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**Méthode axiomatique et formalisme. Essai sur le problème du fondement des mathématiques. Introduction de Jean-Toussaint Desanti, préface de Henri Cartan.**
*(French)*
Zbl 0549.03001

Paris: Hermann. VI, 196 p. (1981).

That somewhat legendary text is the “first thesis” of the author for the doctorate in philosophy (in France that doctorate needs two theses) held up in 1937. It had been published in 1938 (Zbl 0021.28904), but had been long out of print, and it is now reprinted with two new prefaces. The author himself had become a nearly legendary person. Some of his publications, “Remarques sur la formation de la théorie abstraite des ensembles”, his second thesis, published 1938 (Zbl 0021.30201), and the correspondence Cantor-Dedekind, edited in collaboration with Emmy Noether [“Briefwechsel Cantor-Dedekind” (1937; Zbl 0017.38502)] were painstaking historical work. On the other hand, he was non-conformist enough to introduce mathematical logic in France at a time when that subject was practically taboo. And a few years afterwards, in the war, he became a heroic “résistant” who died in 1944 before a German firing squad....- But that ”first thesis” must be evaluated in itself. The larger part of the book is devoted to a semi-historical account of the axiomatization of mathematics and of the development of mathematical logic up to the first half of 1936, with stress upon the Hilbert school. A few curious statements can be found, such as the affirmation that in the propositional calculus there are neither axiom nor rule (”Il n’y a donc ni axiome ni règles de raisonnement”, p. 104), that the calculus of predicates needs no proof of consistency (p. 106), that the universal quantifier means that what is quantified is provable for each individual (”démontrable pour toute individualisation des variables” p. 106), that last statement applied to classical logic (with reference to Tarski),....- And, beside such judgements, which make us doubt whether the author had understood the simpler parts of logic, we find lucid appreciations of much more difficult questions, like the role of Desargues’ theorem in the axiomatization of geometry, the importance of Presburger’s arithmetic, and particularly an account of Gödel’s incompleteness theorems which is made understandable with a minimum of technicalities; even Gentzen’s work is clearly summarized. All that is on the level of good popularization for recent advances of science.- But summarizing of science is not philosophy; and philosophy was the aim. This philosophy, deriving more or less from Husserl (perhaps against Husserl himself), appears chiefly in the last twenty pages, but permeates the whole book, where it is responsible for many obscurities; it may be responsible as well for the incomprehensible mistakes in elementary logic,... The last sentence of the book is the following one, which I cannot translate, for I do not understand it: ”Ici comme ailleurs la nécessité dialectique se masque sous un échec, l’expérience nouvelle n’est donnée que par un effort positif d’authentique aperception”.- Desanti writes that we do not know what philosophy of mathematics Cavaillès would have created if he had lived longer. Of course! But, born in 1903, in 1937 he was not a teen-aged freshman,... When I read his book for the first time, ignorant of recent non-french literature, I saw in it the revelation of several unknown worlds; but 45 years later I ought to say that, even in its time, that book was neither clear history, nor good popularization, nor sensible philosophy.

Reviewer: J.Porte

### MSC:

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

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03-03 | History of mathematical logic and foundations |

00A30 | Philosophy of mathematics |

01A75 | Collected or selected works; reprintings or translations of classics |

03B99 | General logic |

03F07 | Structure of proofs |

03F40 | Gödel numberings and issues of incompleteness |

01A60 | History of mathematics in the 20th century |