The qualitative overcoming of quantity: Hegel`s critique of the mathematical paradigm of modern science

Abstract

It was the goal of the present work to elucidate the historic-philosophical presuppositions of Hegel's critique of infinitesimal calculus and to follow very closely the way in which this critique was effectively carried forth in the intermediary chapter of the Doctrine of Being of the Science of Logic. The first of these two tasks was approached by an interpretative analysis of three chapters of the Phenomenology of Spirit (Force and Understanding, Selfconsciousness and the first of the subdivisions of Reason: Observation of Nature), just as by a confrontation with some aspects of Kant's philosophy (the doctrine of the negative magnitudes) and the further developments of this conceptual starting point in the philosophies of Fichte and Schelling. By doing so it was attempted to show how the transcendental philosophy, due to a fundamental binding to the manner with which Leibniz had invented infinitesimal calculus, lacked a non-quantitative concept of quality; furthermore, it was aimed at showing how the theme of the infinitesimal calculus critique can be observed, already in the Phenomenology of Spirit, as noticeably preparing (in Force and Understanding) and essentially resolving (in Observation of Nature) the dialectical concept of self-consciousness. The second task was, on its turn, carried forth by a detailed reading of the chapters Quality and Quantity of the Science of Logic, through which it became possible to show how the themes brought to light in an introductory manner in the Phenomenology of Spirit were then resolved in the two editions (1812 and 1831) of the first volume of the most important work of Hegel's speculative dialectics. A radical confrontation with the philosophy of Leibniz was, therefore, one of the main red-lines of the present enterprise. In this regard, it became equally necessary to elaborate, departing from Hegel, a reconstruction of the outlines of the historical development of mathematical infinitesimal praxis from Pythagoras to Cauchy, just as to propound - now beyond the scope of Hegel's diagnosis, but essentially based upon it - some observations regarding the development of mathematical analysis in the 19th and 20th centuries.