Richard Dedekind

German mathematician

• Born: October 6, 1831
• Birthplace: Brunswick, Duchy of Brunswick (now in Germany)
• Died: February 12, 1916
• Place of death: Brunswick, Duchy of Brunswick (now in Germany)

Dedekind gave a new definition to the mathematical concept of irrational numbers, based exclusively on arithmetic principles. He helped clarify the notions of infinity and continuity and contributed to the establishment of rigorous theoretical foundations for mathematics.

Early Life

Julius Wilhelm Richard Dedekind (DAY-dih-kihnd) was one of four children born to a well-established professional family in the Germanic duchy of Brunswick. His father was a professor of jurisprudence at the local Collegium Carolinum, and his mother was a professor’s daughter. In school, Dedekind was primarily interested in physics and chemistry, but when he enrolled in the Collegium Carolinum, it was as a student of mathematics. From a résumé, written somewhat later and in Latin, it is clear that this change was based on his dissatisfaction with the lack of rigor in the natural sciences.

In 1850, Dedekind was matriculated at the University of Göttingen, where he followed various courses in mathematics (studying under Carl Friedrich Gauss), astronomy, and experimental physics. In 1852, Dedekind presented his doctoral dissertation, which, in the opinion of Gauss, showed promise. At that time, the standard of mathematics at Göttingen was not very high, and Dedekind spent the following two years studying privately and preparing himself to become a first-class mathematician. No doubt his friendship with the brilliant Georg Friedrich Bernhard Riemann, at Göttingen at the same time, was also a positive influence. In fact, Dedekind attended Riemann’s lectures even after he himself qualified as a university lecturer in 1854. When Gauss died in 1855, Peter Gustav Lejeune Dirichlet, previously professor in Berlin, succeeded him. Dedekind described Dirichlet’s arrival in Göttingen as a life-changing event. Dedekind not only attended Dirichlet’s lectures but also became a personal friend of the new professor.

Life’s Work

In 1858, the Federal Institute of Technology in Zurich, Switzerland, appointed Dedekind as professor of mathematics on Dirichlet’s recommendation. Riemann also applied for the post but his work was considered too abstract. Dedekind stayed in Zurich until 1862 and then accepted an invitation from his old college in Brunswick, which had become a polytechnic by then.

While in Zurich, Dedekind taught differential and integral calculus and was disturbed by having to use concepts that had never been properly defined. In particular, he wrote: “Differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given.” He also deplored accepting without proof the belief that an increasing infinite sequence with an upper bound converges to a limit. He was dissatisfied that the notions of limit and continuity were based solely on geometrical intuition. On November 24, 1858, Dedekind succeeded in securing “a real definition of the essence of continuity.” He waited until 1872 to publish this definition in book form, with the title Stetigkeit und Irrationale Zahlen (“Continuity and Irrational Numbers,” translated in Essays on the Theory of Numbers, 1901).

Dedekind’s problem was essentially that of irrational numbers, known already to the ancient Greeks. Rational numbers are dense in the sense that between any two rational numbers there is always another rational number, although there are infinitely many gaps between them. These gaps can be thought of as irrational numbers, and, before Dedekind began his work, they were characterized by infinite, nonrecurring decimal fractions. Dedekind devised a method, using “cuts,” to define irrational numbers in terms of the rationals. If rational numbers are divided into two sets such that every number in the first set is smaller than every number in the second set, this partition defines one and only one real number. Should there be a largest or smallest number in one of the sets, the Dedekind cut corresponds to that rational number, while an irrational number is defined if neither set has a smallest or largest member.

A Dedekind cut can be imagined as severing a straight line composed of only rational numbers into two parts. Rational and irrational numbers together form the set of real numbers, and this set can now be made to correspond to all the points of a straight line. With this method, Dedekind not only managed to define irrational numbers in terms of rationals without recourse to geometry but also showed that a line, and by implication three-dimensional space, is complete, containing no holes. Furthermore, Dedekind upheld his philosophical principles, according to which numbers do not exist in a Platonic sense but are free creations of the human mind.

Closely connected to this work was the introduction of the concept of “ideals.” Dedekind edited and published Dirichlet’s lectures on number theory after the death of the latter. Dedekind can, in fact, be considered the author of the book, because Dirichlet left only an outline plan for publication, and that was already based on Dedekind’s notes. In the tenth supplement to the second edition of this influential book, Dedekind developed the theory of ideals, following to a certain extent a line Ernst Eduard Kummer had already taken. Dedekind, however, went far beyond Kummer, avoided his mistakes, and made the theory more exact.

Ideals are an extension and generalization of the common number concept. According to the fundamental theory of arithmetic, ordinary integers either are prime numbers or can be uniquely factorized into primes. Unique factorization is a useful feature but does not generally apply to all algebraic integers in a given algebraic number field, algebraic numbers being defined as the roots of polynominal equations with integer coefficients. With the introduction of ideals, unique factorization can be restored. Dedekind subsequently revised and further developed this theory. In an important paper coauthored by Heinrich Weber, the analogy between algebraic numbers and algebraic functions was demonstrated with the help of ideals.

In Was sind und was sollen die Zahlen? (1888; “The Nature and Meaning of Numbers,” translated in Essays on the Theory of Numbers, 1901), Dedekind utilized the concept of what he called systems, which later became known as sets, and developed logical theories of original and cardinal numbers and of mathematical induction. In addition to contributing papers to mathematical journals, Dedekind coedited Riemann’s collected works and supplied a biography of Riemann.

Dedekind stayed at Brunswick until his death and became a director of the polytechnic between 1872 and 1875. It seems that Dedekind was not offered the posts he would have accepted, while he refused the posts, most notably the one at Halle, that he was offered. Dedekind never married but lived with one of his sisters until her death in 1914. Although he lived in relative isolation, he was never a recluse. He was an excellent musician: He played the cello as a young man and the piano in later life. His portraits show a fine-featured man with thoughtful eyes; his character was described as modest, mild, and somewhat shy.

Significance

Although Richard Dedekind was a corresponding member of several academies and an honorary doctor of several universities, he never received the recognition he so fully deserved. It can be seen that his work was one of the most influential in shaping twentieth century mathematics. He is one of only thirty-one mathematicians meriting an individual entry in Iwanami Sugaku Ziten (1954; Encyclopedic Dictionary of Mathematics, 1977), in which he is described as a pioneer of abstract algebra. Transcending pure calculation, Dedekind made an attempt to find theoretical foundations to concepts used in algebraic number theory and in infinitesimal calculus. He defined and thereby created new mathematical structures that generalize the notions of number and serve as examples for further generalization.

Dedekind met Georg Cantor on a holiday in Switzerland and became his friend and also, at times, his frequent correspondent. Cantor submitted his theories to Dedekind for comment and criticism, and Dedekind was one of the first to support set theory in the face of hostility by other mathematicians. Independently of Cantor, he also utilized the concept of the actual, or concrete, infinite—a concept that was then regarded as taboo because there existed no theoretical foundation for its existence. Dedekind’s work assisted in finding just such a foundation.

Bibliography

Bashmakova, I. G., and G. S. Smirnova. The Beginnings and Evolution of Algebra. Translated from the Russian by Abe Schneitzer. Washington, D.C.: Mathematical Association of America, 2000. Chapter 8 in this history of algebra includes information about Dedekind and the birth of numbers theory.

Bell, Eric T. “Arithmetic the Second.” In Men of Mathematics. New York: Simon & Schuster, 1937. Reprint. New York: Penguin Books, 1965. This short chapter in a well-known collective biography of mathematicians discusses the life and work of Kummer and Dedekind. Bell makes a good attempt to explain the abstract and often difficult concepts that are necessary for the understanding and appreciation of Dedekind’s work.

Corry, Leo. Modern Algebra and the Rise of Mathematical Structures. 2d rev. ed. Boston: Birkhäuser, 2004. Traces the development of algebra from the mid-nineteenth century to the present, focusing on ideas concerning algebraic structures. This revised edition includes a revised chapter on Dedekind.

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press, 1979. Not a biography of Cantor, but a study of the emergence of a new mathematical theory. Dedekind’s life, work, and influence on Cantor are featured extensively, but these references are dispersed throughout the book. Readers whose main interest is in Dedekind can rely on the well-constructed index and the twenty-four-page bibliography.

Dedekind, Richard. Theory of Algebraic Integers. Translated by John Stillwell. New York: Cambridge University Press, 1996. Dedekind’s theory, first published in French in 1877, was the genesis of modern algebraic numbers theory. Stillwell provides a detailed introduction offering historical background and outlining the challenges Dedekind faced in devising his theory.

Edwards, Harold M. “Dedekind’s Invention of Ideals.” The Bulletin of the London Mathematical Society 15 (1983): 8-17. Traces the influences on Dedekind’s set theoretic approach mainly to Dirichlet but also to Kummer and Riemann. Évariste Galois’s influence was limited and resulted in steering Dedekind toward conceptual thinking as opposed to mere calculating. Dedekind went beyond Dirichlet, and against the accepted classical doctrine, by using completed infinites. The author stresses the innovative nature of Dedekind’s theories and the analogy between cuts and ideals.

‗‗‗‗‗‗‗. “The Genesis of Ideal Theory.” Archive for History of Exact Sciences 23 (1980): 321-378. Analyzes Kummer’s, Leopold Kronecker’s, and Dedekind’s versions of the theory of ideal factorization of algebraic integers. The author advances the thesis that as Dedekind revised the theory several times to match his philosophical principles, it did not improve from the mathematical point of view, and the first formulation remained the best.

Gillies, D. A. Frege, Dedekind, and Peano on the Foundations of Arithmetic. Assen, the Netherlands: Van Gorcum, 1982. A short paperback with an adequate index and a list of references. Investigates the relationship between logic and arithmetic in the work of the three men. Gillies regards Dedekind as fundamentally a logician and compares him to Gottlob Frege, who denied that a set was a logical notion, and to Giuseppe Peano, who thought that arithmetic could not be reduced to logic.