Alain Badiou’s 1985 seminar on Parmenides initiated his ongoing dialogue with the pre-Socratic philosopher and set out some of the foundational ideas of his own major philosophical works, including Being and Event. The seminar provides both a new understanding of the importance of Parmenides in establishing the fundamental terms of philosophy and a vital account of the intimate relationship between philosophy and mathematics. For Badiou, Parmenides inaugurates the history of philosophy by linking being, thinking, and nonbeing in a “Borromean knot”—a linkage in which the relationship between any two terms requires the mediation of a third. Badiou traces the permutations of this connection among fundamental concepts as it was transmitted to later thinkers, including Plato, Aristotle, Descartes, Hegel, and Heidegger. He also argues that Parmenides’s account of philosophy establishes a new kind of formal reasoning that, like mathematics, breaks with the Greek mythic and poetic traditions