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**Kant and mathematics.
(Kant und die Mathematik.)**
*(German)*
Zbl 1033.00006

Akademische Studien und VortrĂ¤ge 5. Halle: Hallescher Verlag (ISBN 3-929887-30-4). xv, 255 S. (2001).

This volume provides a very original approach to Kant’s philosophy of mathematics. It is based on non-conformist educational material presented to some students of mathematics and physics in the former GDR.

Kant’s ideas concerning an epistemological foundation of mathematics are developed by an exemplary textbook-style introduction to mathematics in the Kantian spirit, illustrated by quotations from Kant’s works, mostly from his “Critique of pure reason”. The author succeeds, on the one hand, in showing the interrelation of Kant’s philosophical wording with what we learn as arithmetic at school, and on the other, in giving an intuitive interpretation of Kant’s philosophy of mathematics. The systematic objective of this volume is to show that F. Klein’s epistemological problem, according to which the consistency proof for the basic laws of arithmetic says nothing about the applicability of these laws to real world situations, has found a solution in Kant (p. XII–XIII, cf. Section 36). This is directed against D. Hilbert’s “intellectual standpoint” (cf. Section 37) as exposed, e.g., in Hilbert’s metamathematics. In a Kantian reconstruction, Hilbert’s (and Leibniz’s) idea of an arithmetical formula such as \(3 + 2 = 5\) consists in a logical determination of a concept by reason (Vernunft), whereas, according to Kant, it is only a schematically regulated single determination by understanding (Verstand). So Hilbert takes the presupposed axiom system as the concept of an object, for the author the expression of a transcendental fallacy.

In sum this volume provides a fresh approach worth to be considered in the philosophy of mathematics and in the interpretative work on Kant’s mathematical epistemology.

Kant’s ideas concerning an epistemological foundation of mathematics are developed by an exemplary textbook-style introduction to mathematics in the Kantian spirit, illustrated by quotations from Kant’s works, mostly from his “Critique of pure reason”. The author succeeds, on the one hand, in showing the interrelation of Kant’s philosophical wording with what we learn as arithmetic at school, and on the other, in giving an intuitive interpretation of Kant’s philosophy of mathematics. The systematic objective of this volume is to show that F. Klein’s epistemological problem, according to which the consistency proof for the basic laws of arithmetic says nothing about the applicability of these laws to real world situations, has found a solution in Kant (p. XII–XIII, cf. Section 36). This is directed against D. Hilbert’s “intellectual standpoint” (cf. Section 37) as exposed, e.g., in Hilbert’s metamathematics. In a Kantian reconstruction, Hilbert’s (and Leibniz’s) idea of an arithmetical formula such as \(3 + 2 = 5\) consists in a logical determination of a concept by reason (Vernunft), whereas, according to Kant, it is only a schematically regulated single determination by understanding (Verstand). So Hilbert takes the presupposed axiom system as the concept of an object, for the author the expression of a transcendental fallacy.

In sum this volume provides a fresh approach worth to be considered in the philosophy of mathematics and in the interpretative work on Kant’s mathematical epistemology.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

00A30 | Philosophy of mathematics |

00-02 | Research exposition (monographs, survey articles) pertaining to mathematics in general |