Formalist Existentialism?

The English translation of Alain Badiou’s Being and Event III: The Immance of Truths has just been published. There is not much in this book that I would recognize as philosophy; neither other philosophers nor questions of interpretation are discussed at any length. Badiou primarily wants to assert that actual infinity is established by classical set theory as an “absolute ontological referent”.

Badiou’s deepest influences are Sartrean existentialism and what at first appears to be a kind of extreme formalist view of mathematics. For Sartre, what distinguishes the human is an ability to make utterly arbitrary choices. Such views have historically been justified by appeals to human likeness to an omnipotent God that, while commonly raised by religious sectarians, actually diverge from more broadly accepted views of orthodoxy in religion, which temper appeals to raw infinite power by emphasizing that God is good and more reasonable than we are, and therefore does not act arbitrarily. Sartre and Badiou, however, are both militant atheists who aim to ground the argument for human arbitrariness in some other, nonreligious way.

I think what we need for ethics is to recognize that we are beings who partake of an active character. We do things, and along the way we make choices between alternatives, but real-world decisions — the only kind there are — are never made in a vacuum. I think activity necessarily involves purposefulness (seeking some good, i.e., something judged by someone to be good in some way, even if we would completely reject the judgment). Any kind of purpose at all is incompatible with complete arbitrariness. (See also Beings.) But Badiou would disqualify this whole line of thought, because he doesn’t believe in ethics or in purposes that are independent of arbitrary decision.

I call Badiou’s appeals to formalism in mathematics extreme because — utterly contrary to the spirit of the early 20th century program of David Hilbert, which is usually taken as the paradigm of mathematical formalism — Badiou claims that his formalist arguments directly apply to the real world. Even so-called mathematical Platonism only asserts the independence of mathematical objects, and nothing like the immediate relevance to politics claimed by Badiou. The whole point of Hilbert’s formalism is that it doesn’t care about the real world at all. For Hilbert, mathematics consisted in purely hypothetical elaboration of the consequences of arbitrary axioms and definitions. He likened this to a kind of game.

Badiou’s use of purely formal elaboration from arbitrary starting points is decidedly not hypothetical; it is combined with an extreme realism. According to Badiou, Paul Cohen’s theorems about generic subsets, for instance, are supposed to directly lead to political consequences that are supposed to be liberating. We are supposed to get some enlightenment from considering, e.g., immigrant workers as a generic subset, and this is supposed to represent a kind of unconditional or “absolute” truth that is nonetheless immanent to our concrete experience. But the treatment of arbitrary hypotheses as unconditional truths is utterly contrary to what Hegel meant by “absolute” knowledge, which I would argue is really supposed to involve the exact opposite of arbitrariness. Hegel’s “absolute” is about as far from Badiou’s “absolute ontological referent” as could be. (See also Hegelian Finitude.)

I am only a moderately well-informed mathematical layman and claim no deep understanding of Cohen’s results, but the basic idea of a generic set or subset seems to be that it is an arbitrary selection of elements from some pre-existing set. Being arbitrary, it has no definition or characteristic function (other than by sheer enumeration of its elements). But in classical set theories, new sets and subsets can be formed from an arbitrary set. Badiou relates this to Georg Cantor’s proof that any set has more subsets than elements. In itself, I find the latter unobjectionable. But Badiou likes classical set theory because it gives a putative mathematical respectability both to arbitrary beginnings and to actual infinity. (See also Categorical “Evil”; Infinity, Finitude.)

According to Badiou, belief in actual infinity is revolutionary and good, whereas disbelief in actual infinity is conservative and bad. Infinity is supposed to be revolutionary precisely because it is unbounded. This just means that it can be used as a putative license for arbitrariness. I want to insist on the contrary that there is nothing socially progressive about arbitrariness! Badiou’s recommended political models are the chaotic Maoist cultural revolution of the 1960s and the ephemeral May 1968 Paris uprising. I don’t see that the oppressed of the world gained any benefit from either.

Badiou explicitly endorses arguments of the notorious Nazi apologist Carl Schmitt that were used to justify a permanent “state of exception” in which absolute political power is asserted. This intellectual red-brown coalition is unfortunately being taken seriously by some academic leftists. The unifying theme is the claim that metaphysical support for arbitrariness is the key to achieving social justice. There are much better ways…