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Encyclopedia > Richard Dedekind
Richard Dedekind
Richard Dedekind

Julius Wilhelm Richard Dedekind (October 6, 1831February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... October 6 is the 279th day of the year (280th in Leap years). ... 1831 was a common year starting on Saturday (see link for calendar). ... February 12 is the 43rd day of the year in the Gregorian Calendar. ... 1916 (MCMXVI) is a leap year starting on Saturday (link will take you to calendar) // Events January-February January 1 - The Royal Army Medical Corps first successful blood transfusion using blood that had been stored and cooled. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Contents


Life

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). His life appears to have been uneventful, consisting of little more than his mathematical teaching and research. Map of Germany showing Braunschweig Braunschweig [ˈbraunʃvaik] (English & French: Brunswick) is a city of 245,500 people (as of December 31, 2004), located in Lower Saxony, Germany. ...


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 theory under 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. 1848 is a leap year starting on Saturday of the Gregorian calendar. ... 1850 was a common year starting on Tuesday (see link for calendar). ... The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... 1852 was a leap year starting on Thursday (see link for calendar). ... In mathematics, there are two types of Euler integral: Euler integral of the first kind: the Beta function Euler integral of the second kind: the Gamma function For positive integers m and n See also Leonhard Euler Factorial ...


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 habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and 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 group for algebra and arithmetic. Bernhard Riemann. ... Habilitation is a term used within the university system in Germany, Austria, and some other European countries such as the German-speaking part of Switzerland, in Poland, the Czech Republic, Slovakia, Hungary, Slovenia, Russia, and other countries of former Soviet Union, such as Armenia, Azerbaijan, Moldova, Kirgizstan, Kazakhstan, Uzbekistan, etc. ... 1854 was a common year starting on Sunday (see link for calendar). ... The word probability derives from the Latin probare (to prove, or to test). ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... Peter Gustav Lejeune Dirichlet. ... In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In mathematics, Galois theory is a branch of abstract algebra. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Algebra is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ...


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. 1858 is a common year starting on Friday. ... ETH Zurich (from its German name Eidgenössische Technische Hochschule Zürich, ETHZ) is the Swiss Federal Institute of Technology in Zürich, Switzerland. ... Location within Switzerland (help· info) (German pronunciation IPA: ; in English often Zurich, without the umlaut) is the largest city in Switzerland (population: 366,145 in 2004; population of urban area: 1,091,732) and capital of the canton of Zürich. ... Technische Hochschule (acronym TH) is, what a university of technology (i. ...


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.


Work

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 number divides 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. Dedekind published his thought on irrational numbers and Dedekind cuts in his paper Stetigkeit und irrationale Zahlen ("Continuity and irrational numbers." Ewald 1996: 766). ETH Zurich (from its German name Eidgenössische Technische Hochschule Zürich, ETHZ) is the Swiss Federal Institute of Technology in Zürich, Switzerland. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards...


In 1874, while on holiday in Interlaken, Dedekind met Cantor. Thus began an enduring 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. 1874 (MDCCCLXXIV) was a common year starting on Thursday (see link for calendar). ... Interlaken is a municipality in the Canton of Bern in Switzerland. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...


If there existed a one-to-one correspondence between 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 equinumerous to one of its proper subsets. 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 N2, (N N2): In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... Two sets A and B are said to be equinumerous if they have the same cardinality, i. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ...

 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 theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... The gauss, abbreviated as G, is the cgs unit of magnetic flux density or magnetic induction (B), named after the German mathematician and physicist Carl Friedrich Gauss. ... Bernhard Riemann. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... 1863 (MDCCCLXIII) is a common year starting on Thursday of the Gregorian calendar (or a common year starting on Saturday of the Julian calendar). ... This article needs to be cleaned up to conform to a higher standard of quality. ... Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ...

"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. (N.B. 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 integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummers 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 Weber applied ideals to Riemann surfaces. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Emmy Noether (born Nöther) (March 23, 1882 – April 14, 1935) was one of the most talented mathematicians of the early 20th century, with penetrating insights that she used to develop elegant abstractions which she formalized beautifully. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekinds definition of ideals for rings. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats last theorem (edition of 1670). ... 1882 (MDCCCLXXXII) was a common year starting on Sunday (see link for calendar). ... Heinrich Martin Weber (1842 - 1913) was a German mathematician who specialized in algebra and number theory. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...


Dedekind made other contributions to algebra. For instance, around 1900, he wrote the first papers on modular lattices. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ...


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. 1888 is a leap year starting on Sunday (click on link for calendar). ... In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... One redirects here. ... A successor function is the label in the literature for what is actually an operation. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ...


See also

In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ... In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekinds definition of ideals for rings. ... Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ...

Bibliography

Primary literature in English:

  • 1963 (1901). Essays on the Theory of Numbers. Beman, W. W., ed. and trans. Dover.
  • 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.

Dedekind (1963) and Ewald (1996) both contain English translations of Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen?


Secondary:

  • Edwards, H. M., 1983, "Dedekind's invention of ideals," Bull. London Math. Soc. 15: 8-17.
  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.

There is an online bibliography of the secondary literature on Dedekind, but apparently no biography. Also consult Stillwell's "Introduction" to Dedekind (1996). Ivor Grattan-Guiness is a prolific contemporary historian of mathematics and logic. ...


External link

  • John J. O'Connor and Edmund F. Robertson. Richard Dedekind at the MacTutor archive.

  Results from FactBites:
 
Richard Dedekind - definition of Richard Dedekind in Encyclopedia (910 words)
Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummer's closest follower in arithmetic.
Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
Dedekind was among the first mathematicians who had accepted Cantor's work on the theory of infinite sets; other mathematicians didn't yet understand their ideas.
Richard Dedekind - Wikipedia, the free encyclopedia (1066 words)
Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the Paris Académie des Sciences (1900).
If there existed a one-to-one correspondence between 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 equinumerous to one of its proper subsets.
Dedekind's study of Dirichlet's work was what led him to his later study of algebraic number fields and ideals.
  More results at FactBites »

 
 

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