Part VI: Modern Mathematics
The 20th century — foundational crises, incompleteness, computing, and the ongoing quest to understand the infinite.
Overview
At the dawn of the 20th century, Hilbert proposed 23 problems to guide mathematics into the future. Within decades, Gödel proved that any sufficiently powerful mathematical system must be incomplete. Turing showed that some problems are fundamentally undecidable. Ramanujan, an untrained genius from India, produced results that still astonish researchers today. And von Neumann helped build the digital computer, opening an entirely new chapter in the history of mathematical thought.
Chapters
Chapter 19: Hilbert & Poincaré
Hilbert's 23 problems, the axiomatization of geometry, and Poincaré's contributions to topology, celestial mechanics, and the philosophy of science.
Chapter 20: Gödel & Turing — Limits of Computation
The incompleteness theorems, the Entscheidungsproblem, Turing machines, and the birth of theoretical computer science.
Chapter 21: Ramanujan & Hardy
The extraordinary partnership between an untrained Indian genius and the leading English mathematician — their work on partitions, primes, and infinite series.
Chapter 22: Von Neumann & the Digital Age
Von Neumann's contributions to set theory, quantum mechanics, game theory, and computer architecture — and the mathematical landscape of the modern world.