Richard Dedekind

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Richard Dedekind
Richard Dedekind 1900s.jpg
Born(1831-10-06)6 October 1831
Died12 February 1916(1916-02-12) (aged 84)
Braunschweig, German Empire
Alma mater Collegium Carolinum
University of Göttingen
Known for Abstract algebra
Algebraic number theory
Real numbers
Scientific career
Fields Mathematics
Philosophy of mathematics
Doctoral advisor Carl Friedrich Gauss

Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.

Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Abstract algebra branch of mathematics

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.



Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).

Braunschweig Place in Lower Saxony, Germany

Braunschweig, also called Brunswick in English, is a city in Lower Saxony, Germany, north of the Harz mountains at the farthest navigable point of the Oker River which connects it to the North Sea via the Aller and Weser Rivers. In 2016, it had a population of 250,704.

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. 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 display the talent evident by Dedekind's subsequent publications.

University of Göttingen university in the city of Göttingen, Germany

The University of Göttingen is a public research university in the city of Göttingen, Germany. Founded in 1734 by George II, King of Great Britain and Elector of Hanover, and starting classes in 1737, the university is the oldest in the state of Lower Saxony and the largest in student enrollment, which stands at around 31,500.

Number theory branch of pure mathematics devoted primarily to the study of the integers

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

Carl Friedrich Gauss German mathematician and physicist

Johann Carl Friedrich Gauss (; German: Gauß[ˈkaɐ̯l ˈfʁiːdʁɪç ˈɡaʊs]; Latin: Carolus Fridericus Gauss; was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.

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 and abelian functions. 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 groups for algebra and arithmetic.

Göttingen Place in Lower Saxony, Germany

Göttingen is a university city in Lower Saxony, Germany. It is the capital of the district of Göttingen. The River Leine runs through the town. At the start of 2017, the population was 134,212.

Bernhard Riemann German mathematician

Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

Habilitation defines the qualification to conduct self-contained university teaching and is the key for access to a professorship in many European countries. Despite all changes implemented in the European higher education systems during the Bologna Process, it is the highest qualification level issued through the process of a university examination and remains a core concept of scientific careers in these countries.

In 1858, he began teaching at the 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.

Zürich Place in Switzerland

Zürich or Zurich is the largest city in Switzerland and the capital of the canton of Zürich. It is located in north-central Switzerland at the northwestern tip of Lake Zürich. The municipality has approximately 409,000 inhabitants, the urban agglomeration 1.315 million and the Zürich metropolitan area 1.83 million. Zürich is a hub for railways, roads, and air traffic. Both Zürich Airport and railway station are the largest and busiest in the country.

<i>Technische Hochschule</i>

A Technische Hochschule is a type of university focusing on engineering sciences in Germany. Previously, it also existed in Austria, Switzerland, the Netherlands, and Finland. In the 1970s and the 1980s, the Technische Hochschule emerged into the Technische Universität (German) or Technische Universiteit (Dutch). Since 2009, several German universities of applied sciences were renamed to Technische Hochschulen.

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 Oslo, Zurich, and Braunschweig.

French Academy of Sciences learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research

The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of scientific developments in Europe in the 17th and 18th centuries, and is one of the earliest Academies of Sciences.

University of Oslo Norwegian public research university

The University of Oslo, until 1939 named the Royal Frederick University, is the oldest university in Norway, located in the Norwegian capital of Oslo. Until 1 January 2016 it was the largest Norwegian institution of higher education in terms of size, now surpassed only by the Norwegian University of Science and Technology. The Academic Ranking of World Universities has ranked it the 58th best university in the world and the third best in the Nordic countries. In 2015, the Times Higher Education World University Rankings ranked it the 135th best university in the world and the seventh best in the Nordics. While in its 2016, Top 200 Rankings of European universities, the Times Higher Education listed the University of Oslo at 63rd, making it the highest ranked Norwegian university.

University of Zurich university in Switzerland

The University of Zurich, located in the city of Zürich, is the largest university in Switzerland, with over 25,000 students. It was founded in 1833 from the existing colleges of theology, law, medicine and a new faculty of philosophy.


Dedekind, c. 1886 ETH-BIB-Dedekind, Julius Wilhelm Richard (1831-1916)-Portrait-Portr 11953.tif (cropped).jpg
Dedekind, c. 1886

While teaching calculus for the first time at the 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 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, 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"); [1] in modern terminology, Vollständigkeit, completeness .

Dedekind's theorem [2] states that if there existed a one-to-one correspondence between two sets, then 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 N, (NN2):

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 Lejeune Dirichlet, Gauss, and Riemann. Dedekind's study of Lejeune Dirichlet's work led him to his later study of algebraic number fields and ideals. 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:

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 integer coefficients. The concept underwent further development in the hands of Hilbert and, 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 Weber applied ideals to Riemann surfaces, 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), [3] which included his definition of an infinite set. He also proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function. The next year, Giuseppe Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones.

Dedekind made other contributions to algebra. For instance, around 1900, he wrote the first papers on modular lattices. 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 very 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. [4]


Primary literature in English:

Primary literature in German:

See also


  1. 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. full text
  2. The Nature and Meaning of Numbers. Essays on the Theory of Numbers. Dover (published 1963). 1901, Open Court. Part V, Paragraph 64, October 2011.Check date values in: |year= (help)
  3. Richard Dedekind (1888). Was sind und was sollen die Zahlen?. Braunschweig: Vieweg. Online available at: MPIWG GDZ UBS
  4. Aczel, Amir D. (2001), The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, Pocket Books nonfiction, Simon and Schuster, p. 102, ISBN   9780743422994 .

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Further reading

There is an online bibliography of the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).