Badiou's Equations--and Inequalities: A Response to Robert Hughes's "Riven"

Robert Hughes’s article offers an unexpected perspective on Alain Badiou’s work and its impact on the current intellectual and academic scene, a cliché-metaphor that (along with its avatars, such as performance or performative, also a pertinent theoretical term) may be especially fitting in this case, given that Badiou is not only a philosopher but also a playwright. What makes Hughes’s perspective unexpected is its deployment of “trauma” as the main optics of this perspective. While the subject and language of trauma have been prominent in recent discussions, they are, as Hughes acknowledges, not found in Badiou’s writings nor, one might add, in the (by now extensive) commentaries on Badiou. Hughes’s reading of Badiou in terms of trauma rearranges the “syntax” of Badiou’s concepts, as against other currently available readings of Badiou, even if not against Badiou’s own thinking, concerning which this type of claim would be more difficult to make. In this respect Badiou’s thought is no different from that of anyone else. One can only gauge it by a reading, at the very least a reading by Badiou himself, for example, in Briefings on Existence (which I shall primarily cite here for this reason and because it offers arguably the best introduction to his philosophy).

Before proceeding to Hughes’s argument, I sketch the conceptual architecture of Badiou’s philosophy from a perspective somewhat different from but, I hope, complementing that offered by Hughes. The language of conceptual architecture follows Gilles Deleuze and Félix Guattari’s view of philosophy, in What is Philosophy?, as the invention, construction of new concepts (5). This definition also entails a particular idea of the philosophical concept. Such a philosophical concept is not an entity established by a generalization from particulars or “any general or abstract idea” (What is Philosophy? 11–12, 24). Instead, it is a conglomerative phenomenon that has a complex architecture. As they state, “there are no simple concepts. Every concept has components and is defined by them . . . . It is a multiplicity” (16). Each concept is a conglomerate of concepts (in their conventional sense), figures, metaphors, particular elements, and so forth, that may or may not form a unity; as such, it forms a singular, unique configuration of thought. As a philosopher, Badiou is an inventor, a builder of new concepts, just as Deleuze and Guattari are, even when these concepts bear old names, as do some of Badiou’s key concepts, such as event, being, thought, and truth.

A distinctive, if not unique, feature of Badiou’s philosophy, as against that of most other recently prominent figures, is the dominant role in it of mathematics, in particular of mathematical ontology. It is true that major engagements with mathematics are also found in Lacan and Deleuze, both of whom had a considerable philosophical impact on Badiou’s work. There are, however, also significant differences among these thinkers in this respect, and it is worth briefly commenting on these differences in order to understand better Badiou’s use of mathematics and mathematical ontology. For Badiou, to use his “equation,” “mathematics=ontology” (Briefings 59), and “ontology=mathematics.” (Badiou’s first identity is not mathematical, and hence these two identities are not automatically the same, but they appear to be in Badiou.) Reciprocally, Badiou wants to give mathematics a dimension of thought, specifically of ontological thought, which he distinguishes from other, most especially logical, aspects of mathematical thinking. “Mathematics is a thought,” a thought concerning Being, he argues in Briefings on Existence (45–62). It is, accordingly, not surprising that Badiou is primarily interested in foundational mathematical theories, that is, those that aim at ontologically grounding and pre-comprehending all of mathematics, such as set theory introduced by Georg Cantor in the late nineteenth century, and in Badiou’s more recent works, the category and topos theories, as developed by Alexandre Grothendieck in the 1960s. The latter offers a more fundamental mathematical ontology, encompassing and pre-comprehending the one defined by set theory. Indeed, it might be more accurate to rewrite Badiou’s equations just stated as “mathematical ontology=ontology...