Le but de cet article est de décrire le point de vue d?Henri Poincaré sur l'axiome du choix, dont 1?explication par Zermelo en 1904, déclencha une vive polémique. Agitant le monde mathématique de l'époque, cette polémique avait ses racines dans la diversité des conceptions philosophiques que les mathématiciens avaient sur les mathématiques. Poincaré avait une position originale; quelques lettres qu?il écrivit à Zermelo (1906?1907), ainsi que les articles publiés à la même époques dans la Revue de métaphysique et morale, éclairent (...) ce point de vue. The aim of this paper is to describe the views of Henri Poincaré concerning axiom of choice, of which the statement by Zermelo in 1904 launched a sharp polemic. Shaking the mathematical society of the time, this controversy had its roots into the diversity of philosophical conceptions that mathematicians held about mathematics. Poincaré had a particular point of view; a few letters from him to Zermelo (1906?1907), together with papers published in Revue de métaphysique et morale at the same time, throw light on this view. (shrink)

Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918 he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show (...) that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal. (shrink)

We show that the name “Lucas-Penrose thesis” encompasses several different theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interesting versions of the much weaker thesis known as “Gö- del disjunction”. Our main conclusions are that (...) pure mathematical theorems cannot decide alone any question which is not purely mathematical, and that an argument that cannot be fully formalized cannot be taken as a mathematical proof. (shrink)