Yvon Gauthier

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.

The author has endeavoured to define two main trends in the research on the foundations of mathematics, constructivism and structuralism. He gives many examples in axiomatic set theory, e.g. the continuum hypothesis, and in intuitionism, e.g. the notion of choice sequence, in order to show that the two approaches are complementary. The paper contains some original ideas, concerning the structure of the continuum and the constructive horizon, and is completed by an appendix. The paper is an attempt at the justification of a constructivist philosophy in the making

Le petit ouvrage de Michel Bourdeau dans cette élégante collection des Presses Universitaires de France a des prétentions modestes, comme l'exige d'ailleurs la vocation pédagogique de la collection. Il s'agit d'une introduction philosophique au programme de l'intelligence artificielle. L'auteur voudra aborder la problématique par la voie historique et dans un esprit informel, tout en espérant faire la part de la pensée symbolique et de l'intuition en deux temps «La préhistoire de l'intelligence artificielle» et «Éléments d'une théorie de l'intuition».