On the evolution of the concept of probability as a mirror of the evolution of reason

Abstract

Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of events to Kolmogorov's axiomatic formalization, probability progressively incorporated uncertainty, time, and coherence into scientific judgment. This trajectory reaches a mature epistemological form in modern Bayesian inference, especially in Tarantola's view of probability as a logic of information, where prior knowledge and data are combined coherently. Yet this framework also exposes a limit: probability quantifies uncertainty about well-defined propositions, but does not by itself formalize the vagueness of the concepts used to describe them. The article therefore examines how rationality extends beyond probability. Fuzzy logic is presented as a rigorous language for graded meaning and qualitative judgment, while deep learning is analyzed as a distinct, powerful mode of prediction based on geometric interpolation and optimization rather than explicit inference. By situating probability, fuzzy logic, and deep learning in a common historical and epistemological perspective, the article clarifies their roles and limits. It argues that contemporary scientific rationality cannot be reduced to data-driven performance alone, but requires the explicit articulation of uncertainty, vagueness, and inference.