Taking as his starting point a juxtaposition of Plato and Aristotle, Badiou explores in this article the relationship between art and mathematics. While for Plato both realms present independent orientations for thought, in Aristotle the relation to being and to the idea of both mathematics and art is withdrawn. Two problematic questions arise from this: That of knowing whether art and mathematics are distinct but nevertheless connected in being oriented toward the idea, and that of knowing whether the problem of form is the point at which art and mathematics intersect. Badiou establishes that there are four different contemporary tendencies that present an answer these problems: The Platonic, the Nietzschean, the Aristotelian, and the Wittgensteinian.
We all know that the relationship between mathematical activity and artistic creation is a very old one. We know that for a start the Pythagoreans tied the science of number not merely to the movements of the stars but to musical modes. We know that Babylonian and Egyptian architecture presupposed elaborate geometrical knowledge, even if the notion of demonstration had still not been won. Further back still, we find formal, or abstract, outlines mixed in with animal representations, in the great prehistoric decorations, without our knowing precisely to what it is that these mixtures refer.
For the philosopher that I am, or that I believe I am, the entry into our question, as so many others, passes through the contrasting disposition between Plato and Aristotle.
For Plato, mathematics is fundamental in the sense that it mediates between, on the one hand, experience, or the relation to the sensory world, and, on the other, pure intellection, or dialectical movement. Plato exalts mathematics from a point of view that relates it to being in itself, the form of which is what he calls the Idea. He nonetheless sees its impurity, which comes from its having to affirm its hypotheses – we say its axioms – without being able to infer them from a supreme general principle. Whence its inferiority relative to the dialectic. Yet this should not conceal its superiority over all forms of empirical knowledge. And, especially as mathematics is more structural, less bound to unverifiable intuitions. Plato would have surely admired the refined constructions of Galois, or of Grothendieck. He would have applauded the objectivist, ontological vision of mathematics, that of Kronecker, for example. In the order of the contemporary philosophy of mathematics, he would have rallied to Gödel’s simple realism, or to Albert Lautman’s dialectical realism.
Art, as for it, Plato holds in suspicion, due to its very own readiness to imitate natural objects. Plato is the first formalist, in the very precise sense of a theory of forms. For him, every movement of genuine thought aims at a Form, which is snatched from the real and transcends it. The imitative arts, descriptive poetry or painting, remain captive to an immediate form, at the edge of the formless, instead of separating themselves from it to exhibit a pure form, of which immediate forms are only weak consequences. Plato stigmatizes the purely decorative, or purely melodramatic, effects of trompe-l’œil painting or blood-soaked tragedies....