# Richard Dedekind

**Julius Wilhelm Richard Dedekind** (October 6, 1831 – February 12, 1916) was one of the major German mathematicians in the late nineteenth century who did important work in abstract algebra, algebraic number theory, and laid the foundations for the concept of the real numbers. He was one of the few mathematicians who understood the importance of set theory developed by Georg Cantor.

Dedekind argued that the numbers system can be independently developed from geometrical notations and that they are grounded in and derived from a certain inherent creative capacity of the mind, which were some of those issues debated by Bolzano, Cantor, Frege, and Hilbert.

## Life

Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).

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 there, 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.

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 groups for algebra and arithmetic.

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.

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.

He died on February 12, 1916, aged 84.

## Work

While teaching calculus for the first time at the ETH Zürich Polytechnic, Dedekind came up with the notion now called a "Dedekind cut" (in German: *Schnitt*), now a standard definition of the real numbers. The basic idea behind this notion is that an irrational number divides the rational numbers into two classes, 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. Based on this idea, Dedekind cuts are defined as pairs of such divided classes of rational numbers. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician. This means that every location on the number line continuum contains either a rational or an irrational number. Thus, the Dedekind cuts are considered real numbers. There are no empty locations, gaps, or discontinuities. Dedekind published his thought on irrational numbers and Dedekind cuts in his paper *Stetigkeit und irrationale Zahlen*^{[1]} ("Continuity and irrational numbers." Ewald 1996: 766. Note that Dedekind's terminology is old-fashioned: in the present context, one now says *Vollständigkeit* instead of *Stetigkeit*, so a modern translation would have *continuity* replaced with *completeness*).

In 1874, 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 on infinite sets, proving a valued ally in Cantor's battles with Leopold Kronecker, who was philosophically opposed to Cantor's transfinite numbers.

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. (This is known as *Dedekind's theorem*.) 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**^{2}, (**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 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:

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 David Hilbert, does not appear in Dedekind's work.) Dedekind defined a ring 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.

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

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, Giuseppe Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones.

## Quotation

"Of all the aids which the human mind has yet created to simplify its life—that is, to simplify the work in which thinking consists—none is so momentous and so inseparably bound up with the mind’s most inward nature as the concept of number. Arithmetic, whose sole object is this concept, is already a science of immeasurable breadth, and there can be no doubt that there are absolutely no limits to its further development; and the domain of its application is equally immeasurable, for every thinking person, even if he does not clearly realize it, is a person of numbers, an arithmetician." (undated fragment in Dedekind Nachlass, translated in Ewald 1996: 837)

## Notes

- ↑ Richard Dedekind. Stetigkeit und irrationale Zahlen. Retrieved January 22, 2011.

## Bibliography

### Primary literature in English

- 1890. "Letter to Keferstein" in Jean van Heijenoort. [1967] 2002.
*From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931*, 98-103. Harvard University Press. ISBN 0674324498 - [1901] 1963.
*Essays on the Theory of Numbers*. Beman, W. W., ed. and trans. Dover Publications. ISBN 0486210103. 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 University Press. ISBN 0521565189. 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 University Press. ISBN 0198532717- 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.

### Secondary

- Edwards, H. M. 1983. "Dedekind's invention of ideals,"
*Bull. London Math. Soc. 15*: 8-17. - Everdell, William. 1998.
*The First Moderns.*Chicago: University of Chicago Press. ISBN 0226224805 - Gillies, Douglas A. [1982] 2011.
*Frege, Dedekind, and Peano on the Foundations of Arithmetic*. Routledge. ISBN 978-0415667098 - Grattan-Guinness, Ivor. 2000.
*The Search for Mathematical Roots 1870-1940*. Princeton University Press. ISBN 069105858X

## External links

All links retrieved December 8, 2022.

- Essays on the Theory of Numbers (English) Project Gutenberg.
- The Early Development of Set Theory Stanford Encyclopedia of Philosophy.

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