Mathematical thought in the Renaissance and the genesis of modern mathematics

This article argues that the Renaissance rediscovery of Greek mathematics functioned both as a stimulus to renewal and as a powerful constraint. The classical paradigm was not simply revived: it had to be reorganized and reinterpreted, yet its internal grammar – marked by tensions between form and extension, number and magnitude – continued to shape what could count as an admissible object and a legitimate proof. Francesco Maurolico provides the central case study, through his effort to construct a renewed framework for Archimedean geometry of measure – an effort that did not culminate in a stable synthesis, but instead exposed internal tensions within the classical framework. Galileo is used as a stress test: his attempt to mobilize the Euclidean theory of proportions for the description of motion reveals both the productivity and the limits of that inherited structure.