Between object construction and structure analysis. On Jules Henri Poincaré’s philosophy of mathematics. (Zwischen Objektkonstruktion und Strukturanalyse. Zur Philosophie der Mathematik bei Jules Henri Poincaré.) (German) Zbl 0846.00002

Heinzmann, Gerhard

Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik. 10. Göttingen: Vandenhoeck & Ruprecht. 166 S. (1995).

This thoroughgoing investigation gives a description and interpretation of Henri Poincaré’s philosophy of mathematics. Clarification is needed because Poincaré often uses even its central terms as “analysis”, “intuition”, “convention” inconsistently due to his missing interest in elaborating his writings accurately (p. 23). An example may be seen in the different states of intuition in geometry and arithmetic (cf. pp. 24-25). As a crucial mark of Poincaré’s philosophy of mathematics the author identifies the unity of describing and constructing mathematical objects (p. 12), mediated by a pragmatism close to Peirce’s (p. 19). The author aims at showing (p. 12) that Poincaré’s pragmatic philosophy can be suitably interpreted in terms of a pragmatic constructivism (Ch. S. Peirce, K. Lorenz). Following J. Vuillemin he prefers to call Poincaré’s philosophical pragmatism not “conventionalism” but “occasionalism” in order to stress the epistemological state of conventions in mathematics, which concerns the relations between experience and theory.
Poincaré’s philosophy of mathematics is considered in geometry (by pointing out the influences of Hermann von Helmholtz’s anti-Kantian theory of intuition [Anschauung]), in logic and set theory, and in arithmetic. The author shows that an important motive for Poincaré’s criticism on logicism can be seen in his opinion that mathematical proofs have a synthetic character (p. 71). His logic is compared with Peirce’s theory of theorematic reasoning.
A main topic in the last chapter is a discussion of Poincaré’s attempts to show the nonlogical character of mathematical induction, complemented by an excursion on transfinite induction.

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
01A55	History of mathematics in the 19th century
01A60	History of mathematics in the 20th century
03-03	History of mathematical logic and foundations
00-02	Research exposition (monographs, survey articles) pertaining to mathematics in general
01-02	Research exposition (monographs, survey articles) pertaining to history and biography

Keywords:

conventionalism; pragmatism; occasionalism; foundations of geometry; logic; set theory; arithmetic induction