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Lacan on the Number 13 and the Logic of Suspicion

[A pdf version is available here.]

Lacan once remarked on the Cartesian cogito that etymologically, “the French verb penser (to think)…means nothing other than peser (to weigh)” (1961-2: 14). Lacan’s mix of bad puns, abuse of notation, cryptic aphorisms & immense erudition has created a new style of writing, and even of thinking. A new language, in short, lacking any method for non-experts to weigh its words.

Of all things, Lacan’s earliest papers take the form of math puzzles—a lucid (albeit horrendously verbose) derivation, then reframing of the problem as metaphor. One such paper—“The Number Thirteen and the Logical Form of Suspicion”—has largely been forgotten. This post aims to recast the puzzle using discrete mathematics, and show how it bears upon Lacan’s later ideas.

What I like about this mathematical allegory is that even if one believes Lacan is a charlatan, here there’s no need to immerse oneself in psychoanalytic concepts, but only to think.

I hope that Lacanians will find working through my derivation a challenging exercise, and that mathematicians will be piqued at the idea of treating a math problem as a philosophical ‘text’. I for one would be glad to see more such texts.

1. Lacan’s Algorithm

We are given 12 identical pieces, and told one of them is ‘bad’—either lighter or heavier than the rest, we’re not sure which. Having only a scale with two plates, and no way to gauge numerical weight, we must find the bad piece in 3 weighings.

If we knew whether the bad piece was lighter or heavier, the problem would be easy: just split the pieces into two groups of 6, then split the ‘bad’ half into two groups of 3, then simply weigh two of the bad three. But we don’t.

Here, we’ll overview Lacan’s account for 12 pieces, and then in the next section we’ll consider n pieces, and try to explain why Lacan’s algorithm works.

Lacan begins by placing on the scales two groups of 4. Suppose they balance. Then the bad piece is in the remaining 4, so we can just weigh any 2 of the 4. If those balance, the 2 left-out pieces are bad; if they don’t, the 2 pieces on the scale are bad. So weigh one of the bad 2 against a good piece: if they balance, the other piece is bad; if they don’t, then the piece on the scale is bad. Simple.

Note how this was equivalent to the sub-problem of finding a bad piece out of 4 pieces, in 2 weighings. The sub-problem is embedded in the larger problem.

If the two groups of 4 don’t balance, we use the method of tripartite rotation.

Tripartite rotation

The scales don’t balance, so one is heavier (H), one lighter (L). So, we select 3 pieces from H, L, and the remainder (R), and rotate them: H → L → R → H.[1]

Case 1: Scales balance — the bad piece is in the 3 moved to R, and too light.
Case 2: Balance shifts — the bad piece is in the 3 moved to L, and too heavy.
Case 3: Unbalance doesn’t change — the bad piece is in the 2 unmoved pieces.

In cases 1 and 2, just weigh 2 of the bad pieces: if they’re equal, the remainder is bad; if not, we know the bad piece is the lighter (case 1) or heavier piece (case 2). For case 3, just pick one and weigh it against a good piece. And we’re done.

Lacan then considers the case of 13 pieces: 4 on each scale, 5 remainders. It’s clear that if the scales don’t balance, the problem is the same as with 12 pieces when the scale didn’t balance—the remainders are all good, whether 5 or 4.

Here, when the scales balance, we have a new problem. Recall how we could treat 4 pieces as a separate problem. So let’s examine the 5-piece sub-problem.

Start with 2 pieces on each scale and 1 remainder. If we’re lucky, the scales balance and the remainder is bad. If not, we have 4 pieces, but we know the 4-piece case takes two weighings, so the 5-piece case must take three weighings.

It’s the same even for 1 piece on each scale and 3 remainders. If we’re unlucky, the scales balance, giving a new sub-problem with 3 pieces—the smallest solvable version of Lacan’s problem. Weigh any 2. If they balance, the remainder is bad. If not, weigh a piece on the scale against the good piece. Total: three weighings.

So both the 3-piece and 4-piece cases take two weighings, 5 pieces takes three weighings, so it would seem that 13 pieces must take four weighings. Nope.

Actually, for 13 pieces, the 5 remainders aren’t truly a separate sub-problem. There’s a difference: we have 8 good pieces. For 3 or 4 pieces, this doesn’t matter, but for 5 pieces, Lacan can introduce a new trick: the ‘by-three-and-one’ position.

The ‘by-three-and-one’ position

Here, we have 2 pieces in each plate, with one of the 4 a good piece, and 2 remainders. If the scales balance, just weigh one remainder against a good piece and we’re done. If they don’t balance, here’s the trick: we can do the smallest possible tripartite rotation, H → L → R → H, where R is a good piece.

Case 1: Scales balance — the bad piece is in R.
Case 2: Balance shifts — the bad piece is in L.
Case 3: No change — the unmoved piece is bad.

Thus, the 5 remainders take two weighings, and the 13-piece case takes three.

In this case, treating the 5 remainders as a sub-problem was the wrong way to go, making it seem impossible to solve in 3 weighings. More pieces means more ways to divide between scales and remainder, increasing the risk of such pitfalls.

Thus, Lacan’s task is to find a general algorithm for any number of pieces, including a uniform way to divide them. The algorithm must minimize the maximum amount of weighings—i.e. find the minimum, assuming we don’t get lucky.

The problem also raises some new questions. The main one is: for a given number of pieces, how many weighings are needed? As in the solutions outlined above, Lacan answers this question, but fails to explain why his solution works.

Hence, the next section will diverge from Lacan’s exposition, using discrete mathematics to give an algorithm for n pieces. This will help us see how Lacan’s problem relates to the logic of suspicion, which we will outline in the final section.

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New Translation of Living Currency Is Now Available

To my great excitement, Jordan Levinson has just posted an independent translation of Pierre Klossowski’s Living Currency (La Monnaie Vivante) on his website, available for perusal and download.

To those who have not read my recent (p)review of the text, Living Currency has been praised by Foucault as “the greatest book of our times” and is purported to provide the missing link from Bataille to Baudrillard and from Lacan to Foucault and Deleuze. It also played a key rôle in Lyotard’s Libidinal Economy, and doubtless (though less explicitly) in Deleuze & Guattari’s Anti-Oedipus.

If anyone is interested but finds Klossowski’s style of writing too abstruse, I intend to engage in a close reading of the text, delineating Klossowski’s thought in more accessible language, in addition to tying it in with other theoretical accounts of political economy (such as those of the authors mentioned above, plus the more contemporary For a New Critique of Political Economy by Bernard Stiegler). Optimally, I hope to figure out the text’s implications for the way we think about the economy & economics in general, and to incorporate these into my honors thesis on the work of Piero Sraffa (which you’ll hear more about in the near future).

(For the record, Levinson’s translation is unaffiliated with the translation by Reena Spaulings which I mention in my review.)

The Stakes of Grammatology

The well-known quarrel between Lacan and Derrida over Poe’s “The Purloined Letter” did not come from nowhere. Consider in this regard Lacan’s formulation from “The Instance of the Letter in the Unconscious,” that one is to grasp the letter à la lettre, that is, literally, and Derrida’s counter in the title to section one of his Of Grammatology, “Writing Before the Letter,” in French, avant la lettre, that is, before the fact, before, that is, the literal. Never to shirk a provocation, Lacan responded in the points edition of the Écrits by instating that his insight into the “instance/agency of the letter preceded any grammatology.” This in turn appears to have prompted The Title of the Letter: A Reading of Lacan by Derrida partisans Jean-Luc Nancy and Phillipe Lacoue-Labarthe. The titular phrase, le titre de la lettre, might also be rendered as “the deed to, or rank of the letter.” Here is not the place to elaborate the stakes of this face-off, but suffice it to say that at issue is the nontrivial problem of whether philosophy can think the general economy of signs that conditions the possibility of language, whether spoken or written.

~John Mowitt, in Lyotard – Discourse, Figure, Editor’s Introduction, pg. 397, endnote 7.

Richard Webster on Intellectuals

The tragic predicament of such intellectuals [as Lacan] is that, driven by terrifying feelings of emotional emptiness and insecurity, they mistakenly conclude that intellectual truths can be an adequate substitute for emotional warmth. Convinced that difficult or abstract intellectual formulations can alone fill the void they feel within them, they develop a voracious appetite for such formulations, anorexically judging their goodness by the degree of difficulty or abstraction they possess. Believing that what they have devoured is intrinsically nourishing and failing to grasp the poverty of the diet they have adopted through their own self-denying ordinances, they now feel impelled to share their ‘truths’ with others. Indeed they are driven by their own generosity to do so. Like a starving man who compels others to eat the diet of stones he believes has saved him, they give abundantly of their poverty out of a genuine conviction that they are enriching others. Because their own most generous impulses have become inextricably entwined with their impulse to self-denial they are unable to discriminate between generosity and cruelty and unable to understand that by compulsively sharing with others (or compelling others to share) their own chosen form of intellectual or spiritual wealth they are merely disseminating their poverty.

~Webster, R. “The Cult of Lacan

[The above is the only thought-provoking paragraph of an otherwise worthless essay.]

A Question For Lacanians

When Jacques-Alain Miller was a philosophy student at L’Ecole Normale Superieure, Althusser told him to read “all of Lacan,” so he did. Then, one day when Lacan was visiting the school, Miller asked him a now-‘famous’ question that supposedly revealed the key to Lacan’s psychoanalysis, and the two from then on entered into an understanding which was to last all their lives. Miller’s question was: “Does your notion of the subject imply an ontology?” (via)

Another account of the exchange (which includes Lacan’s answer), located here, is:

The ontological concerns of the Cahiers pour l’Analyse themselves are prefigured in a key moment in Jacques Lacan’s seminar in the spring of 1964, when Jacques-Alain Miller asked Lacan if his theory of the subject, grounded in an account of lack and its structuring function of the unconscious, presupposed an ontology. In the seminar, Lacan answered this question with the suggestion that the constitutive gap of the unconscious was essentially ‘pre-ontological’. The Cahiers pursue Miller’s original question along multiple lines, identifying points of contact between Lacan’s theory of subjectivity and the ontological concerns to be found in recent and contemporary developments in logic and the sciences.

As far as I can tell, there are two possible interpretations of the word ‘ontology’ in Miller’s question. Ontology can mean ‘study of being’, or more rarely can mean ‘diegesis’ (fictional ‘world’), though I’m not sure if the latter is still true in the French. In the former case, it would seem that Miller is asking “Does your notion of the subject imply that your work is a study of being?” In the latter case, he would be asking if the ego (which is an imaginary construction) subsists in a distinct diegesis (i.e. a plane of reality separate from everyday existence).

Even after hearing Cahiers pour l’Analyse‘s account, I still do not understand the question (nor how Lacan’s answer can be correct). My question to Lacanians, then, is: what does Miller mean, what does Lacan’s answer mean in relation to Miller’s question, and what are the broad implications of this question?

(Note: the above exchange can be found in The Four Fundamental Concepts of Psychoanalysis.)

Trying to Get Immanence Out of a (Philosopher’s) Stone: Archetypes, Sociobiology, Harry Potter, and the UK Riots

Click the picture for a more detailed explanation of the notion of signifier/signified.

[To make the parallel of Harry Potter & the Tottenham riots seem less farfetched, see here.]

There are a number of popular (i.e. non-academic) intellectual movements whose objective is to find an immanent basis for the meaning of signifiers. One such example is Jungian archetypes, which states that various symbols are innate in the human mind, and thus that symbols are “universally recognizable.” As well, the sociobiology of Desmond Morris seeks to ground social phenomena in biological instinct (once again, innate), e.g. he ascribes the tradition of women coloring their lips red to the fact that when a woman becomes aroused, her lips become engorged with blood, appearing fuller and redder; thus lipstick is a display of availability for mating, just as is the peacock displaying its feathers. A third, more contemporary instance of this tendency can be found in the Harry Potter series. In Hogwarts, students of witchcraft & wizardry are taught combinations of signifiers (e.g. a “swish & flick” of one’s wand combined with the words “Wingardium Leviosa” pronounced in a specific way) which are somehow inherently connected to their magical function. There is no talk of ‘inventing’ spells; presumably experimental wizards merely spout out Latin-sounding words in hopes that they’ll bring a result connected to their etymology. This essay will outline the three views described above; show how meaning is in fact not immanent, but for the most part purely arbitrary; and show how this immanent treatment of signifiers resonates within the UK riots, perhaps to the point of precluding any significant cultural change.

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A Brief History of the Real, or, Laruelle’s Niche: Ontological Reification

[Note: After actually reading Laruelle, I disavow everything written in this post. It completely misses Laruelle’s point, and I’m only leaving this post up to let it serve as a bad example.]

The responses by An Und Für Sich to Graham Harman’s review of Laruelle have reminded me of an old argument I had against his ‘Non-Philosophy’. My argument centres around a single aspect of Non-Philosophy―namely, the notion of ‘The One’―largely because my exposure to Laruelle has been limited to Anthony Paul Smith’s “Introduction to Non-Philosophy” (notes) and Alexander Galloway’s “François Laruelle, or The Secret.” Nevertheless, I feel that it adequately situates Laruelle within the tradition of Continental philosophy; to make it more accessible, however, I will preface it with in-depth background information.

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