Paul Livingston

Frege ridiculed the formalist conception of mathematics by saying that the formalists confused the unimportant thing, the sign, with the important, the meaning. Surely, one wishes to say, mathematics does not treat of dashes on a bit of paper. Frege’s idea could be expressed thus: the propositions of mathematics, if they were just complexes of dashes, would be dead and utterly uninteresting, whereas they obviously have a kind of life. And the same, of course, could be said of any proposition: Without a sense, or without the thought, a proposition would be an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in order to make a live proposition is something immaterial, with properties different from all mere signs. But if we had to name anything which is the life of the sign, we should have to say that it was its use. (Blue Book, p.

I consider the relationship of Badiou’s schematism of the event to critical thought following the linguistic turn as well as to the mathematical formalisms of set theory. In Being and Event, Badiou uses formal argumentation to support his sweeping rejection of the linguistic turn as well as much of contemporary critical thought. This rejection stems from his interpretation of set theory as barring thought from the 'One-All' of totality; but I argue that, by interpreting it differently, we can understand this implication in a way that is in fact consistent with the critical and linguistic methods Badiou wishes to reject.

Over a period of several decades spanning the origin of the Vienna Circle, Schlick repeatedly attacked Husserl''s phenomenological method for its reliance on the ability to intuitively grasp or see essences. Aside from its significance for phenomenologists, the attack illuminates significant and little-explored tensions in the history of analytic philosophy as well. For after coming under the influence of Wittgenstein, Schlick proposed to replace Husserl''s account of the epistemology of propositions describing the overall structure of experience with his own account based on the structure of language rather than on the intuition of essences. I discuss both philosophers'' accounts of the epistemology of propositions describing the structure of experience. For both philosophers, this epistemology was closely related to the general epistemology of logic; nevertheless, neither philosopher had a completely coherent account of it. Comparison of the two approaches shows that perennial and severe theoretical obstacles stand in the way of giving an epistemology of the structure of experience, a central requirement for both philosophers'' theories. Consideration of these obstacles sheds a new light on the reasons for the historically decisive split between the continental and the analytic traditions, as well as on the subsequent development of the analytic tradition away from the structural description of experience.

In this essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the One and the Indefinite Dyad provides, in the later Plato, a unitary theoretical formalism accounting, by means of an iterated mixing without synthesis, for the structural origin and genesis of both supersensible Ideas and the sensible particulars which participate in them. As these commentators also argue, this duality furthermore provides a maximally general answer to the problem of temporal becoming that runs through Plato’s corpus: that of the relationship of the flux of sensory experiences to the fixity and order of what is thinkable in itself. Additionally, it provides a basis for understanding some of the famously puzzling claims about forms, numbers, and the principled genesis of both attributed to Plato by Aristotle in the Metaphysics, and plausibly underlies the late Plato’s deep considerations of the structural paradoxes of temporal change and becoming in the Parmenides, the Sophist, and the Philebus. After extracting this structure of duality and developing some of its formal, ontological, and metalogical features, we consider some of its specific implications for a thinking of time and ideality that follows Deleuze in a formally unitary genetic understanding of structural difference. These implications of Plato’s duality include not only those of the constitution of specific theoretical domains and problematics, but also implicate the reflexive problematic of the ideal determinants of the form of a unitary theory as such. We argue that the consequences of the underlying duality on the level of content are ultimately such as to raise, on the level of form, the broader reflexive problem of the basis for its own formal or meta-theoretical employment. We conclude by arguing for the decisive and substantive presence of a proper “Platonism” of the Idea in Deleuze, and weighing the potential for a substantive recuperation of Plato’s duality in the context of a dialectical affirmation of what Deleuze recognizes as the “only” ontological proposition that has ever been uttered. This is the proposition of the univocity of Being, whereby “being is said in the same sense, everywhere and always,” but is said (both problematically and decisively) of difference itself.

Quine’s thesis of translational indeterminacy stands as one of the most central, surprising, and influential results of analytic philosophy in the twentieth century. The suggestion that the meaning of linguistic terms and sentences, as shown in the situation of radical translation, is systematically indeterminate and undetermined by actual speech practice, has for decades engendered thought and reflection on the nature and basis of linguistic meaning. And even beyond this surprising moral itself, Quine’s theoretical use of the radical translation scenario has been wholly or partly responsible for a wide variety of theoretical programs in analytic philosophy since the middle of the last century, including Davidsonian semantics, Dummett’s consideration of the form and shape of knowledge about meaning, Rorty’s celebration of “philosophy without foundations,” and, more recently, Brandom’s pragmatist project of analyzing the roots of meaning and normativity in intersubjective and socially regulated speech behavior. On many retellings of the history of the analytic tradition, the suggestion that the philosophical analysis of language and meaning depends on reflection about the structure of intersubjective linguistic practice marks a general sea change in the methodological self-conception of analytic philosophy, away from the brands of metaphysical, epistemological, or purely syntactic analyses that had characterized its earlier projects, and toward the more concrete, pragmatic, and empirically influenced reflective practice that philosophers inherit today.