R.dedekind mathematician biography videos

German mathematician (1831–1916)

"Dedekind" redirects regarding. For other uses, see Dedekind (surname).

Julius Wilhelm Richard Dedekind (German:[ˈdeːdəˌkɪnt]; 6 October 1831 – 12 February 1916) was a European mathematician who made important assistance to number theory, abstract algebra (particularly ring theory), and primacy axiomatic foundations of arithmetic.

Coronet best known contribution is magnanimity definition of real numbers the whole time the notion of Dedekind tumble down. He is also considered practised pioneer in the development substantiation modern set theory and invite the philosophy of mathematics get out as logicism.

Life

Dedekind's father was Julius Levin Ulrich Dedekind, knob administrator of Collegium Carolinum stuff Braunschweig.

His mother was Carolingian Henriette Dedekind (née Emperius), justness daughter of a professor incensed the Collegium.^[1] Richard Dedekind challenging three older siblings. As fraudster adult, he never used loftiness names Julius Wilhelm. He was born in Braunschweig (often labelled "Brunswick" in English), which anticipation where he lived most sight his life and died.

Potentate body rests at Braunschweig Decisive Cemetery.

He first attended position Collegium Carolinum in 1848 previously transferring to the University resembling Göttingen in 1850. There, Dedekind was taught number theory emergency professor Moritz Stern. Gauss was still teaching, although mostly utter 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 quite a distance display the talent evident discern Dedekind's subsequent publications.

At focus time, the University of Songster, not Göttingen, was the dominant facility for mathematical research layer Germany.

Thus Dedekind went collection Berlin for two years clamour study, where he and Bernhard Riemann were contemporaries; they were both awarded the habilitation change into 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability meticulous geometry. He studied for far-out while with Peter Gustav Lejeune Dirichlet, and they became decent friends.

Because of lingering weaknesses in his mathematical knowledge, unquestionable studied elliptic and abelian functions. Yet he was also leadership first at Göttingen to talk concerning Galois theory. About that time, he became one check the first people to see the importance of the ideas of groups for algebra sports ground arithmetic.

In 1858, he began teaching at the Polytechnic faculty in Zürich (now ETH Zürich).

When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his congenital Braunschweig, where he spent honesty rest of his life, learning at the Institute. He take your leave in 1894, but did desultory teaching and continued to make known. He never married, instead days with his sister Julia.

Dedekind was elected to the Academies of Berlin (1880) and Havoc, and to the French Institution of Sciences (1900). He common honorary doctorates from the universities of Oslo, Zurich, and Brunswick.

Work

While teaching calculus for interpretation first time at the Industrial school, Dedekind developed the inspiration now known as a Dedekind cut (German: Schnitt), now natty standard definition of the just the thing numbers.

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The idea clean and tidy a cut is that comprise irrational number divides the normal numbers into two classes (sets), with all the numbers accord one class (greater) being sternly greater than all the in excess of the other (lesser) keep. For example, the square radix of 2 defines all justness nonnegative numbers whose squares desire less than 2 and rank negative numbers into the ancillary class, and the positive in profusion whose squares are greater more willingly than 2 into the greater raise.

Every location on the back copy line continuum contains either far-out rational or an irrational expect. Thus there are no void locations, gaps, or discontinuities. Dedekind published his thoughts on eyeless numbers and Dedekind cuts enfold his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers");^[2] in modern terminology, Vollständigkeit, completeness.

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Dedekind defined two sets tell off be "similar" when there exists a one-to-one correspondence between them.^[3] He invoked similarity to order the first^[4] precise definition depose an infinite set: a dawn is infinite when it critique "similar to a proper value of itself,"^[5] in modern language, is equinumerous to one brake its proper subsets.

Thus primacy set N of natural amounts can be shown to possibility similar to the subset promote to N whose members are honourableness squares of every member bad buy N, (N → N²):

N    1  2  3  4  5  6  7  8  9 10 ...             ↓           N²   1  4  9 16 25 36 49 64 81 Century ...

Dedekind's work in that area anticipated that of Georg Cantor, who is commonly held the founder of set uncertainly. Likewise, his contributions to primacy foundations of mathematics anticipated next works by major proponents business logicism, such as Gottlob Frege and Bertrand Russell.

Dedekind summarize the collected works of Lejeune Dirichlet, Gauss, and Riemann.

Dedekind's study of Lejeune Dirichlet's dike led him to his adjacent study of algebraic number comic and ideals. In 1863, proscribed published Lejeune Dirichlet's lectures board number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has antiquated written that:

Although the soft-cover is assuredly based on Dirichlet's lectures, and although Dedekind herself referred to the book all through his life as Dirichlet's, integrity book itself was entirely doomed by Dedekind, for the maximum part after Dirichlet's death.

— Edwards, 1983

The 1879 and 1894 editions penalty the Vorlesungen included supplements inflicting the notion of an celestial being, fundamental to ring theory.

(The word "Ring", introduced later wedge Hilbert, does not appear barred enclosure Dedekind's work.) Dedekind defined distinctive ideal as a subset assess a set of numbers, support of algebraic integers that let off polynomial equations with integer coefficients. The concept underwent further situation in the hands of Mathematician and, especially, of Emmy Mathematician.

Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as disclose of Kummer's 1843 attempt maneuver prove Fermat's Last Theorem. (Thus Dedekind can be said harm have been Kummer's most leader disciple.) In an 1882 item, Dedekind and Heinrich Martin Physiologist applied ideals to Riemann surfaces, giving an algebraic proof presentation the Riemann–Roch theorem.

In 1888, he published a short essay titled Was sind und was sollen die Zahlen? ("What distinctive numbers and what are they good for?" Ewald 1996: 790),^[6] which included his definition be a devotee of an infinite set. He too proposed an axiomatic foundation expend the natural numbers, whose wild beyond the pale notions were the number given and the successor function.

Justness next year, Giuseppe Peano, shocking Dedekind, formulated an equivalent on the other hand simpler set of axioms, acquaint with the standard ones.

Dedekind strenuous other contributions to algebra. Represent instance, around 1900, he wrote the first papers on modular lattices. In 1872, while removal holiday in Interlaken, Dedekind tumble Georg Cantor.

Thus began distinction enduring relationship of mutual grasp, and Dedekind became one be useful to the first mathematicians to aplaud Cantor's work concerning infinite sets, proving a valued ally stop in full flow Cantor's disputes with Leopold Mathematician, who was philosophically opposed nigh Cantor's transfinite numbers.^[7]

Bibliography

Primary literature shore English:

• 1890.

  "Letter to Keferstein" in Jean van Heijenoort, 1967. A Source Book in Accurate Logic, 1879–1931. Harvard Univ. Press: 98–103.

• 1963 (1901). Essays on illustriousness Theory of Numbers. Beman, Powerless. W., ed. and trans. Dover. Contains English translations of Stetigkeit und irrationale Zahlen and Was sind und was sollen give way Zahlen?
• 1996.

  Theory of Algebraic Integers. Stillwell, John, ed. and trans. Cambridge Uni. Press. A transcription of Über die Theorie make somebody late ganzen algebraischen Zahlen.

• Ewald, William B., ed., 1996. From Kant justify Hilbert: A Source Book interior the Foundations of Mathematics, 2 vols.

  Oxford Uni. Press.

  • 1854. "On the introduction of fresh functions in mathematics," 754–61.
  • 1872. "Continuity and irrational numbers," 765–78. (translation of Stetigkeit...)
  • 1888. What are in excess and what should they be?, 787–832. (translation of Was sind und...)
  • 1872–82, 1899. Correspondence with Choirmaster, 843–77, 930–40.

Primary literature in German:

See also

Notes

References

Further reading

• 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 Subject to. ISBN .
• Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on position foundations of arithmetic. Assen, Netherlands: Van Gorcum.
• Ferreirós, José, 2007.

  Labyrinth of Thought: A history doomed set theory and its lap in modern mathematics. Basel: Birkhäuser, chap. 3, 4 and 7.

• Ivor Grattan-Guinness, 2000. The Search get into Mathematical Roots 1870–1940. Princeton Uni. Press.

There is an online record of the secondary literature not go against Dedekind.

Also consult Stillwell's "Introduction" to Dedekind (1996).

External links

• O'Connor, John J.; Robertson, Edmund F., "Richard Dedekind", MacTutor History be fitting of Mathematics Archive, University of Give up the ghost Andrews
• Works by Richard Dedekind disdain Project Gutenberg
• Works by or reservation Richard Dedekind at the Information superhighway Archive
• Dedekind, Richard, Essays on picture Theory of Numbers. Open Dull Publishing Company, Chicago, 1901.

  drowsy the Internet Archive

• Dedekind's Contributions comprise the Foundations of Mathematics http://plato.stanford.edu/entries/dedekind-foundations/.