Hegel's analysis (Q2351943)

The extensive and substantial paper discusses mainly a 40page annotation which the influential German philosopher G. W. F. Hegel (1770--1831) added to the first volume of his ``Wissenschaft der Logik'' (Nürnberg 1812). The note was entitled ``The notion of mathematical infinity'', and it discussed philosophically the foundations of infinitesimal analysis. Hegel refers to philosophical precursors such as the Eleatic school and Spinoza which described the dialectic character of the notion of infinity and the problems to define it consistently. Hegel goes in detail into the notion of the differential quotient, opposing its definition as a limit but recognizing the use of the notion in applications. Hegel has much positive to say about L. Carnot's ``Reflections on the Infinitesimal Calculus'' of 1797 and about Lagrange's ``Theory of Functions'' (1797). In Newton he criticizes both the logical insufficiencies of the differential quotient and the lacking distinction of the mathematical realm from physical reality. At the end of his article, Boehme polemicizes against a paper by M. Wolff (1986) according to which the second edition of Hegel's ``Logik'' (1832) contains a discussion of Cauchy's foundations of analysis. He insists that Hegel never mentions Cauchy's name and the only bridge between the two may have been an essay-review by Hegel's colleague in Berlin, H. Dirksen (1788--1850), who, however, does not go much into the details of Cauchy's lectures. Boehm reports that Hegel's discussion of the foundations of mathematical analysis has not had influence on mathematicians who considered it apparently as non-mathematical. Boehm does not discuss the possible influence of Hegel's reflections on the study of the foundations of mathematics by Hegel's follower Karl Marx.