World of Mathematics on (Julius Wilhelm) Richard Dedekind
Richard Dedekind is best known for his work in number theory. He redefined irrational numbers, proposing that rational and irrational numbers form a continuum in which real numbers are located by "cuts" in the realm of rational numbers. He also introduced the notion of an ideal (for example, the collection of all integer multiples of a given integer), which allowed wider application of factorizationand is fundamental to modern ring theory. In addition to Dedekind cuts, about a dozen mathematical concepts carry his name. Although he was among the most capable and original mathematicians of his day, Dedekind was a modest man, spending most of his professional life as a teacher at the technical high school in his hometown of Brunswick. He was a gifted teacher, and his teaching was an integral part of his mathematical thinking.
Dedeking was born Julius Wilhelm Richard Dedekind on October 6, 1831, in Brunswick (Braunschweig), now Germany, the last of four children. As an adult he dropped his first two names. His father, Julius Levin Ulrich Dedekind, was a lawyer and professor at Caroline College in Brunswick and the son of a physician and chemist. His mother, Caroline Marie Hanriette Emperius Dedekind, was a daughter of a professor at the College and a granddaughter of an imperial postmaster. Dedekind's brother, Adolf, became a district court president; his sister, Julie, became a novelist.
From age seven to age sixteen, Dedekind studied at the Gymnasium in Brunswick. At first, he concentrated on physics and chemistry, considering mathematics merely a scientific tool; however, he eventually became enthralled with the logic of mathematics. From 1848 to 1859, Dedekind attended Caroline College where he studied analytic geometry, advanced algebra, the calculus, and higher mechanics, and gave private lessons. In 1850, at age nineteen, he entered the University of Göttingen, where he became the last doctoral student trained by Karl Gauss. At Göttingen, Dedekind studied calculus, elements of higher arithmetic, least squares, higher geodesy, and experimental physics. Dedekind's doctoral thesis on Eulerian integrals, completed after only four semesters at Göttingen, was a solid but uninspired piece of work. However, Gauss praised his knowledge and independence, and predicted future success for him.
Posts at Göttingen, Zürich, and Brunswick
Dedekind continued to study and attend lectures, and in 1854 he was appointed lecturer (privatdozent) at Göttingen, just a few weeks after his friend Georg Riemann, who also studied under Gauss, received a similar appointment. Some time later Dedekind and Riemann traveled together to Berlin to meet with the mathematical community there. When Gauss died in 1855, Dedekind served as a pallbearer at the funeral service. When Peter Gustav Lejeune Dirichlet came to Göttingen from the University of Berlin to take Gauss' place, he and Dedekind became close friends and colleagues. Dedekind attended Dirichlet's lectures, and their discussions inspired Dedekind's investigations in new directions, making a "new man" of him. Dedekind was among the first to recognize the application of Galois groupsin algebra and arithmetic, and in 1857-58 gave a course to two students on Évariste Galois' theory of equations.
In 1858, Dedekind was invited to succeed Joseph Ludwig Raabe at the Polytechnic School in Zürich. In recommending him for the position, Dirichlet described Dedekind as "an exceptional pedagogue." A position in Zürich was traditionally a first step toward a professorship in Germany. However, after five years in Zürich, in 1862 Dedekind succeeded Wilhelm Julius Uhde as professor of higher mathematics at the technical high school in Brunswick. He stayed there the remainder of his life; Dedekind directed the school from 1872 to 1875, and was named professor emeritus in 1894.
In assuming the position at the technical school, which had been created under the auspices of Carolina College, Dedekind was following in his father's administrative footsteps. In Brunswick, he lived in close association with his family and did not aspire to a greater position. He was an accomplished cellist and pianist, and composed a chamber opera for his brother's libretto. Dedekind lived with his sister Julie until her death in 1914. He died in Brunswick on February 12, 1916.
Works Focus on Number Theory
Dedekind's work focused almost totally on the area of numbers. As a result of attempts to answer questions about real numbers, several ideas had been put forth, all involving infinite sets or sequences. The simplest idea was Dedekind's. He defined a real number to be a partition, or "cut," of the rational numbers into two sets, so that each member of one set is less than all numbers of the other. These Dedekind cuts--which he said occurred to him on November 24, 1858--gave a precise model for the continuous number line, since they filled all the gaps in the rationals. Other formulations followed from his definition. In 1872, Dedekind defined an infinite set in his paper Stetigkeit und irrationale Zahlen. In 1888, he expanded these ideas in a book, Was sind und was sollen die Zahlen? With his 1872 paper he had joined Karl Weierstrauss and Georg Cantor in defining a new mathematical area.
In 1879, Dedekind published Über die Theorie der ganzen algebraischen Zahlen, in which he introduced the notion of an ideal, which is fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic number field. His idea was later extended by David Hilbert and Emmy Noether. Dedekind also collected, explained, extended, and published the works of those mathematicians who had influenced him: Gauss (1863), Rie__mann (1876), and Dirichlet (1863, 1871). His work on Dirichlet's lectures led him to a theory of generalized complex numbersand forms that can be resolved into linear factors. His work was characterized by exceptional clarity, and he has been credited with creating a "style" of mathematics.
Dedekind was a corresponding member of the Göttingen Academy (1862), the Berlin Academy (1880), and the Paris Académie des Sciences (1910). He was a member of the Leopoldino-Carolina Naturae Curiosorum Academia and the Academy of Rome. Dedekind also received many honors, including honorary doctorates from Brunswick and the University of Oslo. On one occasion, his death was listed on a Calendar for Mathematicians as September 4, 1899; an amused Dedekind wrote the publisher that he had spent that day talking with his friend Georg Cantor.
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