Richard Dedekind

"Dedekind" redirects here. For other uses, seeDedekind (surname).

Richard Dedekind
Born	(1831-10-06)6 October 1831 Braunschweig,Duchy of Brunswick
Died	12 February 1916(1916-02-12)(aged 84) Braunschweig,German Empire
Nationality	German
Alma mater	Collegium Carolinum University of Göttingen
Known for	Dedekind cut Dedekind-Peano axioms Dedekind's theorem Abstract algebra Algebraic number theory Real numbers Logicism
Scientific career
Fields	Mathematics Philosophy of mathematics
Doctoral advisor	Carl Friedrich Gauss

Julius Wilhelm Richard Dedekind[ˈdeːdəˌkɪnt](6 October 1831 – 12 February 1916) was a Germanmathematicianwho made important contributions tonumber theory,abstract algebra(particularlyring theory), and theaxiomatic foundations of arithmetic.His best known contribution is the definition ofreal numbersthrough the notion ofDedekind cut.He is also considered a pioneer in the development of modernset theoryand of thephilosophy of mathematicsknown asLogicism.

Life

Dedekind's father was Julius Levin Ulrich Dedekind, an administrator ofCollegium CarolinuminBraunschweig.His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium.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 Göttingenin 1850. There, Dedekind was taughtnumber theoryby professorMoritz Stern.Gausswas 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 Riemannwere contemporaries; they were both awarded thehabilitationin 1854. Dedekind returned to Göttingen to teach as aPrivatdozent,giving courses onprobabilityandgeometry.He studied for a while withPeter Gustav Lejeune Dirichlet,and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studiedellipticandabelian 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 ofgroupsforalgebraandarithmetic.

In 1858, he began teaching at thePolytechnicschool 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.

Work

Dedekind, before 1886

While teaching calculus for the first time at thePolytechnicschool, 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 numberdivides therational numbersinto 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 2defines 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" );in modern terminology,Vollständigkeit,completeness.

Dedekind defined two sets to be "similar" when there exists aone-to-one correspondencebetween them.He invoked similarity to give the firstprecise definition of aninfinite set:a set is infinite when it is "similar to a proper part of itself,"in modern terminology, isequinumerousto one of itsproper subsets.Thus the setNofnatural numberscan be shown to be similar to the subset ofNwhose members are thesquaresof every member ofN,(N→N²):

N1 2 3 4 5 6 7 8 9 10...
↓
N21 4 9 16 25 36 49 64 81 100...

Dedekind's work in this area anticipated that ofGeorg Cantor,who is commonly considered the founder ofset theory.Likewise, his contributions to thefoundations of mathematicsanticipated later works by major proponents ofLogicism,such asGottlob FregeandBertrand Russell.

Dedekind edited the collected works ofLejeune Dirichlet,Gauss,andRiemann.Dedekind's study of Lejeune Dirichlet's work led him to his later study ofalgebraic number fieldsandideals.In 1863, he published Lejeune Dirichlet's lectures onnumber theoryasVorlesungen 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 theVorlesungenincluded 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 anidealas a subset of a set of numbers, composed ofalgebraic integersthat satisfy polynomial equations withintegercoefficients. 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 Weberapplied 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),which included his definition of aninfinite set.He also proposed anaxiomaticfoundation for the natural numbers, whose primitive notions were the numberoneand 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.

Bibliography

Primary literature in English:

• 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 ZahlenandWas 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.

Primary literature in German:

• Gesammelte mathematische Werke(Complete mathematical works, Vol. 1–3).Retrieved 5 August 2009.

See also

• List of things named after Richard Dedekind
• Dedekind cut
• Dedekind domain
• Dedekind eta function
• Dedekind-infinite set
• Dedekind number
• Dedekind psi function
• Dedekind sum
• Dedekind zeta function
• Ideal (ring theory)

References

• Biermann, Kurt-R (2008). "Dedekind, (Julius Wilhelm) Richard".Complete Dictionary of Scientific Biography.Vol. 4. Detroit: Charles Scribner's Sons. pp. 1–5.ISBN978-0-684-31559-1.

Further reading

• Edwards, H. M.,1983, "Dedekind's invention of ideals,"Bull. London Math. Soc. 15:8–17.
• William Everdell(1998).The First Moderns.Chicago:University of Chicago Press.ISBN0-226-22480-5.
• Gillies, Douglas A., 1982.Frege, Dedekind, and Peano on the foundations of arithmetic.Assen, Netherlands: Van Gorcum.
• Ferreirós, José, 2007.Labyrinth of Thought: A history of set theory and its role in modern mathematics.Basel: Birkhäuser, chap. 3, 4 and 7.
• Ivor Grattan-Guinness,2000.The Search for Mathematical Roots 1870–1940.Princeton Uni. Press.

There is anonline bibliographyof the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).

External links

• O'Connor, John J.;Robertson, Edmund F.,"Richard Dedekind",MacTutor History of Mathematics Archive,University of St Andrews
• Works by Richard DedekindatProject Gutenberg
• Works by or about Richard DedekindatInternet Archive
• Dedekind, Richard,Essays on the Theory of Numbers.Open Court Publishing Company, Chicago, 1901.at theInternet Archive
• Dedekind's Contributions to the Foundations of Mathematicshttp://plato.stanford.edu/entries/dedekind-foundations/.