John W.P. Phillips: On Mathematical Logic

Photo: John W.P. Phillips

We featured an interview with John Phillips on the TCS Blog in September, in which Simon Dawes asked him about aesthetics, Badiou and Rancière; his responses took us on to Lacan, Plato, Žižek and mathematics. We asked John to clarify some of the complex theories and concepts to which he referred, and over the next few weeks we’ll be posting his responses. Here is his first response: on the significance of cardinal and ordinal numbers, and mathematical logic in general, to philosophy and psychoanalysis.

Simon Dawes: Could you tell us more about the importance of classical mathematical/modern logic to Badiou and Lacan, and to philosophy and the symbolic order, more generally? And could you explain in more detail the interest Lacan had in cardinal and ordinal numbers, and how this links up with the ’empty set’ and Russel’s paradox?

John Phillips: The answer in each case comes out differently and it might be wise to distinguish Badiou from Lacan on this point particularly, maintaining the connection, certainly, but acknowledging the methodological centrality of formal demonstration to Badiou’s philosophy. But it’s also worth saying that the importance of mathematics and mathematical logic have a similar orientation for both of them: these domains (or worlds, as Badiou calls them) supposedly attest to a fundamentally materialist ontology that is measured by something that is inaccessible to its subjects. In other words, the inaccessible measure plays a fundamental role in the construction of ontology. In mainstream philosophy truth in a mathematical sense isn’t about empirical facts but about proofs. Once a truth of a mathematical kind enters into a discourse or a language then interesting things can happen. The inaccessible measure of the world, for instance, may be formalized as a proof in a language intrinsically incompetent to deal with such a measure.

The terminology distinguishing cardinal from ordinal is quite critical. It’s part of Cantor’s set theory.

The difference between cardinal and ordinal ordinarily implies a distinction between the question of how many, e.g., 7 stones, and the question of order and hierarchy, e.g., 7th son. Lacan was interested in an ostensible fact of linguistics, according to which cardinals come into language before ordinals. He was also interested after Levi-Strauss in the fact that classification and explanation follow mathematical combinations (Levi-Strauss’s elementary structures of kinship). So the procession of numbers, in other words, manifests a primitive symbolism (the symbolic) that can be radically distinguished from imaginary identifications (the body of a child reflected, as in the mirror stage, alongside his or her objects).

In set theory cardinals measure the size of a set, or its power, while ordinals measure the shape of a set, or its length. The distinction gets interesting with infinite sets. Ordinals measure the way a set is organized so that its elements are defined by their order in a series. The natural numbers are ordered like this: 1, 2, 3, … . One ordinal is first; each ordinal has a successor; and there is an ordinal that succeeds all others. The smallest infinite set, or the first infinite cardinal, is the set of whole numbers: the series 0, 1, 2, 3, …, n, n+1, … to infinity. There are other ordinals with the same shape and length, like the set of odd numbers: 1, 3, 5, … . These are countable, or denumerable, infinities and thus have a specific size or power: their cardinality. The cardinality of denumerable infinity is given the mark “aleph-nought.” We’ll come back to this in a minute.

In his structuralist moments, Lacan likes the fact that cardinals emerge in language before ordinals. But things get more complicated. In the 1960s Lacan talks of number in terms of the “unitary trait.” By this he means integers (any so called rational number). But he derives his explanation from properties of repetition or “iterative construction.” An integer becomes what it is only by its being repeated (then you get one and two–or one from two, as we’ll see in a minute). The unitary trait is therefore singular only to the extent that it can be repeated (thus becoming plural). In this respect he regards integers as special cases of the more general principle of the relation between signifiers. In Lacan’s terminology signifiers only have existence in their differential relations between each other. The “mark” must be both repeatable (it exists only in its repeatability) and differential (it exists as a non-positive term in a system of differences). The signifier, if we have to remind ourselves of this, can be at once individual (you and me–the signifiers representing a subject for other signifiers), as well as phallus, reflection, mark, body, castration (absence) and, yes, number, which already implies a more complex relation between ordinals and cardinals than structural linguistics and anthropology suggest.

So Lacan draws on mathematics to demonstrate some of the basic principles of his psychoanalytic theory. The most important of which involves the identification of the subject in language. To put it very baldly and simply, a subject is constituted in the gap that occurs when a signifier is repeated. Language, as we all know nowadays, is what it always is thanks to an invariable property: the repeatability of signifiers. Because signifiers are repeatable subjects they arise in a language that in every case always precedes them. These are the concrete languages that people speak. So far so good.

Where better to look for the essential properties of this repeatability of marks than in mathematics? The mathematical formula n + 1 (which underlies the genesis of numbers) seems directly concerned with it. Thanks to the repeatability of signifiers the most rigorous formulations of mathematics always lead to paradox. If we grant then that mathematics is always predicated of that property of language which ensures that each signifier can be repeated, then we can begin to see how it might plausibly be given the ontological status that Badiou does attribute to it.

Repeatability and numbers: a number cannot be conceived unless you have two. Two is what grants one its existence. So it’s clear why Frege can generate number on the basis of classes (as I explained in the previous interview): the first class is the class (the only one, the null set) which has no elements. Zero becomes one. One, consequently, becomes two, thanks to a repetition of one. If two is the repetition then one comes into being by virtue of its being repeated. I am now reading what Lacan said about this in 1965: “The question of the two is for us the question of the subject, and here we reach a fact of psychoanalytic experience in as much as the two does not complete the one to make two, but must repeat the one to permit the one to exist” (Of Structure, 191). Everyone knows that psychoanalytic subjects tend to repeat themselves (compulsively, obsessively, defensively, conservatively, neurotically, deceptively, etc). It’s the evident character of the unconscious. But to say that repetition constitutes the subject is slightly more profound. That “first” repetition constitutes the subject in terms of something that is missing from the language they speak (it’s never going to be first really because a repetition by its essential character is always an afterwards of something, at least secondary). This question of the something missing is beyond what we can deal with here, but it is very important in psychoanalysis. (The notion of “foreclosure” designating the problem of the precise grammatical structure that goes missing in the constitution of a subject cannot be ignored, but we can only acknowledge that here). Anyway, this idea of a subject constituted in the space of a repetition in language underlies so much. The idea of a symptom, and the idea for instance that a child is the symptom of the parents, a chilling notion that is all but unthinkable for most of us, is grounded at once on this same formula and in what Lacan learns from what he beguilingly and disarmingly, but also ironically, calls “the psychoanalytic experience.”

To answer in relation to Badiou we must take a further step into sets. Here’s a demonstration based on the iterative construction of number. The rule is that the members of each set must be obtained from earlier stages. Start with the “empty set,” ∅. It’s the only set that has no members. Now construct the available sets whose members are taken from the earlier stage. {∅}. That as we already know is “One.” The next figure in line therefore amounts to two possible sets: {{∅}} and {∅, {∅}}. There’s Lacan’s “Two.” There’s actually no limit to such a hierarchy, which gives us the simplified formula n + 1.

This is how ordinals are figured as sets. (1 = {∅}; 2 = {∅, {∅}}; and so on). Cardinal numbers, we could then in the manner of Cantor go on to argue, are of two kinds: countable and uncountable. This gets us into problems with infiniteness: the problem of the continuum or what Leibniz called “the labyrinth of the continuum.” The entire field of popular maths books feeds insatiably on this as if a few proofs give us access to something wonderful. I think of Badiou’s handling of it as a basic philosophical fall-back. We get the idea of transcendental numbers from 18th century metaphysics. Leibniz, for instance, used a transcendental method for solving certain otherwise insoluble problems by producing proofs involving infinite series, as in the famous example of π r2 (or “Pi r Squared”–finding the square for the area of a circle). With axiomatic set theory we can trace a movement from a metaphysical notion of infinity (a divine sphere unknowable for finite beings) to something more precise (uncountable infinity inaccessible for structural reasons). To keep it simple: Cantor attempted to prove the existence of infinite numbers with different cardinalities (different powers or sizes). Aleph-nought was Cantor’s designation for the smallest infinite number (the set of all natural numbers and the first trans-finite number). The number c, by contrast, has “the power of the continuum” (cardinality C). Numbers with this “c” cardinality would be the so called real numbers (which include Pi) as well as all those points on a line between 0 and 0.99999 …).

Badiou’s way with axiomatic set theory involves the establishment of proofs of an ontological kind. This “ontology” is supposedly as powerful as Heidegger’s fundamental ontology but with the controversial underpinning of mathematics (which might seem to overturn Heidegger’s strongly held conviction concerning the metaphysical foundations of logic in the ancient Greek notion of ta mathemata; but Badiou suggests that in this respect Heidegger consistently failed to mobilize a philosophical opportunity in the urgency of his deconstruction of the history of metaphysics, from Plato to Leibniz and Kant.

Badiou had already addressed the “continuum hypothesis” in maths in his Theory of the Subject. The “ontology of worlds” in Badiou’s philosophy involves a proof of “inaccessible closure” that follows rather exactly the organization of number by sets to which I’ve just been referring. Statements of the kind “every world is measured by an infinite cardinal” (from Logic of Worlds 332) are posed as hypotheses and then submitted to formal demonstration. “Our only certainty,” he writes there, “is that a world can only be measured by an inaccessible cardinal” (335). What we must acknowledge here is that even the denumerable infinity named aleph-nought is too large to be accessed from a vantage within the world that it measures from a seemingly transcendental standpoint. None of us has access to this measure. Cantor’s cardinality of the continuum would then be a step further even than this–finitizing the infinite itself, as Badiou suggests (335). Badiou–wisely and in the manner of the entire poststructuralist field–offers a “preference” (we should also hear in “preference” the “inclination” or “incline” of the epicurean tradition and the clinamen of Lucretius as well as Marx) for the undenumerable infinity: “it preserves,” he writes, “the horizon of worlds endowed with an extensive power in comparison with which the figures of the inaccessible that are known to us remain derisory” (335). A preference for an inaccessible measure that is inaccessible beyond measure compared with even those figures of the inaccessible that we might think we know … this is an extraordinarily powerful statement of finite possibility.

What Badiou does with mathematics is always remarkable (and entertaining, if I can say that). I’d go back, though, to a moment in Jacques Derrida’s “Introduction” to Husserl’s Origin of Geometry, in which a structure of this kind is identified as a stage on the way to a yet more powerful formulation. And it’s certainly true that mathematics (its paradoxes and infinities) play a role in Derrida’s readings but only to be applied by analogy in what always maps out as a kind of second stage of deconstruction (deconstruction in Derrida’s sense, unlike that of Heidegger, always follows three identifiable stages). Derrida does not, certainly for this reason but among others too, go down the philosophical ways of mathematical formalism. I’d also want to put Cantor’s infinities in touch with those of Hegel, because what is at stake here as always has to do with the power of conceptualization as such.

The connection to Russell’s paradox is more straightforward, if only because it plays such a central role in so many philosophical domains in the last century and a bit. It formalizes the ancient paradox of the liar: the man from Crete who claims that every statement he makes is a lie, including that one. Is it a lie or is it true? If it’s a lie it’s true and if it’s true then it’s a lie. In set theory the formal demonstration is more damaging: if there are sets that include themselves as members and there are sets that do not include themselves as members there must be a set of all those sets that do not include themselves as members. Is that set a member of itself? If it is it isn’t and if it isn’t it is. Everyone has a version of this going, of course. Lyotard’s The Postmodern Condition mobilizes it consistently throughout. Baudrillard’s notion of “simulacra” couldn’t function without it. Lacan had said “this only means that in a universe of discourse nothing contains everything, and here you find again the gap that constitutes the subject” (Of Structure 193). Much of Badiou’s Theory of the Subject is worked out as the extension of this statement. But he goes further when he provides its extended mathematical echo by way of formal demonstration in Logic of Worlds, in which he writes: “it means that it is not true that to a well defined concept there necessarily corresponds the set of the objects which fall under this concept” (153). In a more restricted sense (but also echoing Lacan) it means that “the whole does not exist.”

Useful References:

Badiou, Theory of the Subject, trans. Bruno Bosteels (London: Continuum, 2009).

Badiou, Logic of Worlds, trans. Alberto Toscano (London: Continuum, 2009).

Lacan, “On Structure as an Inmixing of Otherness Prerequisite to any Subject Whatever,” in The Languages of Criticism and the Science of Man, ed. Richard Macksey and Eugenio Donato (Baltimore: Johns Hopkins, 1970).

John W.P. Phillips is Associate Professor in the Department of English Language and Literature at the National University of Singapore, and an editorial board member of TCS. He writes on philosophy, literature, critical theory, aesthetics, psychoanalysis, urbanism and military technology. He is co-author with Ryan Bishop of the forthcoming Modernist Avant-garde Aesthetics and Contemporary Military Technology: Technicities of Perception (Edinburgh University Press, 2010), and he is currently researching a project on autoimmunity in biotechnology and political philosophy.

Simon Dawes is the Editor of the TCS Website and Editorial Assistant for Theory, Culture & Society and Body & Society

Go here for the earlier ‘Interview with John W.P. Phillips on Badiou, Rancière…and Žižek’

You can access the article, ‘Art, Politics, and Philosophy: Alain Badiou and Jacques Rancière’ (TCS 27.4) by John W.P. Phillips here

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