Whatever else one might think of his ontology, two major claims animate Badiou’s conception of philosophy. On the one hand, Badiou argues that philosophy itself produces no truths. For Badiou truth always comes from elsewhere, from the domain of praxis. Where one conception of philosophy has it that philosophy is to think the ultimate nature of reality, for Badiou it is always other fields of engagement, or praxis, that think these things and produce these truths. In this connection, it is science (though science largely gets short shrift in his thought), math, love, politics, and art that produce truths. Love, for example, is the encounter of the Two. His favorite examples in this connection are the encounter between Abelard and Heloise or Sartre and Simone de Beauvoir. In their encounter, it is a difference that is encountered, and an entirely new way of thinking the non-relation, the difference, between the Two; a non-relation that produces a new language. The deep ontological truth Badiou sees in the encounter of the Two or their difference is the logic of the “not-all” or that the “whole” is not, for what we find between the Two in their relation of non-relation, are incommensurable universes communicating with one another. The Two does not become a fusional One, but rather this incommensurability becomes the impetus of their relation, perpetually renewing itself not through its sameness, but through its difference.
If philosophy does not produce truths, then what is the vocation of philosophy, according to Badiou? Philosophy, according to Badiou, does not produce truth, but rather thinks truth. In this thesis there is a deep connection between time or history and philosophy. Badiou contends that the vocation of philosophy is to think our present. The thinking of the present consists in the thinking of the truths that populate our world in the present and striving to think them in what Badiou calls their “compossibility”. The term “compossibility”, of course, comes from Leibniz. One way of fruitfully thinking “compossibility” would be as “co-possibility”. Leibniz argued that there are an infinity of possible worlds that could have existed. In speaking of a possible world we are talking about worlds that are “incompossible”. What Leibniz was trying to get at with his idea of compossibility was the idea of interdependence among events. Thus, for example, my existence, my possibility, is dependent upon a whole host of other possibilities. In this world I have brown hair (that is quickly turning grey). The world in which I have blond or red hair is incompossible with this world because the browness of my hair is related to a set of all sorts of other conditions such as my genetics, when my parents conceived me, the time of their meeting, the evolution of the human species, etc., all of which are necessary for me to have brown hair. For Leibniz our universe is a web of interdependencies in which all things depend upon one another.
When Badiou evokes the notion of truth he is not referring to adequation between thought and object. For example, the statement that “Today, on April 10th at 1:10pm, the sun is shining in Frisco” is not, for Badiou, a truth, but is merely a veridical statement. For Badiou “truth” instead refers to the opening of an entirely new domain of practice. Thus, when Badiou refers to contemporary set theory in mathematics as a “truth”, what he means is that it opened an entirely new domain of mathematical thought that is infinite in character, calling for the rethinking of all prior mathematics. When Galileo declared that nature is mathematical, it opened an entirely new way of relating to nature, moving away from the qualitative description of essences in Aristotlean science. When Badiou claims that that Paris Commune is a truth, this is because it opens a new way of thinking the political and the social realm that is infinite in its scope. When he claims that atonal music is a truth, this is because it opens an infinite domain of musical creation and invention. All of these examples of truth are, for Badiou, examples of praxis. With the Paris Commune we get an engagement with the political sphere that calls us to rethink what the social can be and to transform the nature of our social organizations. Galileo opens an infinite domain of scientific investigation in terms of the mathematization of nature. Set theory opens an infinite domain for the production of new theorems, and so on.
What, then, is the difference between the thinking of truth (philosophy) and the production of truth (these praxes). On the one hand, Badiou contends that while praxis produces truth, it is not self-reflexively aware of the truth that it produces. For example, Badiou notoriously claims that ontology does not belong to philosophy at all, but rather it is mathematics that thinks being-qua-being or what can be said of being independent of any beings. However, he adds, while maths think being-qua-being, it does not know that it is thinking being. Indeed, says Badiou, were maths to be are of itself as thinking being-qua-being this would render the exploration of being impossible for the mathematician. According to Badiou, philosophy thinks what a praxis cannot think. We could say that philosophy registers the truth of the praxis. Philosophy, as it were, discloses the truth of the praxis. Here we might make reference to Plato, one of Badiou’s philosophical heroes. Plato is fundamentally struck by the invention of geometry. What can it mean, wonders Plato, that we are able to know properties of figures through thought alone? What can it mean, Plato asks, that these figures studied by the geometer are to be found nowhere in our sense-experience, but can only be grasped in thought? Plato sees geometry as revealing or disclosing the being of being or the true nature of being and builds his philosophy around geometry accordingly. This does not mean that Plato sits around talking about points, lines, planes, and theorems; though occasionally he does. Rather, what it means is that he thinks ideal form in all domains of human experience and engagement. For example, the questions of politics become questions of ideal form in terms of harmony and proportion. Being becomes this domain of the ideal or the form. Plato registers the ontological truth of geometry. Where the geometer is busy thinking points, lines, and planes, Plato grasps the ontological kernel in these strange entities that are the most objective of all and that are universal, but expands this domain well beyond figures.
For Badiou, then, the vocation of philosophy is to make what is implicit, explicit, or to accomplish the self-reflexivity, the self-awareness, of these truths. For example, one of the burning questions of Marxist thought is that of how the proletariat can be made aware of its key historical role. Marx’s work can be seen as registering the truth of this role through his analysis of the social structure. However, for Badiou, it is not simply a question of registering the truth of these praxis, but rather of thinking the compossibility of these truths in a historical present. Where for Leibniz compossibility refers to the interdependence of all events in the unity of a universe, for Badiou compossibility refers to the unity of truths in the eternity of the present (shades of Kierkegaard’s notion of the present, here). This compossibility of truths is essentially a fractal concept. A fractal is a geometry that reflects self-similarity at all levels of scale. Thus, for example, a fractal triangle would have triangles along all of its three sides and then triangles along the sides of these sides and then triangles along the sides of these sides of these sides, and so on to infinity. To think the compossibility of truths is to think this fractal pattern within truths across heterogeneous domains of praxis.
For Badiou there are four domains of truth or four “truth-procedures”: the matheme (science), politics, art/poem, and love. The truths produced in these four domains are historically variable. Thus, for example, Plato thinks love as the fusion of the Two into One, whereas Lacan thinks love as the Two qua Two without One. What, then, is the fractal unity of a present? Badiou claims that with Cantorian set theory it becomes possible to think being-qua-being as pure multiplicity-qua-multiplicity without One. That is, the sets of set theory are infinitely decomposable without limit, and it is for this reason that the concept of “set” is itself left undefined within ZF set-theory (as to define it would be to identify it or make it subject to the operator of the One). Set theory thus enables us to think being as pure multiplicity and infinite dissemination. It will be noted here that Badiou, with this move, does a bit of judo on the the various philosophies of difference. Where Derrida, for example, had argued that difference or multiplicity is the ruin of ontology (ontology being onto-theology), Badiou shows that in fact multiplicity-qua-multiplicity is being or the truth of ontology. In the domain of politics, May of ’68, the Paris Commune, etc., conceive a form of the political without identity or the One. That is, political participation is no longer about discretely defined identities (as in the case of Plato’s Republic, where everyone has a carefully defined identity assigning their function in the City), but rather we have the political field as a field in excess of all identities or the political field as multiplicity without One. In Love we get the thinking of the Two as Two, the Two per se, without a fusional One where the Two merges into the Same. The Two becomes the logic of the “not-all” or the constitutive incompleteness of the relation. And finally, in art, whether we are speaking of the great poet Pessoa, Schoenberg, Mallarme, etc., we get art as the exploration of infinite dissemination and undoing of identity. Across these four domains of praxis we thus get a fractal pattern unified around a particular ontological vision that captures the unity of the present or what the present is.
I do not know that I share Badiou’s particular characterization of our present, but I think he’s hit upon something fundamental in just what it is that philosophy does or what the vocation of philosophy might be. What Badiou seems to be declaring is that philosophy strives to think that which provokes thought in the present, to reflect on its meaning, and to capture its pulsating essence or what is eternal within it. In doing so, philosophy is not a self-contained discipline, but rather always takes its material, its matter, its “stuff” of thought, from outside and elsewhere. If the identity of philosophy, what philosophy is, is so slippery, if we find that it is so difficult to say just what the questions of philosophy are and what the objects of philosophy are because they are so historically variable, then perhaps this is because philosophy itself has no identity, but is instead always a relation to its outside. Rather, philosophy is the attempt to think that which is new and transformative in its present, or what is unprecedented with regard to its history. Socrates’ dictum– “know thyself!” –remains operative in all of these configurations, but this self-knowledge is not the knowledge of a self, nor amenable to eternal answers, but is instead a response to the question of what our time is, of what it is thinking without reflecting that thought. It appears that we can even read the history of philosophy in these terms. Greek thought becomes a response to the outside of the Geometrical and Political truth of its time. Perhaps Medieval thought becomes a response to the Christ-event and the overturning, against all odds, of the Roman empire by the meekest of the meek. 17th century thought becomes a response to the Galileo event. And in our time? What is it that we are to think today? What is it that provokes thought today? What is it that is unprecedented for us today? Without opening ourselves to the outside and undergoing a destitution of our identities as academic philosophers this question cannot even be posed or initiated as a sequence of thought.