Part V: Age of Rigor
The 18th and 19th centuries — when mathematics became a precise, rigorous, and deeply abstract discipline.
Overview
Euler produced more mathematics than any human in history. Gauss brought unprecedented rigor to number theory, geometry, and analysis. Galois, dead at 20, invented group theory in a single night. Riemann reshaped our understanding of geometry and the distribution of primes. Fourier showed how any function could be decomposed into waves. And Cantor dared to put infinity itself on a rigorous footing — provoking both admiration and fierce opposition.
Chapters
Chapter 14: Euler & Laplace
Euler's astonishing output spanning every branch of mathematics, and Laplace's celestial mechanics and probability theory.
Chapter 15: Gauss — Prince of Mathematicians
Disquisitiones Arithmeticae, the constructibility of the 17-gon, non-Euclidean geometry, and the method of least squares.
Chapter 16: Galois & Abel — Tragedy and Algebra
Two young geniuses who died tragically young but whose work — group theory and the unsolvability of the quintic — reshaped algebra forever.
Chapter 17: Riemann & Fourier
Riemannian geometry, the zeta function, the Riemann hypothesis, Fourier series, and the analysis of heat conduction.
Chapter 18: Cantor & Dedekind — Infinity
The diagonal argument, transfinite numbers, Dedekind cuts, and the controversial foundations of set theory.