Teaching calculus for the first time, Dedekind worked out what a real number is
Preparing his first calculus course in Zurich, Richard Dedekind was bothered that nobody had properly defined the number line. His answer, the Dedekind cut, treats a number like the square root of 2 as a clean split of all fractions into those below it and those above, leaving no gaps anywhere.
Dedekind was born on 6 October 1831 in Braunschweig, where his father ran the Collegium Carolinum, and he spent most of his life there. At Göttingen he became the last doctoral student of the aged Gauss, though his 1852 thesis showed little of his later brilliance. In Berlin he trained alongside Bernhard Riemann, both qualifying to lecture in 1854, and back at Göttingen he befriended Dirichlet and became the first there to teach Galois theory. He grasped early how central groups would be to algebra.
He taught at the Zurich Polytechnic from 1858, then returned home in 1862 when the Collegium became an institute of technology, staying until retirement in 1894. He never married and shared a house with his sister Julia. He published the cut idea in 1872 in a pamphlet on continuity and irrational numbers.
In 1888 his short book asking what numbers are and what they are for gave the first exact definition of an infinite set: one that can be matched item for item with part of itself, as the counting numbers can be paired with their squares. It also proposed axioms for arithmetic built from the number one and a successor operation; a year later Giuseppe Peano, crediting him, streamlined them into the version used today. This anticipated both Georg Cantor's set theory and the logicism of Frege and Russell.
Editing Dirichlet's number theory lectures, which he in fact wrote largely after Dirichlet died, Dedekind added supplements introducing ideals, a concept at the core of ring theory that Emmy Noether later developed. He met Cantor on holiday in Interlaken in 1872 and backed him against Leopold Kronecker, but letters suggest Cantor took one of Dedekind's proofs, and the friendship cracked. He died in 1916.
Source: Richard Dedekind
