Hegel, Concepts, and Computation

Gottfried Ploucquet, a teacher at the Tubingen seminary when Hegel was a student there, had been one of the few philosophers to take up Leibniz’s mathematized logic, including his project of reducing logic, and thought itself, to computational processes. In his Science of Logic, Hegel briefly discusses this project when expanding on his own “subjective” logic. The general tenor of the response is predictable. Computational logic seeks to mechanize conceptual processes, but conceptuality itself distinguishes free spiritual beings from machines. Beneath the surface, however, Hegel’s attitude to the relation of computation to conceptual reasoning is more complex. Here I argue that in Book I of his Logic, Hegel, following the approach of Plato in his late dialogues, treats a certain mathematical conception of number, the Neopythagorean triadic monad, as a model for the concept itself. In the section Quantity, Hegel focuses on the incommensurability between discrete and continuous quantities, the numbers of arithmetic and the lines, areas and volumes of geometry. This incommensurability had been discovered by the Pythagoreans and in his later writings, Plato had adopted a proposal for mediating it, attempting to generalize it to a solution of the conceptual incommensurability between the eternal realm of being and the transient realm of becoming.  In line with Plato’s attempt, Hegel presents an account of the development of mathematical practices in which the concept of number from mere counting unit to a triadic form mediating numbers and geometric continua. This structure will in turn provide a model for his own later syllogism. This role for mathematics for Hegel is to be understood as in line with Plato’s later attempts to mediate being and becoming in ways in which eternal Ideas can be approximated in the form of worldly surrogates manifesting this triune structure. Conceptuality cannot be reduced to computation, but relations among computational processes nevertheless reveal much about the nature of conceptuality.

Keywords:

Hegel , Plato , Mathematics , Computation , Conceptuality , Incommensurabiliy