**Leopold Kronecker**'s parents were well off, his father, Isidor Kronecker, being a successful business man while his mother was Johanna Prausnitzer who also came from a wealthy family. The families were Jewish, the religion that Kronecker kept until a year before his death when he became a convert to Christianity. Kronecker's parents employed private tutors to teach him up to the stage when he entered the Gymnasium at Liegnitz, and this tutoring gave him a very sound foundation to his education.

Kronecker was taught mathematics at Liegnitz Gymnasium by Kummer, and it was due to Kummer that Kronecker became interested in mathematics. Kummer immediately recognised Kronecker's talent for mathematics and he took him well beyond what would be expected at school, encouraging him to undertake research. Despite his Jewish upbringing, Kronecker was given Evangelical religious instruction at the Gymnasium which certainly shows that his parents were openminded on religious matters.

Kronecker became a student at Berlin University in 1841 and there he studied under Dirichlet and Steiner. He did not restrict himself to studying mathematics, however, for he studied other topics such as astronomy, meteorology and chemistry. He was especially interested in philosophy studying the philosophical works of Descartes, Leibniz, Kant, Spinoza and Hegel. After spending the summer of 1843 at the University of Bonn, which he went to because of his interest in astronomy rather than mathematics, he then went to the University of Breslau for the winter semester of 1843-44. The reason that he went to Breslau was certainly because of his interest in mathematics because he wanted to study again with his old school teacher Kummer who had been appointed to a chair at Breslau in 1842.

Kronecker spent a year at Breslau before returning to Berlin for the winter semester of 1844-45. Back in Berlin he worked on his doctoral thesis on algebraic number theory under Dirichlet's supervision. The thesis, *On complex units* was submitted on 30 July 1845 and he took the necessary oral examination on 14 August. Dirichlet commented on the thesis saying that in it Kronecker showed:-

... unusual penetration, great assiduity, and an exact knowledge of the present state of higher mathematics.

It may come as a surprise to many Ph.D. students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.

Jacobi had health problems which caused him to leave Königsberg, where he held a chair, and return to Berlin. Eisenstein, whose health was also poor, lectured in Berlin around this time and Kronecker came to know both men well. The direction that Kronecker's mathematical interests went later had much to do with the influence of Jacobi and Eisenstein around this time. However, just as it looked as if he would embark on an academic career, Kronecker left Berlin to deal with family affairs. He helped to manage the banking business of his mother's brother and, in 1848, he married the daughter of this uncle, Fanny Prausnitzer. He also managed a family estate but still found the time to continue working on mathematics, although he did this entirely for his own enjoyment.

Certainly Kronecker did not need to take on paid employment since he was by now a wealthy man. His enjoyment of mathematics meant, however, that when circumstances changed in 1855 and he no longer needed to live on the estate outside Liegnitz, he returned to Berlin. He did not wish a university post, rather he wanted to take part in the mathematical life of the university and undertake research interacting with the other mathematicians.

In 1855 Kummer came to Berlin to fill the vacancy which occurred when Dirichlet left for Göttingen. Borchardt had lectured at Berlin since 1848 and, in late 1855, he took over the editorship of *Crelle's Journal* on Crelle's death. In 1856 Weierstrass came to Berlin, so within a year of Kronecker returning to Berlin, the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker was in place in Berlin.

Of course since Kronecker did not hold a university appointment, he did not lecture at this time but was remarkably active in research publishing a large number of works in quick succession. These were on number theory, elliptic functions and algebra, but, more importantly, he explored the interconnections between these topics. Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass. On 23 January 1861 Kronecker was elected to the Academy and this had a surprising benefit.

Members of the Berlin Academy had a right to lecture at Berlin University. Although Kronecker was not employed by the University, or any other organisation for that matter, Kummer suggested that Kronecker exercise his right to lecture at the University and this he did beginning in October 1862. The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals. In his lectures [1]:-

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

For the best students his lectures were demanding but stimulating. However, he was not a popular teacher with the average students [1]:-

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.

Berlin was attractive to Kronecker, so much so that when he was offered the chair of mathematics in Göttingen in 1868, he declined. He did accept honours such as election to the Paris Academy in that year and for many years he enjoyed good relations with his colleagues in Berlin and elsewhere. In order to understand why relations began to deteriorate in the 1870s we need to examine Kronecker's mathematical contributions more closely.

We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers. However the topics he studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker is well known for his remark:-

God created the integers, all else is the work of man.

Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature. Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist.

In 1870 Heine published a paper *On trigonometric series* in *Crelle's Journal*, but Kronecker had tried to persuade Heine to withdraw the paper. Again in 1877 Kronecker tried to prevent publication of Cantor's work in *Crelle's Journal*, not because of any personal feelings against Cantor (which has been suggested by some biographers of Cantor) but rather because Kronecker believed that Cantor's paper was meaningless, since it proved results about mathematical objects which Kronecker believed did not exist. Kronecker was on the editorial staff of *Crelle's Journal* which is why he had a particularly strong influence on what was published in that journal. After Borchardt died in 1880, Kronecker took over control of *Crelle's Journal* as the editor and his influence on which papers would be published increased.

The mathematical seminar in Berlin had been jointly founded in 1861 by Kummer and Weierstrass and, when Kummer retired in 1883, Kronecker became a codirector of the seminar. This increased Kronecker's influence in Berlin. Kronecker's international fame also spread, and he was honoured by being elected a foreign member of the Royal Society of London on 31 January 1884. He was also a very influential figure within German mathematics [1]:-

He established other contacts with foreign scientists in numerous travels abroad and in extending to them the hospitality of his Berlin home. For this reason his advice was often solicited in regard to filling mathematical professorships both in Germany and elsewhere; his recommendations were probably as significant as those of his erstwhile friend Weierstrass.

Although Kronecker's view of mathematics was well known to his colleagues throughout the 1870s and 1880s, it was not until 1886 that he made these views public. In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:-

... the introduction of various concepts by the help of which it has frequently been attempted in recent times(but first by Heine)to conceive and establish the "irrationals" in general. Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms(or coefficients), ... certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.

Lindemann had proved that π is transcendental in 1882, and in a lecture given in 1886 Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist. So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments. Kronecker explained his programme based on studying only mathematical objects which could be constructed with a finite number of operation from the integers in *Über den Zahlbergriff* Ⓣ in 1887.

Another feature of Kronecker's personality was that he tended to fall out personally with those who he disagreed with mathematically. Of course, given his belief that only finitely constructible mathematical objects existed, he was completely opposed to Cantor's developing ideas in set theory. Not only Dedekind, Heine and Cantor's mathematics was unacceptable to this way of thinking, and Weierstrass also came to feel that Kronecker was trying to convince the next generation of mathematicians that Weierstrass's work on analysis was of no value.

Kronecker had no official position at Berlin until Kummer retired in 1883 when he was appointed to the chair. But by 1888 Weierstrass felt that he could no longer work with Kronecker in Berlin and decided to go to Switzerland, but then, realising that Kronecker would be in a strong position to influence the choice of his successor, he decided to remain in Berlin.

Kronecker was of very small stature and extremely self-conscious about his height. An example of how Kronecker reacted occurred in 1885 when Schwarz sent him a greeting which included the sentence:-

He who does not honour the Smaller, is not worthy of the Greater.

Here Schwarz was joking about the small man Kronecker and the large man Weierstrass. Kronecker did not see the funny side of the comment, however, and never had any further dealings with Schwarz (who was Weierstrass's student and Kummer's son-in-law). Others however displayed more tact and, for example, Helmholtz who was a professor in Berlin from 1871, managed to stay on good terms with Kronecker.

The *Deutsche Mathematiker-Vereinigung* was set up in 1890 and the first meeting of the Association was organised in Halle in September 1891. Despite the bitter antagonism between Cantor and Kronecker, Cantor invited Kronecker to address this first meeting as a sign of respect for one of the senior and most eminent figures in German mathematics. However, Kronecker never addressed the meeting, since his wife was seriously injured in a climbing accident in the summer and died on 23 August 1891. Kronecker only outlived his wife by a few months, and died in December 1891.

We should not think that Kronecker's views of mathematics were totally eccentric. Although it was true that most mathematicians of his day would not agree with those views, and indeed most mathematicians today would not agree with them, they were not put aside. Kronecker's ideas were further developed by Poincaré and Brouwer, who placed particular emphasis upon intuition. Intuitionism stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.

**Article by:** *J J O'Connor* and *E F Robertson*

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