Background

Richard Dedekind was born on October 6, 1831, in Braunschweig, Lower Saxony (Niedersachsen), Germany. Dedekind’s ancestors (particularly on his mother’s side) had distinguished themselves in services to Hannover and Brunswick. His father, Julius Levin Ulrich Dedekind, the son of a physician and chemist, was a graduate jurist, professor, and corporation lawyer at the Collegium Carolinum in Brunswick. His mother, Caroline Marie Henriette Emperius, was the daughter of a professor at the Carolinum and the granddaughter of an imperial postmaster.

Richard Dedekind was the youngest of four children. His only brother, Adolf, became a district court president in Brunswick; one sister, Mathilde, died in 1860, and Dedekind lived with his second sister, Julie, until her death in 1914, neither of them having married. She was a respected writer who received a local literary prize in 1893.

Education

Between the ages of seven and sixteen Dedekind attended the Gymnasium Martino-Catharineum in Brunswick. His interest turned first to chemistry and physics; he considered mathematics only an auxiliary science. He soon occupied himself primarily with it, however, feeling that physics lacked order and a strictly logical structure.

In 1848 Dedekind became a student at the Collegium Carolinum, an institute between the academic high school and the university level, which Carl Friedrich Gauss had also attended. There Dedekind mastered the elements of analytic geometry, algebraic analysis, differential and integral calculus, and higher mechanics, and studied the natural sciences.

In 1849-1850, he gave private lessons in mathematics to his later colleague at the Carolinum, Hans Zincke (known as Sommer). Thus, when he matriculated at the University of Gottingen at Easter 1850, Dedekind was far better prepared for his studies than were the majority of graduates from the academic high school.

At Göttingen, a seminar in mathematics and physics had just been founded, at the initiative of Moritz Abraham Stern, for the education of instructors for teaching in the academic high school. The direction of the mathematics department was the duty of Stern and Georg Ulrich, while Wilhelm Weber and Johann Benedict Listing directed the physics department. Dedekind was a member of the seminar from its inception and was there first introduced to the elements of the theory of numbers. A year later Bernhard Riemann also began to participate in the seminar, and Dedekind soon developed a close friendship with him.

In the first semester, Dedekind attended lectures on differential and integral calculus, which offered him very little new material. He attended Ulrich’s seminar on hydraulics but rarely took part in the physics laboratories run by Weber and Listing; Weber’s lectures on experimental physics, however, made a very strong impression on him throughout two semesters. Weber had an inspiring effect on Dedekind, who responded with respectful admiration.

In the summer semester of 1850, Dedekind attended the course in popular astronomy given by Gauss’s observer, Carl Wolfgang Benjamin Goldschmidt; in the winter semester of 1850-1851, he attended Gauss’s own lecture on the method of least squares. Although he disliked teaching, Gauss carried out the assignment with his usual conscientiousness; fifty years later Dedekind remembered the lecture as one of the most beautiful he had ever heard, writing that he had followed Gauss with constantly increasing interest and that he could not forget the experience. In the following semester, Dedekind heard Gauss’s lecture on advanced geodesy.

In the winter semester of 1851-1852, he heard the two lectures given by Quintus lcilius on mathematical geography and on the theory of heat and took part in lcilius’ meteorological observations. After only four semesters, he did his doctoral work under Gauss in 1852 with a thesis on the elements of the theory of Eulerian integrals. Gauss certified that he knew a great deal and was independent; in addition, he had prophetically “favorable expectations of his future performance.”

Career

Dedekind completed his studies and determined that this knowledge would have been sufficient for teachers in secondary school service but that it did not satisfy the prerequisite for advanced studies at Göttingen. For instance, Dedekind had not heard lectures on more recent developments in geometry, advanced theory of numbers, division of the circle and advanced algebra, elliptic functions, or mathematical physics, which were then being taught at the University of Berlin by Steiner, Jacobi, and Dirichlet. Therefore, Dedekind spent the two years following his graduation assiduously filling the gaps in his education, attending - among others - Stern’s lectures on the solution of numerical equations.

In the summer of 1854, he qualified, a few weeks after Riemann, as a university lecturer; in the winter semester of 1854-1855 he began his teaching activities as Privatdozent, with a lecture on the mathematics of probability and one on geometry with parallel treatment of analytic and projective methods.

After Dirichlet succeeded Gauss in Gottingen in 1855, Dedekind attended his lectures on the theory of numbers, potential theory, definite integrals, and partial diff erential equations. He soon entered into a closer personal relationship with Dirichlet and had many fruitful discussions with him; Dedekind later remembered that Dirichlet had made “a new man” of him and had expanded his scholarly and personal horizons. When the Dirichlets were visited by friends from Berlin (Rebecca Dirichlet was the sister of the composer Felix Mendelssohn-Barlholdy and had a large circle of friends), Dedekind was invited too and enjoyed the pleasant sociability of, for example, the well-known writer and former diplomat, Karl August Varnhagen von Ense, and his niece, the writer Ludmilla Assing.

In the winter semester of 1855-1856 and in the one following, Dedekind attended Riemann’s lectures on Abelian and elliptic functions. Thus, although an instructor, he remained an intensive student as well. His own lectures at that time are noteworthy in that he probably was the first university teacher to lecture on Galois theory, in the course of which the concept of field was introduced. To be sure, few students attended his lectures: only two were present when Dedekind went beyond Galois and replaced the concept of the permutation group by the abstract group concept.

In 1858, Dedekind was called to the Polytechnikum in Zurich (now the Eidgenössische Technische Hochschule) as the successor to Joseph Ludwig Raabe. Thus Dedekind was the first of a long line of German mathematicians for whom Zurich was the first step on the way to a German professorial chair. The Swiss school counsellor responsible for appointments came to Göttingen at Easter 1858 and decided immediately upon Dedekind - which speaks for his power of judgment.

In 1858, Dedekind also noted the lack of a truly scientific foundation of arithmetic in the course of his Zurich lectures on the elements of differential calculus. On 24 October, Dedekind succeeded in producing a purely arithmetic definition of the essence of continuity and, in connection with it, an exact formulation of the concept of the irrational number. Fourteen years later, he published the result of his considerations, Stetigkeit unci irrationale Zahlen (1872), and explained the real numbers as "cuts" in the realm of rational numbers.

In September 1859, Dedekind traveled to Berlin with Riemann, after Riemann’s election as a corresponding member of the academy there. On this occasion, Dedekind met the initiator of that selection, Karl Weierstrass, as well as other leaders of the Berlin school, including Ernst Eduard Kummer, Karl Wilhelm Borchardt, and Leopold Kronecker.

In 1862, he was appointed successor to August Wilhelm Julius Uhde at the Polytechnikum in Brunswick, which had been created from the Collegium Carolinum. He remained in Brunswick until his death, in close association with his brother and sister, ignoring all possibilities of change or attainment of a larger sphere of activity.

Although completely averse to administrative responsibility, Dedekind nevertheless considered it his duty to assume the directorship of the Polytechnikum from 1872 to 1875 - to a certain extent he was the successor of his father, who had been a member of the administration of the Collegium Carolinum for many years - and to assume the chairmanship of the school’s building commission in the course of the transformation to a technical university. Along with his recreational trips to Austria (the Tyrol), to Switzerland, and through the Black Forest, his visit to the Paris exposition of 1878 should also be mentioned. On April 1, 1894 he was made professor emeritus but continued to give lectures occasionally. Seriously ill in 1872, following the death of his father, he subsequently enjoyed physical and intellectual health until his peaceful death at the age of eighty-four.

When Dedekind is mentioned today, one of the first associations is the “Dedekind cut,” which he introduced in 1872 to use in treating the problem of irrational numbers in a completely new and exact manner.

Achievements

• 

  Richard Dedekind went down in history as one of the most notable mathematicians and scientists of his time. His contribution to mathematics could be measured through the number of ideas that bear his name - about a dozen. His contributions to the fundamentals of the concept of number allowed the progress of the real analysis, developing a deeper knowledge of the real numbers and the concept of continuity; his theorems on algebraic ideals have stimulated much additional activity in the twentieth century. His most notable theorem is "Dedekind cut."
  
  Richard Dedekind received honorary doctorates in Kristiania (now Oslo), in Zurich, and in Brunswick. In 1902 he received numerous scientific honors on the occasion of the fiftieth anniversary of his doctorate.
  
  In 1981 a stamp with Dedekind's image was created to commemorate his numerous achievments in science.
  
  Dedekind was also an accomplished pianist and cellist, and composed a chamber opera to his brother’s libretto.

Works

More photos

• book

  • Theory of Algebraic Functions of One Variable

    (This book is the first English translation of the classic...)

    2012
  • Theory of Algebraic Integers

    (The invention of ideals by Dedekind in the 1870s was well...)

    1996
  • Essays on the Theory of Numbers

    (This volume contains the two most important essays on the...)

    1963

Views

In character and principles, in style of living and views, Dedekind had much in common with Gauss, who also came from Brunswick and attended the Gymnasium Martino-Catharineum, the Collegium Carolinum, and the University of Gottingen. Each led a strictly regulated, simple life without luxury. Both were averse to innovations and turned down brilliant offers for other professorial chairs. In their literary tastes, both numbered Walter Scott among their favorite authors.

It is not astonishing to find their similarity persisting in mathematics in the same preference for the theory of numbers, the same reservations about the algorithm, and the same partiality for "notions" above "notations." Although considerable, significant differences existed between Gauss and Dedekind, what they had in common predominates by far. Their kinship also received a marked visible expression: Dedekind was one of the select few permitted to carry Gauss’s casket to the funeral service on the terrace of the Sternwarte.

Dedekind, agreeing with Gauss, regarded numbers as free creations of the human intellect and defended this viewpoint militantly and stubbornly.

Quotations: Dedekind’s own statement to Zincke is more revelatory of his character than any description could be: "For what I have accomplished and what I have become, I have to thank my industry much more, my indefatigable working rather than any outstanding talent."

Membership

Dedekind was a corresponding member of the Gottingen Academy from 1862. Dedekind also became a corresponding member of the Berlin Academy in 1880 upon the initiative of Kronecker. In 1900, he became a correspondent of the Académie des Sciences in Paris and in 1910 was elected as associé étranger. He was also a member of the Leopoldino-Carolina Naturae Curiosorum Academia and of the Academy in Rome.

• 

  The Göttingen Academy , Germany

  1862 - 1916

• 

  The Berlin Academy , Germany

  1880 - 1916

• 

  The Académie des Sciences , France

  1900 - 1916

• 

  The Leopoldino-Carolina Naturae Curiosorum Academia , Germany

• The Academy of Science , Italy

Personality

The small, familiar world in which Dedekind lived completely satisfied his demands: in it his relatives completely replaced a wife and children of his own and there he found sufficient leisure and freedom for scientific work in basic mathematical research. He did not feel pressed to have a more marked effect in the outside world; such confirmation of himself was unnecessary.

Dedekind is often compared to Gauss - both men were cool and reserved in judgment, both were warm-hearted, helpful people who formed strong bonds of trust with their friends. Both had a distinct sense of humor but also a strictness toward themselves and a conscientious sense of duty. Averse to any excess, neither was quick to express astonishment or admiration. Both impressed by that quality called modest greatness. Both men had a conservative sense, a rigid will, an unshakable strength of principles, and a refusal to compromise.

Aside from Gauss the most enduring influences on Dedekind’s scientific work were Dirichlet and Riemann, with both of whom he shared many inclinations and attitudes. Dedekind was all fully conscious of his worth, but with a modesty bordering on shyness, he never let his associates feel this. Ambition being foreign to him, Dedekind was embarrassed when confronted by the brilliance and elegance of his intellect. He loved thinking more than writing and was hardly ever able to satisfy his own demands. Being of absolute integrity, Dedekind had love for plain, certain truth.

Occasionally Dedekind has been called a "modem Eudoxus" because an impressive similarity has been pointed out between Dedekind’s theory of the irrational number and the definition of proportionality in Eudoxus’ theory of proportions

Dedekind also belonged to the mathematicians with great musical talent.

Interests

• piano, cello

• Writers

  Walter Scott

Connections

Dedekind never married, instead living with his sister Julia.

Father:

Julius Levin Ulrich Dedekind

Julius Levin Ulrich Dedekind was a graduate jurist, professor, and corporation lawyer at the Collegium Carolinum in Brunswick.

Mother:

Caroline Marie Henriette Emperius

Caroline Marie Henriette Emperius was the daughter of a professor at the Carolinum and the granddaughter of an imperial postmaster.

Brother:

Adolf Dedekind

Adolf Dedekind was a district court president in Braunschweig.

Sister:

Mathilde Dedekind

Sister:

Julie Dedekind

She was a respected writer who received a local literary prize in 1893.

Friend:

Heinrich Weber

Heinrich Weber was a German mathematician whose main work was in algebra, number theory, and analysis.

Friend:

Peter Gustav Lejeune Dirichlet

Peter Gustav Lejeune Dirichlet was was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis.

References

• Frege, Dedekind, and Peano on the Foundations of Arithmetic First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic.
  1982
• The First Moderns A lively and accessible history of Modernism, The First Moderns is filled with portraits of genius, and intellectual breakthroughs, that richly evoke the fin-de-siècle atmosphere of Paris, Vienna, St. Louis, and St. Petersburg.
  1998