Poststructuralism and Deconstruction: A Mathematical History

Article excerpt

Examples are not lacking of philosophers whose outlook was inspired or in some way influenced by thinking about the meaning of mathematics: Wittgenstein, Husserl, Russell, before them Peirce, and more recently Badiou, to name only the clearest cases in point. Philosopher and mathematical logician Jean Cavailles was undoubtedly influential on post-war philosophy in France, and the effects of his mathematical critique of the philosophy of the subject, especially of Kant and Husserl, carried an important impulse from formalist mathematics to twentieth-century French philosophy. Even Heidegger, although critical of formal logic and technical reason (like some mathematicians of his day), followed the debate on the nature of mathematical knowledge and may have been provoked by it--especially by the controversy regarding the time-continuum--to attempt in his Being and Time (1927) to "destroy" traditional metaphysics and thus transgress the philosophical options that found themselves at loggerheads over the question of the foundations of mathematics.

The link works in the other direction, too, although it has become something of a rarity to find mathematical articles that contain references to Kant, Fichte, Schopenhauer and Nietzsche. During the foundational debate in the 1920s, one could see this in the articles of Hermann Weyl and L.E.J. Brouwer; Weyl was actively interested in phenomenology and maintained correspondence with Husserl, while Kurt Godel is known to have been a serious reader of Kant and Husserl. Much earlier, Hermann Grassmann (a major influence on Alfred North Whitehead) had built on the ideas of his father, Justus Grassmann, who wrote under the influence of Schelling.(1) These links have not been severed, even if they remain unstated. Thus, for example, the philosophically reticent Bourbaki collective was, according one of its members, a "brainchild of German philosophy". One finds the influence of Husserl and Heidegger in the mathematical essays of Gian-Carlo Rota, and at least implicitly in the work of Petr Vopenka on alternative set theory. It may not be the norm, but examples of cross-fertilization are not as difficult to find as the oversimplified binarism of "two cultures" would have us believe.

My goal here is to indicate the relevance of mathematics to several important points made by Jacques Derrida. A number of Derrida's arguments bear resemblance to critiques of logic and excesses of formalist mathematics. These objections hark back to the ideas of "intuitionist" mathematicians who--some, I think, under the influence of German romantic idealism--rebelled in the early 1900s against the hegemony of formal logic and the symbolic reduction of all thought to computation. The situation is not quite that simple, since Derrida apparently also employs certain ideas of formalist mathematics in his critique of idealist metaphysics: for example, he is on record saying that "the effective progress of mathematical notation goes along with the deconstruction of metaphysics."(2)

Derrida's position can, I think, be interpreted as a sublation of two completely opposed schools in mathematical philosophy. For this reason it is not possible to reduce it to a readily available philosophy of mathematics. One could perhaps say that Derrida continues and critically reworks Heidegger's attempt to "deconstruct" traditional metaphysics, and that his method is more "mathematical" than Heidegger's because he has at his disposal the entire pseudo-mathematical tradition of structuralist thought. He has implied in an interview given to Julia Kristeva that mathematics could be used to challenge "logocentric theology," and hence it does not seem unreasonable to try looking for mathematical analogies in his philosophy.

A word of caution, though. The similarities I will outline here are similarities of argumentative techniques, not of philosophical outlooks. The analogies--which are informed and limited by my own interpretive ability and my belief that mathematics and continental philosophy are deeply related--are not to be confused with the gross misstatement that "mathematicians have done it all. …

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