The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus

Abstract

One of the central tasks of Badioursquo;s Being and Event is to elaborate a theory of the subject in the wake of an axiomatic identification of ontology with mathematics, or, to be precise, with classical Zermelo-Fraenkel set theory. The subject, for Badiou, is essentially a free project that originates in an event, and subtracts itself from both being qua being, as well as the linguistic and epistemic apparatuses that govern the situation. The subjective project is, itself, conceived as the temporal unfolding of a lsquo;truthrsquo;. Originating in an event and unfolding in time, the subject cannot, for Badiou, be adequately understood in strictly ontological, i.e. set-theoretical, terms, insofar as neither the event nor time have any place in classical set theory. Badiou nevertheless seeks to articulate the ontological infrastructure of the subject within set theory, and for this he fastens onto Cohenrsquo;s concepts of genericity and forcing: the former gives us the set-theoretic structure of the truth to which the subject aspires, the latter gives us the immanent logic of the subjective procedure, the ldquo;law of the subjectrdquo;. Through the forcing operation, the subject is capable of deriving veridical statements from the local status of the truth that it pursues. Between these set-theoretic structures, and a doctrine of the event and temporality, Badiou envisions the subject as an irreducibly diachronic unfolding of a truth subtracted from language, a subject which expresses a logic quite distinct from that which governs the axiomatic deployment of his classical ontology. This vision of the subject is not unique to Badioursquo;s work. We find a strikingly similar conception in the thought of L.E.J. Brouwer, the founder of intuitionist mathematics. Brouwer, too, insists on the necessary subtraction of truth from language, and on its irreducibly temporal genesis. This genesis, in turn, is entirely concentrated in the autonomous activity of the subject. Moreover, this activity, through which the field of intuitionistic mathematics is generated, expresses a logical structure that, in 1963, Saul Kripke showed to be isomorphic with the forcing relation. In the following essay, I take up an enquiry into the structure of these two theories of the subject, and seek to elucidate both their points of divergence and their strange congruencies; the former, we will see, primarily concern the position of the subject, while the latter concern its form. The paper ends with an examination of the consequences that this study implies for Badioursquo;s resolutely classical approach to ontology, and his identification of ontology as a truth procedure.