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Leopold Kronecker Edit Profile

mathematician

Leopold Kronecker was a German mathematician who worked on number theory, algebra and logic.

Background

Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now Legnica, Poland) in a wealthy Jewish family. His parents were Isidor and Johanna (née Prausnitzep).

Education

His were wealthy and provided private tutoring at home for their son until he entered the Liegnitz Gymnasium.

At the Gymnasium, Kronecker’s mathematics teacher was E. E. Kummer, who early recognized the boy’s ability and encouraged him to do independent research.

He also attended Schelling’s philosophy lectures; he was later to make a thorough study of the works of Descartes, Spinoza, Leibniz, Kant, and Hegel, as well as those of Schopenhauer, whose ideas he rejected. Kronecker spent the summer semester of 1843 at the University of Bonn, having been attracted there by Argelander’s astronomy lectures.

The following autumn he went to Breslau (now Wroclaw, Poland) because Kummer had been appointed professor there.

He remained for two semesters, returning to Berlin in the winter semester of 1844-1845 to take the doctorate. In his dissertation, “On Complex Units, ” submitted to the Faculty of Philosophy on 30 July 1845, Kronecker dealt with the particular complex units that appear in cyclotomy.

Kronecker took his oral examination on 14 August 1845.

Encke questioned him on the application of the calculus of probabilities to observations and to the method of least squares; Dirichlet, on definite integrals, series, and differential equations; August Boeckh, on Greek; and Adolf Trendelenburg, on the history of legal philosophy.

Career

By 1855, however, kronecker’s circumstances had changed enough to allow him to return to the academic life in Berlin as a financially independent private scholar. This was a momentous time for mathematics in Germany.

In 1856 Weierstrass was called to Berlin and Kronecker and Kummer soon became friends with Borchardt and Weierstrass. Although Kronecker had published some scientific articles before he returned to Berlin, he soon brought out a large number of mathematical tracts in rapid succession.

Among other subjects he wrote on number theory (one of his earliest interests, instilled in him by Kummer), the theory of elliptical functions, algebra, and, particularly, on the interdependence of these mathematical disciplines.

In the winter semester of the following year Kronecker, at Kummer’s suggestion, made use of a statutory right held by all members of the Academy to deliver a series of lectures at the University of Berlin.

He attempted to simplify and refine existing theories and to present them from new perspectives.

His teaching and his research were closely linked and, like Weierstrass, he was most concerned with ideas that were still in the process of development.

Unlike Weierstrass—and for that matter, Kummer—kronecker did not attract great numbers of students.

Only a few of his auditors were able to follow the flights of his thought, and only a few persevered until the end of the semester.

To those students who could understand him, however, Kronecker communicated something of his joy in mathematical discussion.

The names of these men constitute a formidable catalog; they were, in the order in which Kronecker assisted them, Heine, Riemann, Sylvester, Clebsch, E. Schering, H. J. Stephen Smith, Dedekind, Betti Brioschi, Beltrami, C. J. Malmsten, Hermite, Fuchs, F. Carorati, and L. Cremona.

The formal nominations that Kronecker made during this period are of great interest, not least because of their subjectivity.

Kronecker’s influence outside Germany also increased.

In 1883 Kummer retired from the chair of mathematics, and Kronecker was chosen to succeed him, thereby becoming the first person to hold the post at Berlin who had also earned the doctorate there.

He was simultaneously named codirector of the mathematics seminar that Kummer and Weierstrass had founded in 1861.

He was enabled, too, to sponsor his own students for the doctorate; among his candidates were adolf Kneser, Paul Stäckel, and Kurt Hensel, who was to edit his works and some of his lectures. The cause of the growing estrangement between Kronecker and Weierstrass was the following.

Since they had long maintained the same circle of friends, their friends, too, became involved on both levels.

Kronecker read this allusion to physical size—he was a small man, and increasingly self-conscious with age—as a slur on his intellectual powers and broke with Schwarz completely.

At any rate, personal quarrel became scholarly polemic.

Weierstrass, for example, believed (perhaps rightly) that Kronecker’s opposition to Cantor’s views on “transfinite numbers” reflected opposition to his own work. The basis of Kronecker’s objection to Weierstrass’ methods of analysis is revealed in his well-known dictum that “God Himself made the whole numbers—everything else is the work of men. ”

He never actually stated his intention of recasting analysis without irrational numbers, however, and it is possible that he did not take his radical notions altogether seriously himself.

Weierstrass could not afford to regard Kronecker’s demands as merely whimsical; in 1885 he claimed indignantly that for Kronecker it was an axiom that equations could exist only between whole numbers, while he, Weierstrass, granted irrational numbers the same validity as any other concepts. Kronecker’s remarks that arithmetic could put analysis on a more rigorous basis, and that those who came after him would recognize this and thereby demonstrate the falseness of so-called analysis, angered and embittered Weierstrass.

He saw in these words an attempt by Kronecker not only to invalidate his whole life’s work, but also to seduce the younger generation of mathematicians to an entirely new theory.

The two men were further at odds over a Swedish mathematics prize contest and over the editing of Borchardt’s works.

By 188. ., Weierstrass had confided to a few close friends that his break with Kronecker was complete; Kronecker, for his part, apparently did not realize how gravely his opinions and activities had wounded Weierstrass, since on several later occasions he still referred to himself as being his friend. Weierstrass at this time even considered leaving Germany for Switzerland to avoid the constant conflict with Kronecker, but one consideration kept him in Berlin.

Kronecker had remained on good terms with Kummer and with Kummer’s successor, Fuchs; it was therefore likely that Kronecker would have considerable influence in the choice of Weier-strass’ own successor.

Weierstrass believed that all his work would be undone by a successor acceptable to Kronecker; for this reason he stayed where he was.

His boundary formulas are particularly noteworthy in this regard, since they laid bare the deepest relationships between arithmetic and elliptical functions and provided the basis for Erich Hecke’s later analytic-arithmetical investigations.

Kronecker also introduced a number of formal refinements in algebra and in the theory of numbers, and many new theorems and concepts.

Among the latter, special mention should be made of his theorem in regard to the cyclotomic theory, according to which all algebraic numbers with Abelian and Galois groups (over the rational number field) are rational combinations of roots of unity.

His theorem on the convergence of infinite series is also significant. The most important aspects of Kronecker’s work were manifest as early as his dissertation of 1845.

In his treatment of complex units, kronecker sought to present a theory of units in an algebraic number field, and, indeed, to present a whole system of units as a group.

Twenty-five years later he succeeded in constructing an implicit system of axioms to rule finite Abelian groups, although he did not at that time apply it explicitly to such groups.

One criterion that suggests itself is the application of the algorithm, and while few mathematicians have held unalloyed opinions on this matter, two sharply differentiated positions may be distinguished.

One group of mathematicians then—of whom Gauss, Dirichlet, and Dedekind are representative—found the algorithm to be most useful as a concept, rather than a symbol; their work centered on ideas, not calculations.

The other group, which includes Leibniz, Euler, Jacobi, and—as kneser demonstrated—kronecker, stressed the technical use of the algorithm, employing it as a means to an end.

Kronecker’s goal was the perfection of the technique of calculation and he employed symbols to avoid the repetition of syllogisms and for clarity.

He termed Gauss’s contrasting method of presenting mathematics “dogmatic, ” although he retained a great respect for Gauss and for his work. Kronecker’s mathematics lacked a systematic theoretical basis, however, and for this reason Frobenius asserted that he was not the equal of the greatest mathematicians in the individual fields that he pursued.

Achievements

  • The 25624 Kronecker asteroid is named after him.

Works

Religion

He also received Evangelical religious instruction, although he was Jewish; he formally converted to Christianity in the last year of his life.

Membership

He was a member of many learned societies, among them the Paris Academy, of which he was elected a corresponding member in 1868, and the Royal Society of London, of which he became a foreign associate in 1884.

Kronecker was elected as a member of several academies:

Prussian Academy of Sciences (1861)

French Academy of Sciences (1868)

Royal Society (1884).

Connections

In its interest he was required to spend a few years managing an estate near Liegnitz, as well as to dissolve the banking business of an uncle. In 1848 he married the latter’s daughter, his cousin Fanny Prausnitzer; they had six children.

friend:
Weierstrass