Embark on a captivating journey through the history of mathematical proof in this 44-minute talk by John Stillwell at The Royal Institution. Explore the evolution of proof across number theory, non-Euclidean geometry, topology, and logic, while delving into the intriguing gap between natural number arithmetic and real numbers. Learn about Stillwell's favorite proof and the Greeks' insistence on proof, then trace the development of algebra, geometry, and calculus. Examine the nature of logic, set theory, and infinity, including Cantor's diagonal argument. Conclude with insights into the relationship between logic and computation, gaining a deeper appreciation for the vital role of proof in mathematics.