What’s wrong with Badiou I: Mathematical Fragments (1)

The difference between philosophical insight and philosophical nullity is a perspectival difference of about 15 degrees.

In a previous post on style in philosophy, I mentioned that I am — and remain — unimpressed with Badiou’s work. Though, in fairness, I ought to restrict this claim: I remain unimpressed with Badiou’s Being and Event. Now, there are two basic reasons for my reaction: first, I suspect that his work here — and here I should add, insofar as I understand him — is nothing more, and certainly nothing newer than a Transcendental Philosophy.* We need not worry whether ‘transcendental’ designates Kant’s conception of it, Fichte’s, or even that of Husserl’s (or even Lacan’s); for the moment, since I plan to write something on precisely the kind of transcendentalism involved in Badiou’s project, it suffices to say that the task Badiou sets himself remains an inquiry into conditions for the possibility of thought and action, which operates according to a twofold philosophical procedure: in its first moment, he proceeds to explicate a metaphysical structure; in its second moment, and by way of a regressive — one might say even say ’subtractive’ — analysis, his argument foregrounds the conditions that underwrite the previous structure and that legitimate it. Indeed, his invocation of ‘meta’ (as in metaontological), signals precisely such a shift. Moreover, even his appeal to the formalism of set-theory — much like Lacan’s appeal to another kind of mathematical formalism — has it’s roots in the Kantian system, which takes (as we know) a certain formal conception of physics as its base, and tries to regressively articulate the underlying, properly philosophical significance of such a conception for thinking and action in general (A quick glance at Kant’s discussion of the Transcendental and Mathematical uses of Reason [B733/A705-B765/A737] will confirm this). In Badiou’s terminology, then, and despite his glib remark on Kant (Introduction, p. 12), which indicates just how far into The Critique of Pure Reason Badiou managed to penetrate, transcendental philosophy is always conditioned on Mathematics. Indeed, this was one of the insights of the Neo-Kantians.

Second — and this is the issue I want to address here — Badiou’s Math appears to be rather sloppy, and — even worse — misleading. Before I continue, however, I should issue two caveats: First, while I do have a background in mathematical logic (from a previous philosophical incarnation), it’s been a number of years since I’ve actively engaged with “mathematico-logical” philosophy in any direct way. Simply put, I’m rusty. So it may be the case that I’ve inadvertently fudged something. If someone sees anything like this, please point it out! Second, some of the mathematical sloppiness may not be Badiou’s fault: there’s something about the translator’s struggle with Badiou’s mathematical terminology that is highly unsatisfactory — and that isn’t going to be solved by glancing at the original French, since the problem is not one of fidelity to Badiou’s text, but rather with how mathematical terminology in English relates to that of French. My honest position on the matter, for which I will give examples, is that the English translator did not check the consistency of his translation against the contemporary English mathematical vocabulary. And this hinders, I think, the comprehensibility of Badiou’s text. For example, on page 50 of Being and Event (”Technical Note: Conventions of Writing”), where Badiou introduces some of the logical signs and operations that are essential to his work, we find the following:

the relations [to be used in the following] are = (equality),” which “always link[s] two variables: α = β, which reads ‘: α is equal to β’

Now there is no logical language which uses the sign, ‘=’ in the sense of ‘equals.’ To do so is actually fatal, since it conflates identity with equality, or equivalence. Here’s a quick example to differentiate the two: Let A stand for the letter ‘A,’ and let B stand for the letter ‘B’. Now, A and B can be shown to be equivalent to one another because they possess a property that is reflexive, symmetric and transitive (namely being a letter in the English alphabet).** But they are in no way Identical — A ≠ B. Equality, in point of fact, is nothing different than equivalence. But Badiou uses the sign ‘↔‘ to denote equivalence. So ‘=’ must denote Identity, despite its translation.

This potentially misleading translation, however, is not the end of the slippage in technical semantics, since ‘↔.’ is a bi-implication, whose logical field is slightly different than what we typically — again in logic — mean by equivalence (which is usually denoted by ‘≣‘). That is to say, whereas bi-implication denotes a necessary and sufficient condition (an “if and only if” clause), but does not define this condition, equivalence precisely defines the condition(s) under which two ‘things’ (objects, elements, multiplicities, whatever) are equivalent because equivalence establishes and explicates some relationship between relata. If this point is not kept in mind, some of Badiou’s formulations are actually nonsensical.

A Second example of translation-problems can be found in Appendix 6 (459-460), where the translation speaks of “Recurrence on the Length of Formulas.” Maybe it’s just a quirk of my own education, but I have never heard ‘Recurrence’ used as a technical term in Mathematical logic. “Recursion,” “Recursive definition,” “Recursively denumerable” ? Sure. But Recurrence? — Never. Now, While I’m willing to admit that (1) I was never cut out to be a logician, a philosopher of Math, and certainly not a mathematician, and (2) my knowledge of mathematical logic is limited (I got up to the Gödel incompleteness theorems, and promptly quit), from which it follows that there may indeed be some kind of technique or property called ‘recurrence’ that I simply never encountered. But such a worry, however, is immediately dispelled when I realized that what Badiou actually describes is a very basic proof technique: Mathematical induction (based on the degree of a formula). Here, I can think of no mitigating circumstance for this translation. It’s just plain bad.

So, all this to say that one must be doubly careful with Being and Event’s ‘Mathy’ parts, since there’s a translation problem, which seems to make the text less than reliable. And this problem isn’t solved simply by consulting the original French

If it were only a problem of misleading terminology, however, the problem could be easily fixed — Badiou does, after all, provide us with the appropriate formalism. But, because Badiou parcels his formulae in a manner that obscures some of the fundamental background presuppositions of set-theory, even these aren’t as clear as they should be. In other words, because crucial bits of information are not conveyed when they need to be (in particular the Universe of Discourse which delimits the range of quantification), even the formalism is wonky. Specifically, the distinction that Badiou wishes to draw between “supposed existence” and “implied existence” (see Being and Event 45f), relies upon the fact that the boundaries of quantification have already been drawn (by the Universe of Discourse, Badiou’s ‘regime of the count?’ his ’situation”?). This ambiguity manifests itself in the slippage of his use quantification, which speaks against his basic conception of the multiple: sometimes a multiple is quantified over (and thereby treated as multiple) and sometimes a multiple is treated as a constant, as an individual. Consider the following paragraph:

Using the metaphor of elements — itself a perpetually risky substantialization of the relation of belonging — the axiom [of union] is phrased as such: for every set, there exists the set of the elements of the elements of that set. That is, if α is presented, a certain β is also presented to which all the δ’s belong which also belong to some γ which belong to α. In other words: if γ∈α and δ∈γ, there exists a β such that δ∈β. The Multiple β gathers together the first dissemination of α, that obtained by decomposing into multiples of multiples which belong to it, thus un-counting α:

(∀α)(∃β)[(δ∈β) ↔ (∃γ)[(γ∈α) & (δ∈γ)]]

In the first instance, while I can understand Badiou’s worry about ’substantializations,’ there is no metaphorical register here. Insofar as δ remains free (not bound by a quantifier), one must treat it as an individual, as an element. Whether a finer analysis will show that δ is in fact a multiple itself is besides the point (indeed, as far as I understand him, Badiou never address the axioms of multiple construction, which delimit multiples in relation to their predicate structures). Here, within the axiom scheme — and not an axiom! — δ is nothing less than a individual, however indeterminate it may be. Perhaps the easiest way to understand what’s at stake here is by alluding to the grammar of a natural language: the subject-position of a sentence need not be an individual per se, but within the context of a given assertion, what occupies a subject position is an individual. Here too, from a set-theoretical perspective δ is an individual, an element. And unless the strictures for its construction are given, unless that conditions for its satisfaction are clear, and until it is quanitified, δ remains an individual. Moreover, one view of set-theory actually asserts that there are Urelementen — primordial, atomic individuals — from which set-theory is developed, and Badiou nowhere addresses this issue — even where his own formalism leaves open the possibility. Furthermore, this latter theory is equally consistent with ZF Theory with the axiom of choice

But now I’ve expended more time here than I had allotted myself. So I have to leave off here. With luck I’ll be able to complete this soon.

*I’m tempted to say that the only hope we have for a system is through the strictures of transcendental philosophy; but that’s a very long and tangential story.

** The actual proof involves constructing something called equivalence classes (or, if memory serves, Henke-sets), to which A and B respectively belong. While the members of each respective class remain non-identical to one another, the classes to which they belong are equivalent. Equivalence, we might say, is a ’soft-core’ kind of identity.