Richard Dedekind

Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician. He was born in Brunswick, the youngest of the four children of Julius Levin Ulrich Dedekind. He lived with Julia, his maiden sister, until she died in 1914;. In 1848 he entered the Colegium Carolinum in his hometown, and in 1850, with a solid background in mathematics, at the University of Göttingen.

Biography

Dedekind's father was Julius Levin Ulrich Dedekind, trustee of the Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a Collegium professor.Richard Dedekind had three older brothers. As an adult, he never used the names of Julius Wilhelm. He was born in Braunschweig, which is where he lived most of his life and died.

He first attended the Collegium Carolinum in 1848 before moving to the University of Göttingen in 1850. Dedekind learned mathematics in that university's departments of mathematics and physics, one of his main professors being Moritz Abraham Stern, and also physics from the hand of Wilhelm Eduard Weber. His doctoral thesis, supervised by Gauss, was entitled Über die Theorie der Eulerschen Integrale (On the Theory of Eulerian Integrals), and although it did not reflect the talent he displayed in his later work, Gauss he came to appreciate Dedekind's gift for mathematics. Dedekind received his doctorate from him in 1852, being Gauss's last student, and subsequently worked on a habilitation thesis, which was necessary in Germany to obtain the & # 34;venia docendi & # 34; (teacher teaching qualification at German universities).

Over the next few years, he studied number theory and other subjects with Gustav Dirichlet, with whom he became a close friend. To broaden his knowledge, he undertook the study of abelian and elliptic functions at the hands of the brilliant Bernhard Riemann. Only after these experiences, in his formation, he finally found his main fields of work: algebra and algebraic number theory. It is said of him that he was the first to teach university classes on the theory of Galois equations. He was also the first to understand the fundamental meaning of the notions of group, field, Ideal in the field of algebra, number theory and algebraic geometry.

Their cuts definitively settle the problem of the foundation of the analysis by defining the set of real numbers from the rational ones. In his magisterial article of 1872, Dedekind characterized the real numbers as an orderly and complete body, and offered a development of the whole question that is a model of organization and clarity.

His work on the natural numbers was also fundamental, laying the foundations for set theory, together with Frege and Cantor, and giving a very rigorous foundation for the so-called Peano Axioms (published by the Italian a year later).

Important though they were, these were not Dedekind's main contributions to pure mathematics: he worked all his life on algebraic number theory, which he largely created. And in the process, he laid down many of the characteristic methods of modern algebra, to the point that Emmy Noether used to repeat that "everything is already in Dedekind".

Dedekind's correspondence with other mathematicians was especially fruitful and stimulating: above all the correspondence with Cantor, where we witnessed the birth of the theory of transfinite sets; but also the correspondence with H. Weber, which among other things led to a pioneering article on algebraic geometry; and the one he maintained with Frobenius, promoting the development of the theory of group representations.

Work

Dedekind, before 1886

While first teaching calculus at the Polytechnic, Dedekind developed the notion now known as the Dedekind cutoff (German: Schnitt), now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes (sets), with all numbers in one (greater) class being strictly greater than all numbers in the other (lesser) class. For example, the square root of 2 defines all nonnegative numbers whose squares are less than 2 and negative numbers in the smaller class, and positive numbers whose squares are greater than 2 in the larger class. Each location on the continuum of the number line contains a rational or irrational number. Therefore, there are no empty places, gaps or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind's cuts in his pamphlet & # 34; Stetigkeit und irrationale Zahlen & # 34; ("Continuity and Irrational Numbers"); in modern terminology, Vollständigkeit, completeness.

Dedekind defined two sets as "similar" when there is a one-to-one correspondence between them. He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself", in the terminology modern, is equinumerous to one of its proper subsets.