Saturday, August 01, 2009

Leopold Kronecker

Leopold Kronecker was born on December 7, 1823 in the city of Liegnitz which is today part of Poland. At the time of his birth, it was part of the Kingdom of Prussia. His father was a wealthy businessman and his mother had come from a wealthy family. As he was growing up, his education was handled by private tutors.

He entered the Gymnasium at Liegnitz. There, he grew interested in mathematics after attending lectures by the mathematician Ernst Kummer.

In 1841, he enrolled at the Berlin University. There, he studied under Johann Dirichlet and Jakob Steiner. He also became quite interested in the philosophies of Rene Descartes, Wilhelm Leibniz, Benedict Spinoza, and Georg Hegel. He attended one semester at the University of Bonn to study astronomy and one year at the University of Breslau to study under Kummer who had recently been appointed the chair of mathematics.

His doctoral thesis was on algebraic number theory and was very well received. He soon became friends with the mathematicians Carl Gustav Jacobi and Ferdinand Eisenstein. These mathematicians would have a great effect on his thinking about mathematics.

In 1848, he married the daughter of his uncle. He helped to manage the family estate and by this time, he had come to his share of the family fortune. He studied mathematics solely for his own enjoyment.

In 1855, he returned to Berlin in order to continue his work in mathematics among the top mathematicians of his day. Kummer had recently transfered to Berlin to take over a position left open by Dirichlet. Carl Borchardt was also in Berlin at this time as he had recently become the editor of Crelle's Journal. Karl Weierstrass came to Berlin in 1856.

Despite the fact that Kronecker was not a professor, he still wrote numerous well-received mathematical papers. His topics included number theory, elliptic functions, algebra, the theory of determinants, the theory of integrals, and the interrelations between these topics. In 1861, he was elected to the Berlin Academy.

Even though he was not a professor, being a member of the Berlin Academy entitled him to lecture at the university. His lectures were hard to follow and not very popular with the students. Still, between his papers and his lectures, his mathematical reputation shined. He was offered the mathematics chair of the University of Gottingen which he declined because he prefered to stay in Berlin. He was elected as a member of the Paris Academy and in 1883, he became math chair of the University of Berlin. In 1884, he was elected a member of the Royal Society of London.

Kronecker had long had certain radical views on the nature of mathematics. He once said:
God created the integers, all else is the work of man.
By this, he meant that in his view, mathematics should deal only with finite numbers and functions that involved a finite number of operations on those numbers. He did not approve the use of irrational numbers, upper and lower limits, transcendental numbers or any other concept which could not be derived in a finite way. Kronecker opposed the publication of Heinrich Heine's work on trigonometric series and the set theory work done by Georg Cantor. This had significance since Kronecker was on the editorial staff of Crelle's Journal and later, in 1880, became editor of the influential math journal. In 1883, he became one of the codirectors of the mathematical seminar in Berlin.

He made his views public when criticized the theory of irrational numbers in 1886:

... 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.

Kronecker was easily offended and often broke contact with mathematicians who's ideas he did not agree with. For example, he did not get along with Weierstrass, Dedekind, and Cantor. When he became math chair of the University of Berlin, Weierstrass planned to move to Switzerland but changed his mind when he decided that someone needed to oppose the views of Kronecker.

Kronecker died on December 29, 1891. The majority of mathematicians of his day accepted the theory of irrational numbers. Today, his strong rejection of the work by Cantor and Dedekind seems quite eccentric. Still, it is important to remember that his thoughts had great impact on Jules Poincare and Luitzen Brouwer in their work on Intuitionism which exerts a strong influence to this day.