are not included in class X. We are now going to make a very simple transformation of this formula: X² = X. It is enough to move one of the members of this equation to the other side of the “=” sign.

You will have two possibilities, BOOLE chooses only one. One can obviously move X to the side of X² or the opposite.
[X² = X ↔ X – X² = 0, or: X² = X ↔ X² – X = 0]
BOOLE chooses only one of these two possibilities. The other is dropped: he will never speak of it again.

     X – X² = 0

This is the derivation and transformation that BOOLE chooses. And from it he deduces another formula, just as simply:

     X × (1 – X) = 0

There is no intersection between 1 – X and X, which therefore means, just as simply, that it is impossible for a being to possess a quality and not to possess it at the same time.

From this law: X² = X one derives, through this interpretation, the statement of the principle of contradiction, given by BOOLE as a consequence of “the fundamental equation of thought.”

In other words, in this order that he follows, the constitution of thought precedes this principle of contradiction.

One may say that these Xs and Ys have been interpreted as classes, but could be interpreted otherwise. In these conditions, the multiplication which gives us X², this multiplication of X by itself—what is it, if not the operation by which a thing—any thing—comes to signify itself, and by which every sign comes to signify itself?

3. The sign of identity.

This formula X² = X is a more elaborate form than a formulation of the principle of identity. But a formulation such that it reveals this, which should not be indifferent to us: that identity supposes the duality of the element identical to itself in the operation of signifying itself. That means… and for those familiar with Dr. LACAN’s system, this is not a proposition without resonance… there is no identity to self without alterity.

In other words, what is the interest one can take in BOOLE’s equation? It is this: that it reveals, through its formula X = X², that the meaning of an element, within the universe of discourse, implies its reduplication, and that its identity to itself is nothing other than the reduction of its double to itself.

To clarify the ideas, let us say—after BOOLE—that this “law of meaning,” the “fundamental law of thought,” as BOOLE calls it, is a second-degree equation. It is obviously the most concise formulation one could give of a principle that has, in a way, governed a large part of Western philosophy.

That thought operates, in meaning, only according to this second-degree equation means that dichotomy is the process of all analysis in meaning, from which one could deduce—we will not do so here, but it is quite simple—that binarism is not a contemporary avatar of reflection, of analysis, but that it is already inscribed in this duality.

BOOLE refuses to make a supposition, saying that one cannot conceive of a thought that would be governed or expressed by a third-degree equation. One cannot even conceive what that would be. Why is the equation X = X³, for example, not interpretable in the algebra of logic? It is not interpretable because, in whatever way one transforms this equation, it involves two terms that are not interpretable in the algebra of logic:
— on the one hand, the expression—and one must note the word “expression”—“1 + X”,
— on the other hand, the symbol “–1”.

Now, the symbol “–1” can already appear a bit earlier in the derivation that BOOLE did not make from his formula. Indeed, he chose to say: X – X² = 0. If he had said: X² – X = 0, we would have had: X × (X – 1) = 0, the “–1” would already have been present there.

He excluded one of the two possible transformations that could have been made!

It is only at the level of X = X³ that he encounters this “–1” again. Why must the symbol—I am not referring here to the interpretation given to it as “universe”—why must the symbol itself, “–1”, be excluded from the field of logic?

Quite simply because it does not follow the law X² = X. In other words, to draw the most straightforward, immediate conclusion from BOOLE’s text: at the origin of mathematical logic, at the very point where it is founded, the exclusion of the symbol “–1” is consummated.

Why? According to the law: because it is the very symbol of the non-identical to itself, insofar as it does not follow this law of identity, of non-contradiction in the order of meaning.

Why is the expression “1 + X” also excluded? It is excluded because—as BOOLE says—one cannot conceive of the addition of anything to the universe. Now, in “1 + X”, the “1” represents the universe, with X being the element that comes as a surplus to this universe. In fact, in the formula “1 + X”, it is X that represents a unity, a single element.

Therefore, what cannot be accepted in mathematical logic, at the point where it is truly constituted, is the excess of an element over the universe, the excess of what one may call a “+1”, or “1 more”.

Let us say then, just as simply as we previously spoke of –1, that at the origin of mathematical logic, the exclusion of “+1” is consummated—the symbol of that which is outside of meaning, or outside the signified, and of the non-representable insofar as it exceeds the totality of the universe.

Now, it may be apparent that these two exclusions are but one: it is the same position occupied by the “1 in excess” and the “1 in default,” with respect both to meaning and to reality. That is to say, both with respect to the universe of discourse and to the universe of things that corresponds to it.

One may express the conjunction of these two exclusions, their unity, through this formula: “that in the order of meaning, the surplus is lacking.”

Without going any further, one may develop this—let us call it a “law of the sign,” as an element of meaning. It is enough to say that in meaning, signs endowed with meaning are constituted so as to obey BOOLE’s law, but that the signifier, as the material of the sign, or as an element outside the signified, does not obey it.

There we find again an axiom ultimately repeated many times here: “that the signifier does not signify itself,” which is properly the counterpoint of BOOLE’s law, but this allows us to understand that the signifier is not constituted in the image of the meaning it supports.

One may have a very simple formula to remember this, since the multiplication of –1 by itself does not yield –1.

But if one wishes—BOOLE interpreted it thus: –1 × (–1) = 1 + 1—this multiplication inverts the factor; let us interpret it thus: it institutes the order of the signified as the inverse of the order of the signifier, in that the signifier repeats itself and can only repeat itself: –1, –1, –1,… whereas the meaning can be multiplied—that is to say, can be redoubled.

Let us say, to provide what is perhaps no longer a mere image—that the chain of the signifier must be thought as constituted by a concatenation of –1s, of units constituted as –1s, of “catenations,” but let us say they are units, to generalize Dr. LACAN’s term: “units of unary type.”

We have produced or made appear a category that is the + or –1. It is now necessary to understand precisely by what path it imposes itself upon the order of meaning. To connect these two laws—of the meaning of the sign and the meaning of the signifier—it would be necessary to show that the + or –1 is produced by all meaning insofar as it presupposes an operation of duplication.

One can begin, to explain this, with the relations between thought and consciousness and, let us say, with what reflection is. To understand this, one can begin by seeking a mathematical definition of reflection or reflexivity. Let us borrow it from RUSSELL, in the Introduction to Mathematical Philosophy.

What he says is simple: a class… one should perhaps say a collection, or a set… is reflexive if it is a class similar to a part of itself. That means that a part of this collection can mirror the whole, or again, that the similarity between these two sets—the part and the whole—consists in the possibility of joining to every element of the whole an element of its part, putting them in one-to-one correspondence. Reflexivity is a property of an infinite collection. One can exemplify it through the countable infinity of wholes, of natural numbers.

One can pair each natural number with the even numbers. That is, pair 1 with 2, 2 with 4, 3 with 6, and so on to infinity. One can apply the set of all even and odd numbers to the even numbers alone. There is, if you will, the same number of even numbers on one hand, and odd numbers on the other. This property characterizes the infinite collection.

Let us say that what characterizes the cardinal number of this collection, to give a simple characteristic, is that it remains unchanged by the addition or subtraction of a unit or several.

Take a unit: what characterizes, let us say, the number n of such a collection is that n = n + 1 just as well as n = n – 1. Moreover, the two propositions mean exactly the same thing. All this is elementary in theory. I recall it only to highlight and emphasize these +1s and –1s.

If in SPINOZA there are thoughts that think themselves within the divine understanding, it is precisely because the divine understanding is infinite. So there are as many ideas as there are ideas of ideas, etc. In the same way that the even numbers are ideas of ideas, the even and odd numbers are the sum of ideas and of the ideas that reflect them.

GOD, if he is conscious of his ideas, is not conscious of himself—that is, he is not a person. He is conscious of his ideas by the reflexive property of that infinite set of his infinite understanding. Yet, if there is something one calls a “whole” and something one calls a “part,” there must at least be a small difference between the two—the simple difference that maintains the opposition of the part to the whole.

This set must indeed obey the law: n = n – 1. Let us say, for greater clarity, that there is no reflection unless something of the “whole” falls outside of the reflection—a single element of the whole. This is what one sees when placing all the natural numbers in correspondence with all the natural numbers minus 1.

One must necessarily drop at least one element at the beginning for this inflection to take place, for it to make sense. We will not focus here on the fact that often it is the 0 of the sequence that is paired with 1. Thus, 0 no longer has reflection. It is enough to say that one element falls. And what does this fallen element represent? It represents the difference between the whole and the part. That is to say, in a way, the whole itself falls—or the totality of the whole.

In other words, to be conscious of one’s ideas in the Spinozist mode implies that there is no consciousness and that there is an infinite understanding. Of course, this rests on that type of reflection that SARTRE calls “the requirement of reflection as positional consciousness.” Which presupposes this model of a one-to-one link between an idea and the consciousness of the idea. Which presupposes a one-to-one correspondence between the idea and the idea of the idea, modeled on SPINOZA’s reflection.

Now, in Being and Nothingness—pp. 8–19—SARTRE demands that we avoid what he calls an “infinite regression.” He has no other word to condemn this “infinite regression” than the word “absurd.”

“It is necessary,” he says, “if we want to avoid infinite regression, that self-consciousness be an immediate and non-cognitive relation of the self to itself.”

One may formulate this in terms that are not exactly SARTRE’s and even noticeably displace them.

SARTRE says: “if we want to avoid…” If one excludes the possibility of “an infinite understanding” and if one wishes to obtain “self-consciousness,” one must produce reflection: an element such that it relates to itself without duplicating itself. It is, said SARTRE, the non-thetic consciousness of self, non-positional, of a type opposite to the Spinozist type, which no longer presupposes one element here and another there.

And he writes:

“If the primary consciousness of primary consciousness…
—which is somewhat mysterious here—
is not positional, it is because it is one with the consciousness of which it is conscious.”

By taking this text forcefully, literally, imposing on SARTRE a schema that is not his own—the schema of the univocal—if one tries to think SARTRE’s text from the perspective of the one-to-one correspondence in reflection, one must say that if the element called “consciousness of consciousness” is one with the consciousness of which it is conscious, if there truly is the possibility of unity between the one and the other, then this element called “consciousness of consciousness,” or non-positional consciousness of self, is constituted as an ego, an ego that—as SARTRE said—takes its disguises of style from what it lacks in being, another formulation that I have not cited.

At the same time, if something like a “consciousness of consciousness” manifests itself, it must be said that in the field of reflection it is a phenomenon of aberration, a misstep or a surplus element breaking the one-to-one correspondence of ideas and ideas of the idea.

What can be said of this element “consciousness of consciousness,” if not that it occupies the position of a point of reflection, such that it alone must bear the difference between the whole and the part. By itself alone, it assumes the reflective property of the infinite collection. This point is, in a way, within conscious thought, within its space, a point at infinity. It is there that the infinite collection posited by SPINOZA collapses.

And the aberrations, and the absence of this point, are clearly marked by a category SARTRE uses here and there in relation to bad faith, which is the category of evanescence. This point is evanescent… We shall rather say that this point, in reflection, necessarily wavers from +1 to –1.

And that, in this wavering, one must recognize a being evidently heterogeneous, as much with respect to reality as to reflection, a being:
— always in excess of reality and reflection when it comes to identify itself,
— always in default of them when it separates itself.
This heterogeneous being, let us call it the being of the subject.

It was among our intentions to complete this a bit by examining the principle of the vicious circle, where one can grasp, let us say in its naked state, the birth of this “+1,” product of that “one too many” produced by meaning. To go very quickly, let us say that this principle, and all that pertains to the totality of a collection, must not be an “element of the collection.” That which disposes the totality of a collection cannot be internal to that collection.

Which means:
— one cannot predicate about a collection except from outside it,
— or again, one can only think the unity of a collection from outside that collection.

To grasp a collection as a whole presupposes that one encircles it. This encirclement itself is the unity of the collection. The boundary of every collection is an element produced in excess by all predication, all discourse on the collection. The collection can only be signified as such from the standpoint of “the one in excess.”

From this formula, one may also derive the following:

“That the one in excess is lacking from the elements of the collection for that collection to be closed.”

One may interpret it as an uncountable, a beyond-signified, to which meaning refers, insofar as it superimposes a duplication. This is to indicate in what way one must contradict BOOLE’s equation, which nevertheless remains fundamental.

And one could complete it with an examination of RUSSELL’s theory of types. But this examination has already been partially carried out by Dr. LACAN on the “I lie,” which he sees produced, through RUSSELL’s theory of types, by a division of the subject: the “I lie” may be understood within truth, within the element of truth, on the condition that the “I” is doubled.

This division of the subject produced by truth, this division of the subject which responds in a slightly inflected sense to BACHELARD’s formula: “All value divides the valuing subject.”

This division of the subject, I believe I have said enough for it not to be confused—this matters for theory—with duplication within meaning.