On Conditions (1992), Ch. 1 "The (Re)turn of Philosophy Itself," featuring Mark, Wes, Dylan, and Seth.
Badiou is arguing against contemporary post-structuralist French philosophers like Derrida and Foucault whom he characterizes as denying the existence of truth. Philosophy as a profession has consequently devolved into just being a history of philosophy. Post-structuralists, along with Nietzsche and Wittgenstein among others, play the same role for philosophy now as the sophists did for Plato: Sophists are essential in any age to give philosophy a conversation partner against which it can argue for the positive Truth that comes out of the individual truths of what Badiou calls philosophy's "truth conditions," which are politics, art, science (including mathematics), and love.
So it's not philosophy itself that generates truths, and in fact when it thinks it does, then we get "disaster": the type of dogmatism that led to Stalinism and Nazism. Badiou also distinguishes truths from knowledge, which is routinized, part of something considered established and communicated around. A truth instead emerges through an "event," which is something unexpected, inspirational, special... Essentially something that can't actually be articulated, yet we try to do so anyway. Think about a political revolution, an artistic creation, an act of falling in love, a startling scientific discovery: All these can be events, as described in Badiou's previous book Being and Event (1988).
So why would we need philosophy if these other activities actually generate truths? Philosophy weaves them together into a unified vision of the world, which is something we need in order to be fully human. Philosophy lets us "think truths as compossible," i.e. existing together. Contra Wittgenstein, who famously said, “whereof one cannot speak, thereof one must be silent,” Badiou claims that philosophy exactly tries to speak about what can't strictly be spoken about. He regards Jacques Lacan as his master, and from Lacan we get the idea of the "real," which is the unarticulated residue between language, something that's of vital psychological importance. Badiou refers to "the void" as where Truth resides, and philosophy's role is not to fill that void (again, that would result in dogmatism), but to use that void to sustain philosophy as the "pincers of Truth" that grab individual truths and bring them together. Philosophy, Badiou insists, is "subtractive," in that Truth is subtracted from the labyrinth of meaning, kind of like when a sculptors say they create a thing by cutting away everything that is is not that thing. So the sophists put out arguments, definitions, metaphors -- the whole philosophers' toolkit -- and the philosopher stakes out the truth by subtracting from the sophists' positions, showing how they don't make sense and thereby pointing to something better.
Are you bewildered by this language? So are we! Our goal in this episode is to try to decode all this terminology in order to then in ep. 282 explain what Badiou has to say about one of the particular conditions, love (described in ch. 11 of this same book).
Purchase Conditions or try this online version. A helpful secondary source is Alain Badiou: Key Concepts. The Roger Scruton (very critical) book that Wes brings up is Fools, Frauds and Firebrands: Thinkers of the New Left.