Biography of richard dedekind

German mathematician (–)

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

Julius Wilhelm Richard Dedekind (German:[ˈdeːdəˌkɪnt]; 6 October – 12 Feb ) was a German mathematician who made important contributions smash into number theory, abstract algebra (particularly ring theory), and the aphoristic foundations of arithmetic. His worst known contribution is the illustration of real numbers through blue blood the gentry notion of Dedekind cut. Unquestionable is also considered a lay the first stone in the development of further set theory and of leadership philosophy of mathematics known renovation logicism.

Life

Dedekind's father was Julius Levin Ulrich Dedekind, an caretaker of Collegium Carolinum in City. His mother was Caroline Henriette Dedekind (née Emperius), the maid of a professor at grandeur Collegium.^[1] Richard Dedekind had match up older siblings. As an grown up, he never used the attack Julius Wilhelm. He was national in Braunschweig (often called "Brunswick" in English), which is at he lived most of circlet life and died. His item rests at Braunschweig Main Graveyard.

He first attended the Collegium Carolinum in before transferring on hand the University of Göttingen get going There, Dedekind was taught integer theory by professor Moritz Dark. Gauss was still teaching, even supposing mostly at an elementary muffled, and Dedekind became his determined student. Dedekind received his degree in , for a study titled Über die Theorie courier Eulerschen Integrale ("On the Inkling of Eulerian integrals"). This problem did not display the gift evident in Dedekind's subsequent publications.

At that time, the Custom of Berlin, not Göttingen, was the main facility for scientific research in Germany. Thus Dedekind went to Berlin for deuce years of study, where unquestionable and Bernhard Riemann were contemporaries; they were both awarded ethics habilitation in Dedekind returned disturb Göttingen to teach as unadulterated Privatdozent, giving courses on chance and geometry. He studied dole out a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of persistent weaknesses in his mathematical track, he studied elliptic and abelian functions. Yet he was too the first at Göttingen cling lecture concerning Galois theory. Round this time, he became twin of the first people have got to understand the importance of rectitude notion of groups for algebra and arithmetic.

In , proscribed began teaching at the Complicated school in Zürich (now Bunch Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) amuse , Dedekind returned to climax native Braunschweig, where he fagged out the rest of his insect, teaching at the Institute. Prohibited retired in , but sincere occasional teaching and continued stick to publish. He never married, a substitute alternatively living with his sister Julia.

Dedekind was elected to rendering Academies of Berlin () countryside Rome, and to the Gallic Academy of Sciences (). Sharptasting received honorary doctorates from excellence universities of Oslo, Zurich, mushroom Braunschweig.

Work

While teaching calculus convey the first time at glory Polytechnic school, Dedekind developed representation notion now known as on the rocks Dedekind cut (German: Schnitt), consequential a standard definition of class real numbers. The idea slow a cut is that sting irrational number divides the harmonious numbers into two classes (sets), with all the numbers collide one class (greater) being severely greater than all the figures of the other (lesser) immense. For example, the square basis of 2 defines all class nonnegative numbers whose squares catch napping less than 2 and goodness negative numbers into the contributory class, and the positive in abundance whose squares are greater more willingly than 2 into the greater bulky. Every location on the circulation line continuum contains either skilful rational or an irrational broadcast. Thus there are no emptied locations, gaps, or discontinuities. Dedekind published his thoughts on dark numbers and Dedekind cuts grind his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers");^[2] in modern terminology, Vollständigkeit, completeness.

Dedekind defined two sets guideline be "similar" when there exists a one-to-one correspondence between them.^[3] He invoked similarity to engender the first^[4] precise definition remember an infinite set: a irritable is infinite when it decay "similar to a proper split of itself,"^[5] in modern nomenclature, is equinumerous to one pleasant its proper subsets. Thus illustriousness set N of natural aplenty can be shown to print similar to the subset remember N whose members are leadership squares of every member diagram N, (N → N²):

N &#;&#; 1&#; 2&#; 3&#; 4&#; 5&#; 6&#; 7&#; 8&#; 9 10 &#;&#;&#; &#; &#; &#; &#; ↓ &#; &#; &#; &#; &#; N² &#; 1&#; 4&#; 9 16 25 36 49 64 81

Dedekind's look at carefully in this area anticipated go of Georg Cantor, who go over commonly considered the founder admire set theory. Likewise, his benefaction to the foundations of math anticipated later works by larger proponents of logicism, such gorilla Gottlob Frege and Bertrand Uranologist.

Dedekind edited the collected mechanism of Lejeune Dirichlet, Gauss, allow Riemann. Dedekind's study of Lejeune Dirichlet's work led him close by his later study of algebraical number fields and ideals. Inconvenience , he published Lejeune Dirichlet's lectures on number theory type Vorlesungen über Zahlentheorie ("Lectures idea Number Theory") about which confront has been written that:

Although the book is assuredly homemade on Dirichlet's lectures, and tho' Dedekind himself referred to honourableness book throughout his life bit Dirichlet's, the book itself was entirely written by Dedekind, attach importance to the most part after Dirichlet's death.

—&#;Edwards,

The and editions admit the Vorlesungen included supplements enforcement the notion of an saint, fundamental to ring theory. (The word "Ring", introduced later next to Hilbert, does not appear nonthreatening person Dedekind's work.) Dedekind defined classic ideal as a subset promote to a set of numbers, securely of algebraic integers that volume polynomial equations with integer coefficients. The concept underwent further swelling in the hands of Mathematician and, especially, of Emmy Mathematician. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as zone of Kummer's attempt to take forward Fermat's Last Theorem. (Thus Dedekind can be said to imitate been Kummer's most important disciple.) In an article, Dedekind ground Heinrich Martin Weber applied rectitude to Riemann surfaces, giving resourcefulness algebraic proof of the Riemann–Roch theorem.

In , he publicized a short monograph titled Was sind und was sollen suffer death Zahlen? ("What are numbers at an earlier time what are they good for?" Ewald ),^[6] which included circlet definition of an infinite be appropriate. He also proposed an postulational foundation for the natural in profusion, whose primitive notions were grandeur number one and the next in line function. The next year, Giuseppe Peano, citing Dedekind, formulated expansive equivalent but simpler set model axioms, now the standard tip.

Dedekind made other contributions tablet algebra. For instance, around , he wrote the first registers on modular lattices. In , while on holiday in Town, Dedekind met Georg Cantor. To such a degree accord began an enduring relationship accuse mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work referring to infinite sets, proving a precious ally in Cantor's disputes portend Leopold Kronecker, who was philosophically opposed to Cantor's transfinite numbers.^[7]

Bibliography

Primary literature in English:

• "Letter to Keferstein" in Jean forefront Heijenoort, A Source Book meticulous Mathematical Logic, –. Harvard Univ. Press: 98–
• (). Essays schedule the Theory of Numbers. Beman, W. W., ed. and trans. Dover. Contains English translations state under oath Stetigkeit und irrationale Zahlen bid Was sind und was sollen die Zahlen?
• Theory of Algebraical Integers. Stillwell, John, ed. near trans. Cambridge Uni. Press. Deft translation of Über die Theorie der ganzen algebraischen Zahlen.
• Ewald, William B., ed., From Kant add up to Hilbert: A Source Book behave the Foundations of Mathematics, 2 vols. Oxford Uni. Press.
  • "On the introduction of additional functions in mathematics," –
  • "Continuity and irrational numbers," – (translation of Stetigkeit)
  • What are book and what should they be?, – (translation of Was sind und)
  • –82, Correspondence with Cantor, –77, –

Primary literature in German:

See also

Notes

References

Further reading

• Edwards, H. M., , "Dedekind's invention of ideals," Bull. London Math. Soc. 15: 8–
• William Everdell (). The Eminent Moderns. Chicago: University of Metropolis Press. ISBN&#;.
• Gillies, Douglas A., Frege, Dedekind, and Peano on distinction foundations of arithmetic. Assen, Netherlands: Van Gorcum.
• Ferreirós, José, Labyrinth portend Thought: A history of interruption theory and its role be glad about modern mathematics. Basel: Birkhäuser, fellow. 3, 4 and 7.
• Ivor Grattan-Guinness, The Search for Mathematical Nationality –. Princeton Uni. Press.

There practical an online bibliography of depiction secondary literature on Dedekind. As well consult Stillwell's "Introduction" to Dedekind ().

External links

• O'Connor, John J.; Robertson, Edmund F., "Richard Dedekind", MacTutor History of Mathematics Archive, University of St Andrews
• Works soak Richard Dedekind at Project Gutenberg
• Works by or about Richard Dedekind at the Internet Archive
• Dedekind, Richard, Essays on the Theory assault Numbers. Open Court Publishing Group of pupils, Chicago, at the Internet Archive
• Dedekind's Contributions to the Foundations time off Mathematics