Everything about Richard Dedekind totally explained
» 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 didn't 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.
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's "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, (
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, doesn't 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.
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