Luis Cornelio Recalde

Fréchet and the Logic of the Constitution of Abstract Spaces from Concrete Reality

SPACES FROM CONCRETE REALITY 1. FRÉCHET AND POINCARÉ’S CONVENTIONALISM In several of his works Fréchet made important commentaries on Poincaré’s ideas on such topics as conventionalism, mathematical language, intuition and logic, and geometry and experience. In this part of our study we will refer to the relation between these ideas, how both of them have been conceptualised, mainly with regard to the nature and function of conventions in mathematics. Nevertheless, beyond the record of a similarity of ideas that becomes evident, what interests us most at the present time, is to identify the difficulties encountered (at least those faced by Fréchet), when trying to defend the conventionality position, researchers were gradually led to elucidate more fundamental problems, particularly the problem of understanding those acts of reasoning that allow the cognitive subject to formulate and to study mathematical entities as conventions. They were also led to attempt to clarify the genetic relation between mathematical conventions and empirical reality. We will begin by a quick revision of some publications which throw light on the subject in order to locate this issue within the philosophical and intellectual context of the time. 1.1. The Practical Value of Science Down the Centuries Poincaré belongs to the generation of philosophers and scientists that, at the end of the XIX century and the beginning of the new millennium, openly participated in the debates about the new philosophical and epistemological tendencies in science and mathematics, in particular, on the problem of the genesis of knowledge from experience, and the debates on the practical value of science. They also reflected on the relationships between intuition and logic that were gaining such importance at a time in which the movement of the arithmetisation of analysis was becoming consolidated. We will begin by referring to the first two of these tendencies. Synthese 134: 245–272, 2003.