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{{Short description|Немачки математичар (1831–1916)}}
{{Научник
| име = Рихард Дедекинд
Линија 25 ⟶ 26:
Дедекинд је у свом делу ''-{Was sind und was sollen dir Zahlen}-?'' (Природа и значење бројева, [[1888]]) понудио [[аксиома]]тски приступ [[природан број|природним бројевима]]. Касније, дефинисао је [[ирационалан број|ирационалне]] помоћу [[Дедекиндов пресек|Дедекиндовог пресека]].<ref>{{cite web |title=Richard Dedekind German mathematician |url=https://www.britannica.com/biography/Richard-Dedekind |website=Britannica |access-date=15. 1. 2021}}</ref>
[[Датотека:Stamp GDR 1981 Richard Dedekind.jpg|мини|250п|Поштанска марка са ликом Дедекинда издата у [[Немачка Демократска Република|НДР]].]]
== Живот ==
{{рут}}
Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of [[TU Braunschweig|Collegium Carolinum]] in [[Braunschweig]]. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium.<ref>{{cite book |last=James |first=Ioan |date=2002 |title=Remarkable Mathematicians |publisher=Cambridge University Press |page=196 |isbn=978-0-521-52094-2}}</ref> 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.
He first attended the Collegium Carolinum in 1848 before transferring to the [[University of Göttingen]] in 1850. There, Dedekind was taught [[number theory]] by professor [[Moritz Stern]]. [[Carl Friedrich Gauss|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 [[Euler integral|Eulerian integrals]]"). This thesis did not display the talent evident by Dedekind's subsequent publications.
At that time, the [[University of Berlin]], not [[Göttingen]], was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and [[Bernhard Riemann]] were contemporaries; they were both awarded the [[habilitation]] in 1854. Dedekind returned to Göttingen to teach as a ''[[Privatdozent]]'', giving courses on [[probability]] and [[geometry]]. He studied for a while with [[Peter Gustav Lejeune Dirichlet]], and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied [[elliptic function|elliptic]] and [[abelian variety|abelian function]]s. Yet he was also the first at Göttingen to lecture concerning [[Galois theory]]. About this time, he became one of the first people to understand the importance of the notion of [[Group (mathematics)|groups]] for [[algebra]] and [[arithmetic]].
In 1858, he began teaching at the [[ETH Zürich|Polytechnic]] school in [[Zürich]] (now ETH 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 sister Julia.
Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the [[French Academy of Sciences]] (1900). He received honorary doctorates from the universities of [[University of Oslo|Oslo]], [[University of Zurich|Zurich]], and [[Technical University at Brunswick|Braunschweig]].
== Рад ==
[[Датотека:ETH-BIB-Dedekind, Julius Wilhelm Richard (1831-1916)-Portrait-Portr 11953.tif (cropped).jpg|thumb|250п|Dedekind, before 1886]]
While teaching calculus for the first time at the [[ETH Zürich|Polytechnic]] school, Dedekind developed the notion now known as a [[Dedekind cut]] (German: ''Schnitt''), now a standard definition of the real numbers. The idea of a cut is that an [[irrational number]] divides the [[rational number]]s into two classes ([[Set (mathematics)|sets]]), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, the [[square 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");<ref>Ewald, William B., ed. (1996) "Continuity and irrational numbers", p. 766 in ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols. Oxford University Press. [http://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html full text]</ref> in modern terminology, ''Vollständigkeit'', ''[[complete space|completeness]]''.
Dedekind defined two sets to be "similar" when there exists a [[one-to-one correspondence]] between them.<ref>{{cite book|work=Essays on the Theory of Numbers|publisher=Dover|publication-date=1963|date=1901 |title=The Nature and Meaning of Numbers|at=Part III, Paragraph 32}}</ref> He invoked similarity to give the first precise definition of an [[Dedekind-infinite set|infinite set]]: a set is infinite when it is "similar to a proper part of itself,"<ref>{{cite book|work=Essays on the Theory of Numbers|publisher=Dover|publication-date=1963|date=1901 |title=The Nature and Meaning of Numbers|at=Part V, Paragraph 64}}</ref> in modern terminology, is [[equinumerous]] to one of its [[subset|proper subsets]]. Thus the set '''N''' of [[natural number]]s can be shown to be similar to the subset of '''N''' whose members are the [[square (algebra)|square]]s of every member of '''N''', ('''N'''<span style="font-size:140%; color:darkgreen;"> → </span> '''N'''<sup>2</sup>):
'''N''' 1 2 3 4 5 6 7 8 9 10 ...
<span style="font-size:140%; color:darkgreen;"> ↓ </span>
'''N'''<sup>2</sup> 1 4 9 16 25 36 49 64 81 100 ...
Dedekind's work in this area anticipated that of [[Georg Cantor]], who is commonly considered the founder of [[set theory]]. Likewise, his contributions to the [[foundations of mathematics]] anticipated later works by major proponents of [[Logicism]], such as [[Gottlob Frege]] and [[Bertrand Russell]].
Dedekind edited the collected works of [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]], [[Carl Friedrich Gauss|Gauss]], and [[Bernhard Riemann|Riemann]]. Dedekind's study of Lejeune Dirichlet's work led him to his later study of [[algebraic number field]]s and [[ideal (ring theory)|ideal]]s. In 1863, he published Lejeune Dirichlet's lectures on [[number theory]] as ''[[Vorlesungen über Zahlentheorie]]'' ("Lectures on Number Theory") about which it has been written that:
{{quote|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 (algebra)|ring theory]]. (The word "Ring", introduced later by [[David Hilbert|Hilbert]], does not appear in Dedekind's work.) Dedekind defined an [[ring ideal|ideal]] as a subset of a set of numbers, composed of [[algebraic integer]]s that satisfy polynomial equations with [[integer]] coefficients. The concept underwent further development in the hands of Hilbert and, especially, of [[Emmy Noether]]. Ideals generalize [[Ernst Eduard Kummer]]'s [[ideal number]]s, 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 Weber]] applied ideals to [[Riemann surface]]s, giving an algebraic proof of the [[Riemann–Roch theorem]].
In 1888, he published a short monograph titled ''Was sind und was sollen die Zahlen?'' ("What are numbers and what are they good for?" Ewald 1996: 790),<ref>{{cite book | author=Richard Dedekind | title=Was sind und was sollen die Zahlen? | location=Braunschweig | publisher=Vieweg | year=1888 }} Online available at: [http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?pn=1&url=%2Fmpiwg%2Fonline%2Fpermanent%2Feinstein_exhibition%2Fsources%2F8GPV80UY%2Fpageimg&viewMode=images&tocMode=thumbs&tocPN=1&searchPN=1&mode=imagepath&characterNormalization=reg&queryPageSize=10 MPIWG] [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN23569441X&DMDID=dmdlog55 GDZ] [http://www.digibib.tu-bs.de/?docid=00063941 UBS]</ref> which included his definition of an [[infinite set]]. He also proposed an [[axiom]]atic foundation for the natural numbers, whose primitive notions were the number [[1 (number)|one]] and the [[successor function]]. The next year, [[Giuseppe Peano]], citing Dedekind, formulated an equivalent but simpler [[Peano axioms|set of axioms]], now the standard ones.
Dedekind made other contributions to [[abstract algebra|algebra]]. For instance, around 1900, he wrote the first papers on [[modular lattice]]s. In 1872, while on holiday in [[Interlaken]], Dedekind met [[Georg 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 with [[Leopold Kronecker]], who was philosophically opposed to Cantor's [[transfinite numbers]].<ref>{{citation|title=The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity|series=Pocket Books nonfiction|first=Amir D.|last=Aczel|publisher=Simon and Schuster|year=2001|isbn=9780743422994|page=102|url=https://books.google.com/books?id=nQinWBLQG3UC&pg=PA102}}.</ref>
== Библиографија ==
{{refbegin|30em}}
Основна литература на енглеском:
* 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 ''[https://web.archive.org/web/20051031071536/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.
Остновна литература на немачком:
*[https://gdz.sub.uni-goettingen.de/id/PPN235685380 Gesammelte mathematische Werke] (Complete mathematical works, Vol. 1–3).<ref>{{cite journal|doi=10.1090/S0002-9904-1933-05535-0|title=Book Review: ''Richard Dedekind. Gesammelte mathematische Werke''|year=1933|last1=Bell|first1=E. T.|author-link=Eric Temple Bell|journal=Bulletin of the American Mathematical Society|volume=39|pages=16–17}}</ref> Retrieved 5 August 2009.
{{refend}}
== Референце ==
{{reflist}}
== Литература ==
{{refbegin|30em}}
* {{cite encyclopedia
| last = Biermann
| first = Kurt-R
| title = Dedekind, (Julius Wilhelm) Richard
| encyclopedia = [[Dictionary of Scientific Biography|Complete Dictionary of Scientific Biography]]
| volume = 4
| pages = 1–5
| publisher = Charles Scribner's Sons
| location = Detroit
| year = 2008
| isbn = 978-0-684-31559-1
}}
*[[Harold Edwards (mathematician)|Edwards, H. M.]], 1983, "Dedekind's invention of ideals," ''Bull. London Math. Soc. 15'': 8–17.
*{{cite book |author = William Everdell |author-link = William Everdell |year = 1998
|title = The First Moderns
|publisher = [[University of Chicago Press]]
|location = Chicago
|isbn = 0-226-22480-5
|url-access = registration
|url = https://archive.org/details/firstmodernsprof00ever
}}
*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.
* {{Citation|last=Bourbaki|first=Nicolas|title=Commutative Algebra|publisher=Addison-Wesley|year=1972}}
* {{Citation|last=Claborn|first=Luther|title=Dedekind domains and rings of quotients|journal = Pacific J. Math.|volume = 15|year = 1965|pages = 59–64|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102995991|doi=10.2140/pjm.1965.15.59|doi-access=free}}
* {{Citation|last=Claborn|first=Luther|title=Every abelian group is a class group|journal = Pacific J. Math.|volume = 18|issue=2|year = 1966|pages = 219–222|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102994263|doi=10.2140/pjm.1966.18.219|doi-access=free}}
* {{Citation|last=Clark|first=Pete L.|title=Elliptic Dedekind domains revisited|journal = L'Enseignement Mathématique|volume = 55|issue=3| year = 2009 | pages = 213–225 | url=http://math.uga.edu/~pete/ellipticded.pdf| doi=10.4171/lem/55-3-1|arxiv=math/0612469}}
* {{cite book | last=Cohn | first=Paul M. |authorlink=Paul Cohn | title=Further algebra and applications |year=2003 |publisher=Springer |isbn=1-85233-667-6 }}
* {{citation | last1=Fröhlich | first1=A. | author1-link=Albrecht Fröhlich | last2=Taylor | first2= M.J. | author2-link=Martin J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X | zbl=0744.11001 | chapter=II. Dedekind domains | pages=35–101 }}
* {{Citation|last=Gomez-Ramirez|first=Danny|title=Conceptual Blending as a Creative meta-generator of mathematical concepts: Prime Ideals and Dedekind Domains as a blend|journal = In: T.R. Besold, K.U. Kühnberger, M. Schorlemmer, A. Smaill (eds.) Proceedings of the 4th International Workshop on Computational Creativity, Concept Invention, and General Intelligence (C3GI) PICS|volume = 2|year = 2015}}[http://ikw.uni-osnabrueck.de/en/system/files/02-2015.pdf ]
* {{Citation|last=Leedham-Green|first=C.R.|title=The class group of Dedekind domains|journal = Trans. Amer. Math. Soc.|volume = 163|year = 1972|pages = 493–500|doi=10.2307/1995734|jstor=1995734|doi-access=free}}
* {{Citation|last=Milne|first=J.S.|title=Algebraic Number Theory (v3.00)|year=2008|url=http://jmilne.org/math/CourseNotes/ant.html}}
* {{Citation|last=Nakano|first=Noburu|title=Idealtheorie in einem speziellen unendlichen algebraischen Zahlkörper|journal = J. Sci. Hiroshima Univ. Ser. A|volume = 16|year = 1953|pages = 425–439}}
* {{Citation|last=Rosen|first=Michael|title=Elliptic curves and Dedekind domains|journal = Proc. Amer. Math. Soc.|volume = 57|year = 1976 | pages = 197–201|doi=10.2307/2041187|jstor=2041187|issue=2|doi-access=free}}
* {{Citation|last=Steinitz|first=E.| authorlink=Ernst Steinitz | title=Rechteckige Systeme und Moduln in algebraischen Zahlkörpern|journal = Math. Ann.|volume = 71|year = 1912| pages = 328–354|doi=10.1007/BF01456849|issue=3|url=https://zenodo.org/record/2367807}}
* {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative Algebra, Volume I | year=1958 | publisher=D. Van Nostrand Company}}
{{refend}}
== Спољашње везе ==
{{портал|Биографија}}▼
{{Commonscat|Richard Dedekind}}
* {{MacTutor Biography|id=Dedekind}}
* {{MathGenealogy |id=18233}}
* [http://finanz.math.tugraz.at/~predota/old/history/mathematiker/dedekind.html Биографија Рихарда Дедекинда] {{De}}
* {{Gutenberg author |id=Dedekind,+Richard | name=Richard Dedekind}}
* {{Internet Archive author |sname=Richard Dedekind}}
* [https://archive.org/details/essaysintheoryof00dedeuoft Dedekind, Richard, ''Essays on the Theory of Numbers.'' Open Court Publishing Company, Chicago, 1901.] at the [[Internet Archive]]
* Dedekind's Contributions to the Foundations of Mathematics http://plato.stanford.edu/entries/dedekind-foundations/.
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{{DEFAULTSORT:Дедекинд, Јулиус Вилхелм Рихард}}
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