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**Philosophie mathématique.**
*(French)*
Zbl 0105.24408

Histoire de la pensée. 6. Paris: Hermann & Cie. 274 p. (1962).

This book comprises three separate contributions to the philosophy of mathematics: I: an essay (150 p.) “Remarques sur la formation de la théorie abstraite des ensembles”, originally published in 1938 and reviewed in Zbl 0021.30201; II: “the Cantor-Dedekind correspondence”, edited by Emmy Noether and Jean Cavaillès; III: a further essay (21 p.) “Transfini et Continu,” written in 1940 or 1941 and now published, posthumously, for the first time.

The “Remarques” is still, after twenty-five years, a most valuable contribution to the literature of the theory of sets. In it the author makes a thorough and penetrating analysis of the historical origins of the principal notions of set theory in the researches that were being made in the middle of the nineteenth century into the behaviour of real functions and the structure of the real continuum; and he goes on to show how these notions crystallized by degrees, in the mind of Cantor, into the material of a new mathematical discipline. The essay is thus a much-needed complement to the many books that are now available on the formal axiomatic treatment of sets; and besides this, it offers to the philosopher of mathematics a wealth of stimulating reflections on questions concerning foundations. The reprinting, in the same volume, of the letters exchanged by Cantor and Dedekind during the years 1872–1899 gives the argument of the author added weight as a historical study.

The volume’s second essay, “Transfini et Continu”, may best be looked upon as an appendix to the “Remarques”’, prompted by Gödel’s publication in 1939 of his results on the independence of the axiom of choice and the continuum hypothesis. This essay incorporates an interesting discussion of the work of Church and Kleene on general recursiveness and constructivity, and of the philosophical implications of attempts such as Gentzen’s to enlarge Hilbert’s conception of finitary reasoning to cover transfinite induction over some suitably constructive segment of the second number class. The author maintains that no absolute separation can ever be made between a domain of mathematics that is intuitively secure and a further domain where justification is called for. All thought is relative to the conceptual resources of the tradition in which it is rooted, and “les liaisons intellectuelles dépassent l’histoire empirique: c’est leur développement dialectique qui assure à la fois le mouvement de celles-ci et par elles-mêmes la permanence de leur validité”.

The “Remarques” is still, after twenty-five years, a most valuable contribution to the literature of the theory of sets. In it the author makes a thorough and penetrating analysis of the historical origins of the principal notions of set theory in the researches that were being made in the middle of the nineteenth century into the behaviour of real functions and the structure of the real continuum; and he goes on to show how these notions crystallized by degrees, in the mind of Cantor, into the material of a new mathematical discipline. The essay is thus a much-needed complement to the many books that are now available on the formal axiomatic treatment of sets; and besides this, it offers to the philosopher of mathematics a wealth of stimulating reflections on questions concerning foundations. The reprinting, in the same volume, of the letters exchanged by Cantor and Dedekind during the years 1872–1899 gives the argument of the author added weight as a historical study.

The volume’s second essay, “Transfini et Continu”, may best be looked upon as an appendix to the “Remarques”’, prompted by Gödel’s publication in 1939 of his results on the independence of the axiom of choice and the continuum hypothesis. This essay incorporates an interesting discussion of the work of Church and Kleene on general recursiveness and constructivity, and of the philosophical implications of attempts such as Gentzen’s to enlarge Hilbert’s conception of finitary reasoning to cover transfinite induction over some suitably constructive segment of the second number class. The author maintains that no absolute separation can ever be made between a domain of mathematics that is intuitively secure and a further domain where justification is called for. All thought is relative to the conceptual resources of the tradition in which it is rooted, and “les liaisons intellectuelles dépassent l’histoire empirique: c’est leur développement dialectique qui assure à la fois le mouvement de celles-ci et par elles-mêmes la permanence de leur validité”.

Reviewer: G. T. Kneebone

### MSC:

00A30 | Philosophy of mathematics |

00B60 | Collections of reprinted articles |

03A05 | Philosophical and critical aspects of logic and foundations |