Heidegger’s preference for poetry as the philosophical form par excellence is well known. For Heidegger, science, mathematics, and all other disciplines which attempt to circumscribe and dominate truth by means of the proposition must surely fail, for the Being of beings cannot be submitted to the intellect in this way (hence, the forgetting of being). Poetry, on the other hand, speaks and acts in such a way that being is presented ever anew, so it consequently becomes the locus of truth, or the process of revealing and concealing, aletheia.
Necessarily then, the pinnacle of the intellectual hubris of mankind, 20th century analytic philosophy and mathematics, since it is essentially a large-scale attempt to submit all of being to the language of the proposition, should be understood as a massive failure. The entire field of analytic philosophy is a sort of banging of one’s head up against a wall, expecting that eventually one will be able to pass through. What Badiou points out, however, is that this is not at all what occurred.
Rather than viewing truth as judgment, or as a comment on the adequation of a proposition to the world, Badiou prefers to think of truth as what he calls, following Lacan, “a process in the real.” This means that, no matter what historical situation one finds oneself in, a truth can be had. Truth is not a respecter of locations. For Heidegger, truths may only be located where propositions are absent, and it is not out of bounds to label this as a sort of totalization by restriction. The discoveries of 20th century mathematics cannot have, for Heidegger, produced any level of truth because they were directionally misappropriated from the beginning. Heidegger has determined this area a philosophical crime scene, and no one is allowed in but the forensic specialists, the phenomenologists.
Badiou’s “process in the real”, however, gives us a chance to find a truth in any given situation, should we at least look hard enough. In early 20th century mathematics and the advent of set theory, Badiou finds exactly such a truth in Godel’s Incompleteness theorem (as well as other instances, but we’ll focus just the second Incompleteness theorem for now). For Godel, it is impossible to show from within a mathematical theory that it is itself consistent and non-contradictory, thus any grand theory will be in the end incomplete, non-total. Badiou uses this event to postulate a level of mathematical untotalizability, something first presented the very same mathematicians who advanced the fact there could not be conceived a set of all sets, i.e. a totalization, in the wake of Godel. In this sense, the realm of mathematical set theory, that which has forced being into the cage of the proposition for so long, underwent a truth procedure. This would have been impossible for Heidegger, but for Badiou, there is always “a real” in every situation, and therefore always the potential for a truth.
What I like most about this construal of ontology is that it opens up many more possiblities, a wealth of new sites for philosophy. Badiou often makes the point that his “mathematics as ontology” maxim is meant to point out that ontology is not about describing or revealing the fundamental nature of being, but about the relations amongst elements. Science preforms the former, while philosophy’s job is to analyze the relations of the elements of particular situations, something that is inherently possible in any situation. There’s a level of equality of situations that is conspicuously lacking in the hubris of much post-Kantian philosophy. There is no fruit to be found in a battle amongst the academic elite as there is always a space of compossibility between actors in this play.