The Age of Rigor: Cauchy and Analysis

Augustin-Louis Cauchy brought unprecedented rigor to mathematical analysis. His 1821 "Cours d'Analyse" established standards for mathematical proof that remain influential. Concepts like limits, continuity, and convergence received precise definitions replacing intuitive understanding. This rigorization—making mathematics logically watertight—exemplified French emphasis on foundations.

Cauchy's productivity was legendary—over 800 papers covering every mathematical domain. But his greater influence came through establishing how mathematics should be done: clearly stated definitions, carefully proven theorems, explicit assumptions. This methodological contribution shaped mathematical practice globally. Rigor became not pedantry but liberation—secure foundations enabled bolder exploration.

The resistance Cauchy faced illustrated tensions between intuition and rigor. Contemporaries complained his precision destroyed mathematical poetry. But Cauchy showed rigor enhanced rather than diminished beauty. By clarifying exactly what was assumed and proved, mathematics became more not less elegant. French mathematics would consistently demonstrate that precision and beauty align.

Women's exclusion from formal mathematical institutions didn't prevent contributions. Sophie Germain's work on elasticity theory and number theory, conducted through correspondence since she couldn't attend university, showed exceptional insight. Her results on Fermat's Last Theorem remained best for decades. French mathematics' loss through gender exclusion was partially recovered through exceptional individuals overcoming barriers.