name = PAGENAME
caption =Richard Dedekind, c. 1850
October 6, 1831
birth_place = Braunschweig
February 12, 1916
death_place = Braunschweig
nationality = German
field = mathematician
abstract algebra algebraic number theory real numbers
Julius Wilhelm Richard Dedekind (
October 6, 1831– February 12, 1916) was a German mathematician who did important work in abstract algebra, algebraic number theoryand the foundations of the real numbers.
Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. As an adult, he never employed the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).
In 1848, he entered the Collegium Carolinum in Braunschweig, where his father was an administrator, obtaining a solid grounding in mathematics. In 1850, he entered the
University of Göttingen. Dedekind studied number theoryunder Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled "Über die Theorie der Eulerschen Integrale" ("On the Theory of Eulerian integrals"). This thesis did not reveal the talent evident on almost every page Dedekind later wrote.
At that time, the University of Berlin, not Göttingen, was the leading center for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Riemann were contemporaries; they were both awarded the
habilitationin 1854. Dedekind returned to Göttingen to teach as a "Privatdozent", giving courses on probabilityand geometry. He studied for a while with Dirichlet, and they became close friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions. Yet he was also the first at Göttingen to lecture on Galois theory. Around this time, he became one of the first to understand the fundamental importance of the notion of groups for algebraand arithmetic.
In 1858, he began teaching at the Polytechnic in
Zürich. When the Collegium Carolinum was upgraded to a " Technische Hochschule" (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his unmarried sister Julia.
Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the Paris Académie des Sciences (1900). He received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig.
While teaching calculus for the first time at the Polytechnic, Dedekind came up with the notion now called a
Dedekind cut(in German: "Schnitt"), now a standard definition of the real numbers. The idea behind a cut is that an irrational numberdivides the rational numbers into two classes (sets), with all the members of one class (upper) being strictly greater than all the members of the other (lower) class. For example, the square root of 2 puts all the negative numbers and the numbers whose squares are less than 2 into the lower class, and the positive numbers whose squares are greater than 2 into the upper class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thought on irrational numbers and Dedekind cuts in his paper " [http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen] " ("Continuity and irrational numbers." Ewald 1996: 766. Note that Dedekind's terminology is old-fashioned: in the present context, one now says "Vollständigkeit" instead of "Stetigkeit", so a modern translation would have "continuity" replaced with "completeness").
In 1874, while on holiday in
Interlaken, Dedekind met Cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the very first mathematicians to admire Cantor's work on infinite sets, proving a valued ally in Cantor's battles with Kronecker, who was philosophically opposed to Cantor's transfinite numbers.
If there existed a
one-to-one correspondencebetween two sets, Dedekind said that the two sets were "similar." 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 modern terminology, is equinumerousto one of its proper subsets. (This is known as Dedekind's theorem.) Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N, (N → N2):
N 1 2 3 4 5 6 7 8 9 10 ... ↓ N2 1 4 9 16 25 36 49 64 81 100 ...
Dedekind edited the collected works of
Dirichlet, Gauss, and Riemann. Dedekind's study of Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Dirichlet's lectures on number theoryas " Vorlesungen über Zahlentheorie" ("Lectures on Number Theory") about which it has been written that:
"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)
The 1879 and 1894 editions of the "Vorlesungen" included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by
Hilbert, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integercoefficients. The concept underwent further development in the hands of Hilbertand, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind and Heinrich Martin Weberapplied ideals to Riemann surfaces, giving an algebraic proof of the Riemann-Roch theorem.
Dedekind made other contributions to algebra. For instance, around 1900, he wrote the first papers on
In 1888, he published a short monograph titled "Was sind und was sollen die Zahlen?" ("What are numbers and what should they be?" Ewald 1996: 790), which included his definition of an infinite set. He also proposed an
axiomatic foundation for the natural numbers, whose primitive notions were one and the successor function. The following year, Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones.
Dedekind eta function
Dedekind zeta function
Ideal (ring theory)
Vorlesungen über Zahlentheorie
Primary literature in English:
*1890. "Letter to Keferstein" in
Jean van Heijenoort, 1967. "A Source Book in Mathematical Logic, 1879-1931". Harvard Univ. Press: 98-103.
* 1963 (1901). "Essays on the Theory of Numbers". Beman, W. W., ed. and trans. Dover. Contains English translations of " [http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen] " and "Was sind und was sollen die Zahlen?"
* 1996. "Theory of Algebraic Integers". Stillwell, John, ed. and trans. Cambridge Uni. Press. A translation of "Über die Theorie der ganzen algebraischen Zahlen".
* Ewald, William B., ed., 1996. "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. Oxford Uni. Press.
**1854. "On the introduction of new functions in mathematics," 754-61.
**1872. "Continuity and irrational numbers," 765-78. (translation of "Stetigkeit...")
**1888. "What are numbers and what should they be?", 787-832. (translation of "Was sind und...")
**1872-82, 1899. Correspondence with Cantor, 843-77, 930-40.
*Edwards, H. M., 1983, "Dedekind's invention of ideals," "Bull. London Math. Soc. 15": 8-17.
year = 1998
title = _es. "The First Moderns"
University of Chicago Press
location = Chicago
id = ISBN 0-226-22480-5
*Gillies, Douglas A., 1982. "Frege, Dedekind, and Peano on the foundations of arithmetic". Assen, Netherlands: Van Gorcum.
Ivor Grattan-Guinness, 2000. "The Search for Mathematical Roots 1870-1940". Princeton Uni. Press.
There is an [http://www-groups.dcs.st-and.ac.uk/~history/References/Dedekind.html online bibliography] of the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).
* [http://www.archive.org/details/essaysintheoryof00dedeuoft Dedekind, Richard, "Essays on the Theory of Numbers." Open Court Publishing Company, Chicago, 1901.] at the
NAME=Dedekind, Julius Wilhelm Richard
SHORT DESCRIPTION=Mathematician specializing in
abstract algebraand real numbers
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