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Richard Dedekind | |
---|---|

Born | |

Died | 12 February 1916 84) Braunschweig, German Empire | (aged

Nationality | German |

Alma mater | Collegium Carolinum University of Göttingen |

Known for | Abstract algebra Algebraic number theory Real numbers Logicism |

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.

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

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**, 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.

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** 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.

**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** 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.

**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** 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.

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

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.

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.

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.

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.

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**, (**N** → **N**^{2}):

N1 2 3 4 5 6 7 8 9 10 ... ↓N^{2}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:

- 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*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.

Primary literature in German:

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

- ↑ 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 - ↑
*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) - ↑ Richard Dedekind (1888).
*Was sind und was sollen die Zahlen?*. Braunschweig: Vieweg. Online available at: MPIWG GDZ UBS - ↑ 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 .

**Georg Ferdinand Ludwig Philipp Cantor** was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

**Mathematical logic** is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In mathematics, a **transcendental number** is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer coefficients. The best-known transcendental numbers are π and *e*. Though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation *x*^{2} − 2 = 0. Another irrational number that is not transcendental is the golden ratio, or , since it is a solution of the polynomial equation *x*^{2} − *x* − 1 = 0.

In the philosophy of mathematics, the abstraction of **actual infinity** involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with **potential infinity**, in which a non-terminating process produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps.

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called **global L-functions**, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these

**Algebraic number theory** is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

**Johann Peter Gustav Lejeune Dirichlet** was a German mathematician who made deep contributions to number theory, and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

In mathematics, **analytic number theory** is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet *L*-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers and additive number theory.

**Leopold Kronecker** was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by Weber (1893) as having said, "* Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk*". Kronecker was a student and lifelong friend of Ernst Kummer.

In number theory 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 Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is *principal* if it consists of multiples of a single element of the ring, and *nonprincipal* otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

** Vorlesungen über Zahlentheorie** is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.

In mathematics, the **Dedekind zeta function** of an algebraic number field *K*, generally denoted ζ_{K}(*s*), is a generalization of the Riemann zeta function. It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane **C** with only a simple pole at *s* = 1, and its values encode arithmetic data of *K*. The extended Riemann hypothesis states that if *ζ*_{K}(*s*) = 0 and 0 < Re(*s*) < 1, then Re(*s*) = 1/2.

**Heinrich Martin Weber** was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He is best known for his text *Lehrbuch der Algebra* published in 1895 and much of it is his original research in algebra and number theory. His work *Theorie der algebraischen Functionen einer Veränderlichen* established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann-Roch theorem. Weber's research papers were numerous, most of them appearing in *Crelle's Journal* or *Mathematische Annalen*. He was the editor of Riemann's collected works.

In number theory, a **Hecke character** is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of *L*-functions larger than Dirichlet *L*-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

In mathematics, the **irrational numbers** are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being *incommensurable*, meaning that they share no "measure" in common, that is, there is no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using **Cantor's first uncountability proof**, which differs from the more familiar proof using his diagonal argument. The title of the article, "**On a Property of the Collection of All Real Algebraic Numbers**", refers to its first theorem: the set of real algebraic numbers is countable. In 1879, Cantor modified his uncountability proof by using the topological notion of a set being dense in an interval.

In set theory, the **Schröder–Bernstein theorem** states that, if there exist injective functions *f* : *A* → *B* and *g* : *B* → *A* between the sets *A* and *B*, then there exists a bijective function *h* : *A* → *B*. In terms of the cardinality of the two sets, this means that if |*A*| ≤ |*B*| and |*B*| ≤ |*A*|, then |*A*| = |*B*|; that is, *A* and *B* are equipotent. This is a useful feature in the ordering of cardinal numbers.

- Biermann, Kurt-R (2008). "Dedekind, (Julius Wilhelm) Richard".
*Complete Dictionary of Scientific Biography*.**4**. Detroit: Charles Scribner's Sons. pp. 1–5. ISBN 978-0-684-31559-1.

- 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. ISBN 0-226-22480-5. - 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.

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

Wikiquote has quotations related to: Richard Dedekind |

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- O'Connor, John J.; Robertson, Edmund F., "Richard Dedekind",
*MacTutor History of Mathematics archive*, University of St Andrews . - Works by Richard Dedekind at Project Gutenberg
- Works by or about Richard Dedekind at Internet Archive
- 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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