Actual Infinite – The Postmodern Peripatetic

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…

Here is another area where I find myself with mixed sympathies.

Plato seems to have regarded infinity — or what he called the Unlimited — as something bad. Aristotle argued that infinity exists only in potentiality and not in actuality, a view I find highly attractive. I think I encounter a world of seemingly infinite structure but only finite actualization.

Some time in the later Hellenistic period, notions of a radical spiritual infinity seem to have appeared in the West for the first time, associated with the rise of monotheism and the various trends now commonly called Gnostic. This kind of intensive rather than extensive infinity sometimes seems to be folded back on itself, evoking infinities of infinities and more. The most sophisticated development of a positive theological infinite in later Western antiquity occurred in the more religious rethinking of Greek philosophy by neoplatonists like Plotinus, Proclus, and Damascius.

In 14th century CE Latin Europe, Duns Scotus developed an influential theology that made infinity the principal attribute of God, in contrast to the pure Being favored by Aquinas. Giordano Bruno, burned at the stake in 1600, was a bombastic early defender of Copernican astronomy and notorious critic of established religion who espoused a curious hybrid of Lucretian atomistic materialism, neoplatonism, and magic. He proclaimed the physical existence of an infinity of worlds like our Earth.

Mathematical applications of infinity are a later development, mainly associated with Newton and Leibniz. Leibniz in particular enthusiastically endorsed a speculative reversal of Aristotle’s negative verdict on “actual infinity”. Nineteenth century mathematicians were embarrassed by this, and developed more rigorous reformulations of the calculus based on limits rather than actual infinity. The limit-based formulation is what is generally taught today. Cantor seemingly went in the opposite direction, developing infinities of infinities in pure mathematics. I believe there has been another reformulation of analysis using category theory that claims to equal the rigor of 19th century analysis while recovering an approach closer to that of Leibniz, which might be taken to refute an argument against infinity based solely on lack of rigor according to the standards of contemporary professional mathematicians. One might accept this and still prefer an Aristotelian interpretation of infinity as not applicable to actual things, though it is important to recall that for Aristotle, the actual is not all there is.

The philosophy of Spinoza and even more so Leibniz is permeated with a positive view of the infinite — both mathematical and theological — that in a more measured way was later also taken up by Hegel, who distinguished between a “bad” infinite that seems to have been an “actual” mathematical infinite having the form of an infinite regress, and a “good” infinite that I would gloss as having to do with the interpretation of life and all within it. Nietzsche’s Eternal Return seems to involve an infinite folding back on itself of a world of finite beings. (See also Bounty of Nature; Reason, Nature; Echoes of the Deed; Poetry and Mathematics.)

On the side of the finite, I am tremendously impressed with Aristotle’s affirmative development of what also in a more Kantian style might be termed a multi-faceted “dignity” of finite beings. While infinity may be inspiring or even intoxicating, I think we should be wary of the possibility that immoderate embrace of infinity may lead — even if unwittingly — to a devaluation of finite being, and ultimately of life. I also believe notions of infinite or unconditional power (see Strong Omnipotence; Occasionalism; Arbitrariness, Inflation) are prone to abuse. In any case, ethics is mainly concerned with finite things.