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

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ABSTRACT: One of the central tasks of Badiou'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 'truth'. 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 Cohen'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 'law of the subject: 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 Badiou'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 Badiou's resolutely classical approach to ontology, and his identification of ontology as a truth procedure.

KEYWORDS: Badiou; Brouwer; Intuitionism; Forcing; Genericity; Subject

'There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source--the infinite'. (1)

--G.W. Leibniz


One of the principal stakes of Alain Badiou's Being and Event is the articulation of a theory of the subject against the backdrop of the thesis that ontology, the science of being qua being, is none other than axiomatic set theory (specifically, the Zermelo-Fraenkel axiomatization ZF). In accordance with this thesis, every presentation of what there is--every situation--is held to be thought 'in its being' when thought has succeeded in formalizing that situation as a mathematical set. (2) The formalization of the subject, however, proceeds somewhat differently. Badiou insists that set theory alone cannot furnish a complete theory of the subject, and that for this task one needs the essentially non-mathematical concepts of time and the event. It is nevertheless possible, Badiou maintains, to determine the set-theoretical form of the subject's ontological infrastructure--the form of its 'facticity', to borrow a term from Sartre. (3) In Being and Event, the sought-after structures are declared to be found in the two concepts that Paul J. …


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