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Consequences of Diagonalization

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*Metamathematics *(alternately *metalogic*) may best be regarded as naming an experiment in self-reference: what happens when we realize not only that mathematics is encoded, but that the encoding has, in turn, mathematical properties, raising the problem of* the mathematics of the code of mathematics*? Is there knowable truth in this abyssal reflection, or do we have to be content here with the usual duplicity of the understanding and its imaginary supplement? Cantor’s discovery of diagonalization, arguably the major event of logical history since Plato, shows that there is some knowledge that can be articulated in this place, while at the same time the truth of diagonalization protects itself in a completely novel way from reduction to the knowledge we acquire of it. For while diagonalization is the metamathematical “technique” *par excellence*, it is also something more and less than a technique, something which functions and dysfunctions at the very edge of formalization. If we cannot avoid assimilating diagonalization to *tekhne *at a first pass, neither should we avoid the rebound of what diagonalization says and shows upon the concept of *tekhne*, and likewise on the possibilities and limits of mathematics, logic, rationality, technology, system, science, *etc*. This reversal and speaking out of turn—calling into question the terms in which you characterize it—is perhaps one of the signature behaviors of diagonalization and of what happens at this extreme point of formalization to complicate and redouble the link of form to existence which philosophy always has to rethink.

Diagonalization introduces a novel form of temporality, neither historicist nor transcendental, into mathematics. Thus, Cantor’s discovery initializes a series of diagonal theorems, each at once *negative *and irreducibly *creative *(renewing immediately this disparaged potential of dialectical thought), and each characterized both by the form

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