Raison et infini
Institution:
Université Pierre Mendès France (Grenoble)Disciplines:
Directors:
Abstract EN:
The subject of the study is limited to pure reason and its interaction with the question of infinity as defined from a mathematical, philosophical; and religious point of view. The thesis aims at showing that the various rational principles (identity, non contradiction. . . ) come down to the principle of symetry which is the operating version of the principle of identity. It shows that the various rationalities which refer to various types of existence (actual, constructive, logical. . . ) can come down to a single one, the most general, as these existences are included. So a weaker logic than classical logic is put forward, able to deal with infinite and impredicative definitions with their paradoxes. Cantor's theorem and the scale of infinite numbers are contested. Infinity, owing to the absence of limits, is a weaker notion than finity. So both metaphysical infinite and mathematical infinite must be approached with that same logic. Thus reason can use an open formalism, providing a broader coherence to mathematics and which includes, as particular cases, the other rational theories. It also includes infinity which then appears as a unitary pole structuring thought.
Abstract FR:
Le sujet d'etude se limite a la raison pure et a son interaction avec la question de l'infini mathematique, philosophique et religieux. La these vise a montrer que les differents principes rationnels (identite, non-contradiction. . . ) se ramenent au principe de symetrie, qui est la version operatoire du principe d'identite. Elle montre que les rationnalites differentes, qui se referent a differents types d'existence (effective, constructive, logique. . . ), peuvent se ramener a une seule : la plus generale, puisque ces existences sont emboitees. Une logique plus faible que la logique classique est ainsi proposee, apte a traiter l'infini et les definitions impredicatives avec leurs paradoxes. Le theoreme de cantor et l'echelle des nombres transfinis sont contestes. L'infini, par l'absence de limite, est une notion plus faible que le fini. Donc l'infini metaphysique, comme l'infini mathematique doit etre aborde avec cette meme logique. Ainsi la raison dispose d'un formalisme ouvert, assurant une consistance plus large aux mathematiques, et qui inclut, en cas particulier, les autres theories rationnelles. Elle inclut egalement l'infini qui se revele alors comme un pole unitaire structurant la pensee.