Paul Redding

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.

In this paper I return to the familiar territory of the Lord-Bondsman "dialectic" in Hegel's Phenomenology of Spirit in order to raise the question of the relation of Hegel's use of the theme of recognition there to Fichte's. Fichte had introduced the notion of recognition in his Foundations of Natural Right , to "deduce" the social existence of humans within relations of mutual recognition as a necessary condition of their very self-consciousness. However, there it also functioned as part of a solution to a problem within the work on which the theory of rights was meant to be based, the earlier Foundation of the Complete Wissenschaftslehre of 1794-5. In Hegel's classic account in chapter 4 of the Phenomenology we find recognition offered as a solution to a problem within an account of "self-consciousness" that has a number of clearly Fichtean features. But I suggest that to the degree that the lord-bondsman episode there expresses any "theory of recognition", it is not Hegel’s own theory but rather his interpretation of Fichte's , a theory of which he is critical. Freed from this misleading assumption that the "lord-bondsman dialectic" represents something deep about Hegel's own philosophy, we might then be more able to get clearer about Hegel's actual views about recognition and the role it plays in his own philosophy.