Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummer's closest follower in arithmetic.
Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind. He later rejected his first names Julius Wilhelm. He lived with his unmarried sister Julia until her death in 1914, he himself also never married. In 1848, he entered Collegium Carolinum in Brunswick and in 1850 with the solid knowledge in mathematics he entered the University of Göttingen.
In Göttingen, Gauss taught mathematics mostly at an elementary level. In the departments of mathematics and physics, Dedekind learnt about number theory. Among Dedekind's main professors was Moritz Abraham Stern who at that time wrote many works on number theory. He made his short doctoral thesis supervised by Gauss Über die Theorie der Eulerschen Integrale (On the Theory of Eulerian integrals). His thesis was dexterous and autonomous but it didn't show any of the special talent which was visible on almost every page of Dedekind's later works. Nevertheless, Gauss must certainly have seen Dedekind's gift for mathematics. Dedekind received his doctorate in 1852 and he was Gauss's last student.
After that he spent two years in Berlin. In 1854 he was awarded with his habilitation degree almost as the same time as Riemann. Dedekind began teaching as Privatdozent in Göttingen and he gave courses on probability and geometry. He studied some time with Dirichlet, and they became close friends. Because of the lack of mathematical knowledge he was still studying elliptic functions and abelian functions. At the same time he was the first to lecture on Galois theory. He was one of the earliest to understand the fundamental meaning of the notion of group, in algebra and in arithmetic.
In 1858 he went to Zürich to teach at the Polytechnikum. At this time he defined the Dedekind cut (German: Schnitt), a new idea to represent the real numbers as a divisions of the rational numbers. An irrational number is a cut separating all rational numbers into two classes, an upper and lower class (set) For example, the square root of 2 is a cut putting the negative numbers and the numbers with square smaller than 2 into the lower, and the positive numbers with square greater than 2 into the higher class. This is now one of the standard definitions for the real numbers. After Collegium Carolinum had been upgraded to the Technical High School, Dedekind started to teach there in 1862. He remained there for the remaining 50 very productive years of his life.
In 1863, he published Dirichlet's lectures on number theory in Vorlesungen über Zahlentheorie (Lectures on Number Theory). In 1872 he published his cognitions on his major rigorous redefinition of irrational numbers in terms of Dedekind cut in a paper titled Stetigkeit und irrationale Zahlen (Continuity and irrational numbers). In the year 1874 he met Cantor in the Swiss city Interlaken. Dedekind was among the first mathematicians who had accepted Cantor's work on the theory of infinite sets; other mathematicians didn't yet understand their ideas. His help was salutary for Cantor against Kronecker's objections to the general infiniteness in number theory. In above work he gave the first precise definition of an infinite set. A set is infinite, he argued, when it is "similar to a proper part of itself." Thus the set N of natural numbers can be shown to be 'similar', that is, matched or put into a one-to-one correspondence with a proper part, in this case with the set of their squares N2, (N → N2):
N 1 2 3 4 5 6 7 8 9 10 ...
N2 1 4 9 16 25 36 49 64 81 100 ...
In his third edition of the previous book Über die Theorie der ganzen algebraischen Zahlen (On the Theory of algebraic whole numbers) 1879 he proposed the notion of an ideal. He based his work on Kummer's ideas from his previous work on Fermat's last theorem from 1843. An ideal is a collection of numbers that may be separated out of a larger collection, composed of algebraic integers that satisfy polynomial equations with ordinary integers as coefficients. The term is fundamental to later ring theory as formulated by Hilbert and a little later by Emmy Noether. An ideal number is not a number but it is an infinite class of numbers, consisting of a number and all its multiples. We can easily see that for arbitrary whole numbers m and n and if for their such 'classes' 'class' (m) is part of 'class' of (n) (we write then as (m)/(n)) only and only then if m divide n.
1882 with Heinrich Martin Weber he published an article where they applied Dedekind's theory of ideals to the theory of Riemann surfaces. 1888 he published a work Was sind und was sollen die Zahlen? (What are numbers and what should they be?) where he defined an infinite set in his own way. Here he demonstrated how arithmetic could be derived from a set of axioms. A simpler, but equivalent version, formulated by Peano a year later in 1889, is much better known today.