Žižek via Lacan: Formule no° 4 (1962) — Hal Odetta

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One pluckable conceptual gem in Žižek’s Less Than Nothing is in that interlude on Quentin Meillassoux (“Correlationism and Its Discontents”). It argues that the starting question of Hegel/Lacan is what made all the difference in their way of doing a materialist stance in philosophy as compared to all other thinkers of today claiming the same stance. Being Lacanian, the key idea that Žižek mobilizes here is, of course, the objet a in relation to the concept of the “Real”.

One sentence in that interlude stands out for being placed in parenthesis. It is like a moment when Žižek, who mostly uses Lacan in a “working mode”, paused and historicized Lacan. This is the sentence: “(Once Lacan got this point, he changed the status of the objet a from imaginary to Real.)” So when exactly can we date textually the trace of this event when Lacan finally “got this point”?

There is a set of “formulae” (les formules), composed of four formal statements (1), in Lacan’s lecture on the 21st of November 1962. This is the second lecture for this annual series and there is an adumbration of this formula-set in his first lecture of that year: “I thought it necessary … to launch, in advance, a formula indicating the essential relationship between anxiety and the desire of the Other” (November 14, 1962). And so midway in his second lecture, and without much introduction, Lacan simply said:

Shall I now introduce the formulae I’ve written up? I’m not claiming, far from it, that the bag of tricks will spill right open. I ask you, today like last time, to jot them down. That’s why this year I’m noting things up on the blackboard. You’ll be seeing how they function afterwards.

These formulae did not fare well in their afterlife as compared to the more well-known and studied “formulae of sexuation”, perfected a decade later. What perhaps added to its neglect is that while the first three equations (2) have accompanying discussions (albeit short and introductory), the fourth formula (labelled as Formule no° 4 in the 2004 published version of this lecture) (3), was not given an explanation nor elaborated in the course of this annual seminar. This is the only thing said that day that is directly related to that formula:

The formula labelled fourth … is not Hegel’s truth, but the truth of anxiety which, for its part, can only be grasped with reference to the second formula, which concerns desire at the psychoanalytic level. (Emphasis mine.)

I tried checking if this formula was taken up again in the succeeding years, down to his last seminar, but it does not seem to have been so (4).

Lacan’s “les formules”

Figure 1 gives the four equations (formal statements) of Lacan. There are two statements comprising the fourth equation, but as Lacan said: “those that are bracketed together are just two ways of writing the same thing, in one direction, then in the palindromic direction”.

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Figure 1. The four/five les formules: T1, p. 31; T2, p. 34-35; T3, p. II 11; T4, p. 24-25. This figure is largely lifted from T1, but corrected on the term placed on gray square-lines (no. 4 above). T1, 3 ’s d (A) missed the mark for “barred A” and was corrected in T2, 4. T1 is used here, with my incorporated correction, because it presented the formulae in one group-list and with the inequality operator lined up together, clearly separating the two columns.

What are these equations about? These are statements that express in compact forms the four important relations in Lacan’s central thematic project: the dynamics in the “field of desire”. As seen in the central operator lined-up in Figure 1 (<, >), these are statements about inequalities of the given terms on both sides of the columns; in short, these are “inequality equations” (5).

“The side of the Other” and “my side”

Although these set-equations were given first before Lacan presented what is then called as the “first” and “second” ‘tables of division’ (see Figure 2, which fuses the two tables), these two tables set the general framework for the set-equations.

Let us quickly check Figure 2. The two columns show the structural locations occupied by “A” (the “Other”) and “S” (the yet-to-emerge “Subject”). Figure 2 diagrams the “division” of one term by the other term—“S” is “divided” by “A”, resulting to what is written as “barred S” (we will write it here as ~S); and likewise for “A” (as ~A). The dual “division” that happened is marked by the transposed locations for both (as seen in the second line of the table). One can now say that the left column is “the desiring/speaking subject as situated in the location of the Other” and the right column is “the (subjectified) Other as situated in the location of the Subject”. There is a “remainder” in the instantiated process of “division”: for ~S, the objet a. Below the emergent ~A is the symbol 0 (“zero”) (6). We will see below what function this “zero” will subsequently play in the process of the fourth formal statement of Lacan.

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Figure 2. The opération/schéma/table of division (l’opération de la division; schéma de la division): T1, p. 34 [p. 141]; T2, p. 37 [p. 135]; T3, p. II 11 [p. IX 77]; T4, p. 26 [p. 114]. Left column: côté de l’Autre, “the side of the Other”; mon côté, “my side”.
First equation: Violence in unmediated desire

This is how the first equation is written: d (a) : d (A) < a. Here, as subsequently, the operator d is explicitly mentioned as a notation for “desire” (in strict Lacanian vocabulary, this is always “desire for desire”): so that, d (a) is “desire of/for a”, and so on. The “colon” [:] symbol functions like the ratio/fraction form of relation (one desire relative to another desire). The first equation formalizes desire in the sense of Lacan’s “Hegel” (also Kojeve’s Hegel, but not Žižek’s). Desire—d (a)—is directly in “ratio” with d (A) and both never measures up to a or objet a. Desire here appears in its brutal simplicity: there is direct confrontation of structurally divided desires and there is no “mediation” but, as Lacan notes, “that of violence”:

I can’t stand myself acknowledged in the only type of acknowledgement I can obtain. Therefore it has to be settled at any cost between our two consciousnesses. There’s no longer any mediation but that of violence. Such is desire’s lot in Hegel.

Second equation: Mediating desire by an image

This is how the second equation is written: d (a) < i (a) : d (~A). The second formal statement contains again three terms, distributed (as in all the other statements) on both sides of the inequality operator: a, ~A, and one first-appearing term: i (a). This last term is well-discussed in many earlier Seminar lectures: it is an “image” or fantasy-image of the objet a. In this mode, the “field of desire” is no longer given as a field with a totalizing symbolic order (A is replaced by ~A). In contrast to the first mode of desiring, desire now stands in unequal relation to the dialectic of the “image” or fantasy in relation to the “barred Other” (7). One can say that here desire is oriented towards a mediating (fantasy-generating) image.

Third equation: Anxiety grounds desire

This is how the third equation is written: d (x) : d (A) < x. One can see intuitively the homology of the forms of the two statements (the first and the third). The third statement simply replaces the term objet a with the notation for “anxiety”: x. According to Lacan, because the truth of “anxiety” is not recognized or conceptualized in the first mode, the fault/limit of that structure is that it is:

too tightly focused on the imaginary. It’s very nice to say that the slave’s servitude is brimming with the whole future right up to absolute knowledge, but politically this means that till the end of time the slave will remain a slave (Emphasis mine).

We can, without exaggeration, say that the whole seminar on “anxiety” that Lacan is unpacking at this point in time is an expanded treatment of this third equation, including the need to discuss the varied forms of the objet a since these also have corresponding modes of “anxiety”. Lacan also added, with great clarificatory economy, this contrastive line for this third equation: “Kierkegaard is the one who imparts the truth of the [the first] formula.”

What is Formule no° 4 about? Drive!

Our entry-point for our reading of the fourth formula is to note a puzzle. Figure 3 shows the supposed ‘palindromic’ statements of the fourth formula. If Lacan treats these two as “two ways of writing the same thing”, the puzzle is why the second statement is not in this expected form, d (~A) : 0 > d (0)? But the “change” here should be taken in as it was written: we then have to affirm the equivalence of these two terms: ~A = a. Why is this conceptually correct?

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Figure 3. Puzzle in Formule no° 4’s dual forms: ~A becomes a. Given that Lacan explicitly said that the second statement is just a ‘palindromic’ inverse form of the first, why is the term d (~A) here in the first statement transformed into d (a) in the second statement?

Before resolving that puzzle, we will first make some transformations (Figure 4)—using the standard ratio/fraction rules (a/b < c/d = a × d <  b × c)—so as to see some dimensions clearly, in simplified terms, if these are not readily visible in the original forms. We will skip discussing the rest of the transformations. What is immediately of value for us at present, in Figure 4, is the fourth equation on the right that we have derived: d (a) > 0.

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Figure 4. Transformed equations. The form expressing “barred A” in Figures 1 and 2 above are here written in this form: ~A. Technically, the last statement in the right column above should be written as d (a) > 0 . d (0). The terms 0 . d (0) are here, for conceptual reasons, simplified notationally into 0 (“zero”, or point zéro). The terms can be read as: “the point zéro multiplied by the desire for point zéro”.

Given the equivalence noted above (a = ~A) we can, therefore, write in two forms the derived equation, thus: d (a) > 0 and d (~A) > 0.

These two forms correspond to the two sides of the “table of division” and the sides of the lined-up columns of the inequality equations: d (a) > 0 is the form under the horizon of the self/subject (“my side”); d (~A) > 0 is the form under the horizon of “the side of the Other”. But what are these equations or “images” (8) conceptualizing?

“Masculine” drive: d (a) > 0, “Feminine” drive: d (~A) > 0

This is this essay’s argument. Although derived from Lacan’s unexplained fourth formula of 1962, these two equations makes sense when interpreted as “drive”, especially as it is explicitly defined by Žižek in his 2017 work, Incontinence of the Void. There Žižek differentiated “the two aspects of objet a”. Objet a is not only this “fascinating element that fills in this void”, as it works in the dynamics of desire; objet a can also function “as the void [itself] around which desires and/or drives circulate” (Incontinence of the Void, p. 220, Emphasis mine).

So when Žižek says that, in order to “break the spell of objet a”, one has “to recognize beneath the fascinating agalma, the Grail of desire, the void that it covers” (ibid., Emphasis mine), is there a more economical way of writing this “point of recognition” than by this formula: d (a) > 0? The equation can be read as saying that, at this point, desire for a is now oriented towards—and gaining the upper hand over—the “zero point” and not anymore towards objet a as a ‘filler of the void’.

And, moving on, when Žižek says that the above point “is homologous to the feminine subject’s shift from [the phallic signifier] to the signifier of the barred Other in Lacan’s graph of sexuation” (ibid., Emphasis added), is there another way of writing that than by this formula: d (~A) > 0? The equation can be read as saying that, at this point, desire is now not only oriented towards—and gaining the upper hand over—the “zero point” (and not anymore towards objet a as a ‘filler of the void’), the objet a is here also replaced by (the signifier of) “the barred Other”.

Here is how Žižek defined “drive” relative to objet a (9):

The Lacanian objet a as the object which overlaps with its loss, which emerges at the very moment of its loss (so that all its fantasmatic incarnations, from breast to voice and gaze, are metonymic figurations of the void, of nothing), remains within the horizon of desire—the true object-cause of desire is the void filled in by its fantasmatic incarnations. While, as Lacan emphasizes, the objet a is also the object of the drive, the relationship is here thoroughly different: although, in both cases, the link between object and loss is crucial, in the case of the objet a as the object-cause of desire, we have an object which is originally lost, which coincides with its own loss, which emerges as lost, while, in the case of the objet a as the object of the drive, the “object” is directly the loss itself-in the shift from desire to drive, we pass from the lost object to loss itself as an object. In other words, the weird movement called “drive” is not driven by the “impossible” quest for the lost object; it is a push to directly enact the “loss”—the gap, cut, distance—itself. There is thus a double distinction to be drawn here: not only between the objet a in its fantasmatic and post-fantasmatic status, but also, within this post-fantasmatic domain itself, between the lost object-cause of desire and the object-loss of the drive (Emphasis mine).

Reading Žižek via Lacan’s fourth formula gives us this pointer: “object-loss of the drive” (as a mode of objet a) → (“points to”) un point zéro (10). The simplicity of Lacan’s fourth formula, which we take here as the formula for “drive”, comes from its use of the symbol 0 (“zero point”).

Echoing that parenthetical sentence of Žižek that we cited at the beginning of this essay, this 0 (un point zéro) can be taken as Lacan’s notational mark for objet a, specifically at the fourth phase/equation of the dialectics of desire, when—as Žižek would be putting it more than a half-century later—“it has changed its status from imaginary to Real”.

A brief note on mathemes (and lalangues) at the end of the wor(l)d

What is the nature of a Lacanian matheme or “algebraic formula”? Lacan is quite careful in distinguishing here what he is doing relative to standard mathematics and so the qualifiers, “our algebraic equation”, “my algebraic notation” (notre algèbre, mon algèbre; in later seminars, d’algèbre lacanien); “this way of embellishing it with an image through mathematical forms” (dans cette façon de l’imager dans les formes mathématiques [13 March 1963, my emphasis]). This is how he puts it explicitly:

But in order to grasp it, to see clearly its consequences, it seems to me that our algebra brings us here a ready-made instrument. If the demand comes here improperly at the place of what is eluded, o the object, this explains to you, on condition that you make use of my algebra – what is an algebra if not something very simple[,] designed to make us achieve a mechanical state in its handling, without you having to understand something very complicated, and it is much better like that: I have always been told: in mathematics, it is enough for the algebra to be correctly constructed … (T3, 12 December 1962, my emphasis; in T4: “the equation just has to be correctly put together”). (Mais pour le saisir, pour en bien voir les conséquences, il me semble que notre algèbre nous apporte là un instrument tout trouvé. Si la demande ici vient indûment à la place de ce qui est escamoté, a l’objet, ceci vous explique, à condition que vous vous serviez de mon algèbre — qu’est-ce que c’est qu’une algèbre si ce n’est pas quelque chose de très simple destiné à nous faire passer dans le maniement à l’état mécanique, sans que vous ayez à le comprendre, quelque chose de très compliqué, et ça vaut beaucoup mieux ainsi, on me l’a toujours dit en mathématiques, il suffit que l’algèbre soit correctement construite … [T1, 12 décembre 1962])

Lacanian equation is not strictly mathematics (even if it can use some mathematical rules in its design): perhaps one could say, it is post-linguistics but pre-mathematics. In the Lacanian grammar, a matheme is to mathematics as a lalangue is to linguistics. As lalangue takes the position of (wild) playfulness to language so as to un-fascinate us with linguistics, the matheme takes the position of seriously constructing—like making a programming language—a different symbolic “system”: “the equation just has to be correctly put together”. Aside from doing practical/cognitive ends, Lacan’s matheme also function to reveal an “it” in our “fascinating” relation to numbers, math, and any quantifying symbols. A matheme, therefore, aims also to provide a separative distancing between mathematics and fetishism.

Hal Odetta is a pen name. He is an anthropologist who teaches at the University of the Philippines and works with the Manobo indigenous peoples of Mindanao.

Endnotes

(1) Formal statements or “algebraic” equations, which Lacanians also call as mathèmes. This distinct name (mathème) given by Lacan to many of his formal statements is not used in Ecrits and, in the Seminaires, was not used in this seminar; it was first used during his December 2, 1971 lecture.

(2) Of course these are technically “inequalities”, but they can still be called by the paradoxical name “inequality equations”.

(3) There are four texts-sources I used for this Seminar on Anxiety, Book X: [T1] L’angoisse, Les Seminaires (1952-1978), l’Association Lacanienne Internationale, I.S.I./2000; [T2] Le Seminaire livre X – L’angoisse, Editions du Seuil/2004; [T3] Book X, Anxiety, Translated by Cormac Gallagher; accessible here, bit.ly/2WTo1we; [T4] Anxiety, English edition, translated by A.R. Price, Polity Press/2014. The relevant lecture-dates for this essay are the following: 21 November 1962; 10 and 23 January 1963; 27 February 1963.

(4) I can only speculate that what happened to him after this seminar—his “Excommunication” from the psychoanalytic Association wherein he belonged (see his opening lecture in Book XI)— made him branch off to a path that, although not logically disconnected from the themes he is pursuing, gave less emphasis on the ideas related to the enigmatic fourth formula.

(5) The basic form for the above statements is really very elementary (two terms are set in a relation of inequality): a < b for the first four lines, and a > b (a reversed relation) for the fifth line (Figure 1). What makes the statements unusual (“weird”) to look at, perhaps from the point of view of “standard” mathematical formalization of this type, is that one of the terms (or variables) in each line is expressed in ratio form (the meaning of the colon [:] symbol), as in the standard ratio/proportion statement: a : b :: c : d (read as, “a is to b, as c is to d”). As a “ratio” can also be expressed as a “fraction”, terms in this sample statement can also be written thus: a/b and c/d. The terms in ratio-relation are in relations of “division”. Keeping in mind that the statements are simply expressions of the a < b inequality statement, the terms expressed in ratio or fraction forms would not be strange. There are two more things, directly mathematical in concern, to clarify in the given set of equations. One is the status of the “zero” (in the fourth formula); the other is the question on the nature of the variables/terms being given as notations: are they quantifications? The direct answer to the first is, the “zero” here is more conceptually to be aligned with the idea of “a zero point” (un point zéro). This does not relate to the mathematical number “zero” but functions as a “variable”. The second question is more difficult to answer but certainly not “un-answerable” in standard terms. Are these quantifications? It should suffice for now to note that the concept of “quantification” can take a broad scope even in mathematics, beyond arithmetical numbers. Perhaps it is also relevant to note that the concept of “measurement” (classically developed, for example by Stanley Stevens with his well-known “levels/scales of measurement”) includes the concept of a “nominal” or categorial level.

(6) At the side of this term “zero”, the original paper of Lacan contains a kind of “marker” element: “< d >”, here rendered as “(d)”. This was edited out in the published version, T4. Is this marking “desire” or “drive”?

(7) This is Lacan’s straightforward explanation of the dynamics of the second mode:

We don’t yet know when, how or why this i (a) can be the specular image, but it most certainly is an image. It’s not the specular image, though it belongs to the realm of the image. Here, it’s the fantasy which I don’t hesitate on occasion to overlap with the notation of the specular image. I’m saying therefore that this desire is desire inasmuch as its image-support is equivalent to the desire of the Other. This is why the colon that was here is now here. This Other is connoted by a barred A because it’s the Other at the point where it’s characterized as lack.

(8) Lacan thinks of his formula/equation/matheme as a kind of “image”: “this way of embellishing it with an image through mathematical forms” (dans cette façon de l’imager dans les formes mathématiques [13 March 1963]); while also emphasizing that “the equation just has to be correctly put together” (il suffit que l’algèbre soit correctement construite [12 December 1962]).

(9) The quotation here can be found four times—in its exact wordings—in three books: Parallax View/2006: p. 60-61; In Defense of Lost Causes/2008: p. 328; Less Than Nothing/2012: pp. 497-498, 638-639 (twice!).

(10) For Lacan’s idea of this “zero point”, see the blog-post: Lacan’s spatial materialism: Un point zéro (1963)

Acknowledgments. I gained useful speculative insights from my discussions with Işık Barış Fidaner on the drafts of this essay. I am also thankful for some early impressionistic feedback on the abstract, formal aspects of the above equations (and their transformations) from my university colleague, Rommel Real.

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