- For the 16th century humanist, see Friedrich Dedekind.
Julius Wilhelm Richard Dedekind (October 6, 1831 –
February 12, 1916) was a German mathematician who did important work in abstract algebra, algebraic number theory and the
foundations of the real numbers.
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).
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.
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.
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.
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. 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 thought on irrational numbers and Dedekind
cuts in his paper Stetigkeit und irrationale Zahlen ("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
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
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 N2, (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 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 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.
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, Peano, citing Dedekind, formulated an equivalent but simpler set of
axioms, now the standard ones.
See also
| “ |
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.
|
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.
Secondary:
- 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).
External links
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