This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of (...) modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)

In 1887–1894, Richard Dedekind explored a number of ideas within the project of placing mappings at the very center of pure mathematics. We review two such initiatives: the introduction in 1894 of groups into Galois theory intrinsically via field automorphisms, and a new attempt to define the continuum via maps from ℕ to ℕ in 1891. These represented the culmination of Dedekind’s efforts to reconceive pure mathematics within a theory of sets and maps and throw new light onto the nature (...) of his structuralism and its specificity in relation to the work of other mathematicians. (shrink)

La « crise » des fondements des mathématiques (1902-1931) fut l’agent de l’avènement du paradigme axiomatico-ensembliste dans lequel la plupart des tensions et stratégies fondationnelles continuent d’être formulées. Au terme d’une synthèse de ces interrogations et de leurs traductions techniques, qui suit le fil conducteur de l’opposition constructif / non-constructif, on montre quelques-unes des subversions, encore minoritaires, que subit ce paradigme. On insiste alors sur les voies possibles de son dépassement, en pointant l’essoufflement conceptuel qui transparaît derrière le dynamisme de (...) l’une de ses branches actuelles, la théorie descriptive des ensembles. L’objectif est ainsi de contribuer à l’examen distancié, proprement épistémologique, de la nature des étapes du développement des mathématiques au 20ème siècle. (shrink)