Despite physicists, proof is essential in mathematics. (English) Zbl 1052.00512

Contributions to the recent debate about the significance of proofs [see M. F. Atiyah and et al., Bull. Am. Math. Soc., New Ser. 30, No. 2, 178–207 (1994; Zbl 0803.01014)] are clarified or corrected, among them new advances in physics and their contribution to mathematical insights, and the case of algebraic geometry in Italy after 1880. The main part of the paper is a whirlwind tour of the author’s philosophy. He claims that foundational studies took a wrong turn back in 1908 with the Zermelo axioms and Russell’s theory of types, eventually leading to set-theoretic Platonism and to the fact that many philosophers of mathematics depend too much on mathematical logic, and too little on any acquaintance with mathematics. Proof theory as it exists is too narrow to supply insights in the construction of mathematical proof. In contrast to all this, mathematics must be seen as that part of science which applies in more than one empirical context. The same formal structure may arise in different uses, but a theorem must be formally proved to be of use in other cases. Unlike Platonism, this view accounts for the usefulness of mathematics in other sciences.


00A35 Methodology of mathematics
00A30 Philosophy of mathematics
03A05 Philosophical and critical aspects of logic and foundations


Zbl 0803.01014
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