Richard Dedekind

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Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

,
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
(particularly
ring theory In algebra, ring theory is the study of ring (mathematics), 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 ...

), and the axiomatic foundations of arithmetic. His best known contribution is the definition of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s through the notion of
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, p ...
. He is also considered a pioneer in the development of modern
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
and of the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such quest ...
known as ''
Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reduction (philosophy), reducible t ...
''.

Life

Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in
Braunschweig Braunschweig () or Brunswick ( , from Low German ''Brunswiek'' , Braunschweig dialect: ''Bronswiek'') is a List of cities and towns in Germany, city in Lower Saxony, Germany, north of the Harz Mountains at the farthest navigable point of the riv ...

. 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. He first attended the Collegium Carolinum in 1848 before transferring to the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...

in 1850. There, Dedekind was taught
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

by professor Moritz Stern.
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

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. At that time, the
University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin Berlin ( , ) is the capital and largest city of Germany ...
, not
Göttingen Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
, was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...
were contemporaries; they were both awarded the
habilitation Habilitation is the highest academic degree, university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, us ...
in 1854. Dedekind returned to Göttingen to teach as a ''
Privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...
'', giving courses on
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. He studied for a while with
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
, and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied
elliptic In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
and abelian functions. Yet he was also the first at Göttingen to lecture concerning
Galois theory In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. About this time, he became one of the first people to understand the importance of the notion of groups for
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
and
arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their chang ...
. In 1858, he began teaching at the Polytechnic school in
Zürich Zürich () is the list of cities in Switzerland, 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. As of January 2020, the municipality has 43 ...
(now ETH Zürich). When the Collegium Carolinum was upgraded to a ''
Technische Hochschule A ''Technische Hochschule'' (, plural: ''Technische Hochschulen'', abbreviated ''TH'') is a type of university focusing on engineering sciences in Germany. Previously, it also existed in Austria, Switzerland, the Netherlands (), and Finland (, ). ...
'' (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 The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
(1900). He received honorary doctorates from the universities of
Oslo Oslo ( , , or ; sma, Oslove) is the Capital city, capital and List of towns and cities in Norway, most populous city of Norway. It constitutes both a Counties of Norway, county and a Municipalities of Norway, municipality. The municipality o ...
, Zurich, and
Braunschweig Braunschweig () or Brunswick ( , from Low German ''Brunswiek'' , Braunschweig dialect: ''Bronswiek'') is a List of cities and towns in Germany, city in Lower Saxony, Germany, north of the Harz Mountains at the farthest navigable point of the riv ...
.

Work

While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, p ...
(German: ''Schnitt''), now a standard definition of the real numbers. The idea of a cut is that an
irrational number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
divides the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s 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 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
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"); in modern terminology, ''Vollständigkeit'', '' completeness''. Dedekind defined two sets to be "similar" when there exists a
one-to-one correspondence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
between them. He invoked similarity to give the first precise definition of an
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...
: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is
equinumerous In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
to one of its proper subsets. Thus the set N of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s can be shown to be similar to the subset of N whose members are the
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
s of every member of N, (N N2): N    1  2  3  4  5  6  7  8  9 10 ...                       N2   1  4  9 16 25 36 49 64 81 100 ... Dedekind's work in this area anticipated that of
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...
, who is commonly considered the founder of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
. Likewise, his contributions to the
foundations of mathematics Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
anticipated later works by major proponents of
Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reduction (philosophy), reducible t ...
, such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
. Dedekind edited the collected works of Lejeune Dirichlet,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

, 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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

as ''
Vorlesungen über Zahlentheorie (German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the i ...
'' ("Lectures on Number Theory") about which it has been written that: The 1879 and 1894 editions of the ''Vorlesungen'' included supplements introducing the notion of an ideal, fundamental to
ring theory In algebra, ring theory is the study of ring (mathematics), 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 ...
. (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 integer In algebraic number theory, an algebraic integer is a complex number which is Integral element, integral over the Integer#Algebraic properties, integers. That is, an algebraic integer is a complex root of a polynomial, root of some monic polyno ...
s that satisfy polynomial equations with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
coefficients. The concept underwent further development in the hands of Hilbert and, especially, of
Emmy Noether Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...
. Ideals generalize Ernst Eduard Kummer's
ideal number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or ...
s, devised as part of Kummer's 1843 attempt to prove
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...
. (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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
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), which included his definition of an
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...
. He also proposed an
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
atic foundation for the natural numbers, whose primitive notions were the number one and the
successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
. The next year,
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concer ...
, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones. Dedekind made other contributions to
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. For instance, around 1900, he wrote the first papers on modular lattices. In 1872, while on holiday in
Interlaken Interlaken (; lit.: ''between lakes'') is a Switzerland, Swiss List of towns in Switzerland, town and Municipalities of Switzerland, municipality in the Interlaken-Oberhasli (administrative district), Interlaken-Oberhasli administrative district i ...
, Dedekind met
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...
. 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 Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a Germans, German mathematician who worked on number theory, abstract algebra, algebra and mathematical logic, logic. He criticized Georg Cantor's work on set theory, and was quoted b ...
, who was philosophically opposed to Cantor's transfinite numbers.

Bibliography

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.

* List of things named after Richard Dedekind *
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, p ...
*
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every Ideal (ring_theory)#Examples and properties, nonzero proper ideal factors into a product of prime ideals. It can be shown t ...
*
Dedekind eta function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
* Dedekind-infinite set * Dedekind number * Dedekind psi function * Dedekind sum * Dedekind zeta function *
Ideal (ring theory) In ring theory, a branch of abstract algebra, an ideal of a ring (mathematics), ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction ...

References

*

* Edwards, H. M., 1983, "Dedekind's invention of ideals," ''Bull. London Math. Soc. 15'': 8–17. * *Gillies, Douglas A., 1982. ''Frege, Dedekind, and Peano on the foundations of arithmetic''. Assen, Netherlands: Van Gorcum. *
Ivor Grattan-Guinness Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a History of mathematics, historian of mathematics and History of logic, logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and ed ...
, 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Uni. Press. There is a
online bibliography
of the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).