Dedekind's father was Julius Levin Ulrich Dedekind, an administrator ofCollegium Carolinum inBraunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium.[2] Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests atBraunschweig Main Cemetery.
He first attended the Collegium Carolinum in 1848 before transferring to theUniversity of Göttingen in 1850. There, Dedekind was taughtnumber theory by professorMoritz 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 titledÜber die Theorie der Eulerschen Integrale ("On the Theory ofEulerian integrals"). This thesis did not display the talent evident in Dedekind's subsequent publications.
At that time, theUniversity of Berlin, notGöttingen, was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he andBernhard Riemann were contemporaries; they were both awarded thehabilitation in 1854. Dedekind returned to Göttingen to teach as aPrivatdozent, giving courses onprobability andgeometry. He studied for a while withPeter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studiedelliptic andabelian functions. Yet he was also the first at Göttingen to lecture concerningGalois theory. About this time, he became one of the first people to understand the importance of the notion ofgroups foralgebra andarithmetic.
In 1858, he began teaching at thePolytechnic school inZürich (now ETH Zürich). When the Collegium Carolinum was upgraded to aTechnische 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 sister Julia.
Dedekind was elected to the Academies of Berlin (1880) and Rome, and to theFrench Academy of Sciences (1900). He received honorary doctorates from the universities ofOslo,Zurich, andBraunschweig.
While teaching calculus for the first time at thePolytechnic school, Dedekind developed the notion now known as aDedekind cut (German:Schnitt), now a standard definition of the real numbers. The idea of a cut is that anirrational number divides therational numbers into two classes (sets), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, thesquare root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater 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 thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers");[3] in modern terminology,Vollständigkeit,completeness.
Dedekind defined two sets to be "similar" when there exists aone-to-one correspondence between them.[4] He invoked similarity to give the first[5] precise definition of aninfinite set: a set is infinite when it is "similar to a proper part of itself,"[6] in modern terminology, isequinumerous to one of itsproper subsets. Thus the setN ofnatural numbers can be shown to be similar to the subset ofN whose members are thesquares of every member ofN, (N →N2):
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
It was in this time frame that he had developed the Dedekind cut in 1872.[7] The 1879 and 1894 editions of theVorlesungen included supplements introducing the notion of an ideal, fundamental toring theory. (The word "Ring", introduced later byHilbert, does not appear in Dedekind's work.) Dedekind defined anideal as a subset of a set of numbers, composed ofalgebraic integers that satisfy polynomial equations withinteger coefficients. The concept underwent further development in the hands of Hilbert and, especially, ofEmmy Noether. Ideals generalizeErnst Eduard Kummer'sideal numbers, devised as part of Kummer's 1843 attempt to proveFermat's Last Theorem. (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind andHeinrich Martin Weber applied ideals toRiemann surfaces, giving an algebraic proof of theRiemann–Roch theorem.
In 1888, he published a short monograph titledWas sind und was sollen die Zahlen? ("What are numbers and what are they good for?" Ewald 1996: 790),[8] which included his definition of aninfinite set. He also proposed anaxiomatic foundation for the natural numbers, whose primitive notions were the numberone and thesuccessor function. The next year,Giuseppe Peano, citing Dedekind, formulated an equivalent but simplerset of axioms, now the standard ones.
Dedekind made other contributions toalgebra. For instance, around 1900, he wrote the first papers onmodular lattices. In 1872, while on holiday inInterlaken, Dedekind metGeorg Cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes withLeopold Kronecker, who was philosophically opposed to Cantor'stransfinite numbers.[9]
1890. "Letter to Keferstein" inJean 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 ofStetigkeit und irrationale Zahlen andWas sind und was sollen die Zahlen?
1996.Theory of Algebraic Integers.Stillwell, John, ed. and trans. Cambridge Uni. Press. A translation ofÜber 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 ofStetigkeit...)
1888.What are numbers and what should they be?, 787–832. (translation ofWas sind und...)
1872–82, 1899. Correspondence with Cantor, 843–77, 930–40.
^James, Ioan (2002).Remarkable Mathematicians. Cambridge University Press. p. 196.ISBN978-0-521-52094-2.
^Ewald, William B., ed. (1996) "Continuity and irrational numbers", p. 766 inFrom Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press.full text
^"The Nature and Meaning of Numbers".Essays on the Theory of Numbers. Dover. 1963 [1901]. Part III, Paragraph 32 – via Google Books –.1901 edition, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
^Moore, G.H. (17 November 1982).Zermelo's Axiom of Choice. New York: Springer.ISBN978-0-387-90670-6.
^"The Nature and Meaning of Numbers".Essays on the Theory of Numbers. Dover. 1963 [1901]. Part V, Paragraph 64 – via Google Books –.1901 edition, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
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