The paper is a full-scale study on Giuseppe Peano’s philosophy of mathematics, published with several rare photographs and a list of Peano’s writings (pp. 322-335) taken (with corrections) basically from H. C. Kennedy’s pioneer work ‘Life and work of Giuseppe Peano’ [Dordrecht, Reidel (1980; Zbl 0429.01015)]. It concentrates on the mathematical and logical contexts of Peano’s axiomatization of arithmetic (1893). The author shows convincingly that Peano’s axiomatization was not intended to contribute to the logical foundations of mathematics, but that it was a reaction to the 19th century mathematics’ quest for rigor caused by the discovery of non-Euclidean geometry which undermined the trust in Euclidean geometry as guiding mathematical doctrine, and by the discovery of monster formulae in analysis. In particular the author tries to defend three theses (cf. p. 205): (1) Peano formulated his axioms to give a clear and rigorous presentation of arithmetic, and, thus, of mathematics in general. (2) Peano was not interested in reducing mathematics to logic. (3) Even limited to the rigor in symbolism, Peano’s sort of “foundationism” was a failure.
The material is organized in eight chapters. Chapter 1 (pp. 208-224) gives an outline of the history of the concept of mathematical rigor from ancient times up to Cauchy. Chapter 2 (pp. 225-241) is devoted to Cauchy’s rigorous calculus and Weierstrass’s refinements. Chapter 3 (pp. 242-250) gives a brief presentation of Peano’s life and work following Kennedy’s biography. Chapter 4 (pp. 251-265) sketches the state of art in symbolic logic at Peano’s time and its emergence since Leibniz. Chapter 5 (pp. 266-280) gives a detailed presentation of the stages in Peano’s work which led him to the formulation of his axioms. Chapter 6 treats his mathematical work following the axiomatization of arithmetic, especially his discovery of “Peano’s curve” which showed the limited reliability of intuition in mathematics. Chapter 7 (pp. 287-300) discusses Peano’s axioms in detail. The final Chapter 8 relates Peano’s conception to found mathematics to formalism, logicism and intuitionism.
The author stresses that Peano tried to use logic as an instrument in mathematics (p. 206). For him this indicates the originality of Peano’s approach compared with the logical systems of his predecessors and fellow logicians. This judgement can be doubted, since almost every mathematician doing logic up to the publication of Whitehead’s and Russell’s ‘Principia mathematica’ (1910/13) regarded the mathematization of logic as a conditio sine qua non for its applicability in mathematics. The best example might be Ernst Schröder who thought to use his algebra of relatives as a universal language being a device to reformulate all branches of mathematics and physics. The paper under review is an impressive example of contextual research in the history of mathematics. It proves the need to investigate the history of logic and foundations within a broader perspective on the development of mathematics and philosophy in general.