Richard Dedekind

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Richard Dedekind
Dedekind-Peano axioms
Abstract algebra
Algebraic number theory
Real numbers
Logicism
Scientific career
FieldsMathematics
Philosophy of mathematics
Doctoral advisorCarl Friedrich Gauss

Julius Wilhelm Richard Dedekind [ˈdeːdəˌkɪnt] (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as Logicism.

Life

Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of

Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium.[1] 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. His body rests at Braunschweig Main Cemetery
.


He first attended the Collegium Carolinum in 1848 before transferring to the

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 in Dedekind's subsequent publications.

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
.

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.

Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the

Braunschweig
.

Work

Dedekind, before 1886

While teaching calculus for the first time at the

completeness
.

Dedekind defined two sets to be "similar" when there exists a

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 → 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, who is commonly considered the founder of set theory. Likewise, his contributions to the foundations of mathematics anticipated later works by major proponents of Logicism, such as Gottlob Frege and Bertrand Russell.


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

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),

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

transfinite numbers.[7]

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:

See also

Notes

  1. ISBN 978-0-521-52094-2
    .
  2. ^ 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
  3. ^ "The Nature and Meaning of Numbers". Essays on the Theory of Numbers. Dover. 1963 [1901]. Part III, Paragraph 32 – via Google Books –. 1901 edition, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
  4. ISBN 978-0-387-90670-6
    .
  5. ^ "The Nature and Meaning of Numbers". Essays on the Theory of Numbers. Dover. 1963 [1901]. Part V, Paragraph 64 – via Google Books –. 1901 edition, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
  6. ^ Richard Dedekind (1888). Was sind und was sollen die Zahlen?. Braunschweig: Vieweg. Online available at: MPIWG GDZ UBS
  7. ISBN 9780743422994
    .
  8. doi:10.1090/S0002-9904-1933-05535-0
    .

References

Further reading

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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