Kronecker

Kronecker's primary contributions were in the theory of equations. He made major contributions in elliptic functions and the theory of algebraic numbers.

Kronecker was taught mathematics at school by Kummer and it was due to him that Kronecker became interested in mathematics. Kronecker became a student at Berlin University in 1841 where he studied under Jacobi, Dirichlet and Eisenstein. After spending time at Bonn and Breslau he returned to Berlin to write a Ph. D. thesis on algebraic number theory under Dirichlet's supervision.

Kronecker then left Berlin and returned to Silesia where he was to make a private fortune as a banker. He remained there until 1855 when he returned to Berlin where he was to remain for the rest of his life. In 1855 Kummer came to Berlin to fill the vacancy left when Dirichlet left for Göttingen. Kronecker was not to become a professor until Kummer retired in 1883.

Kronecker was of very small stature and extremely self-conscious about his height. In fact he attacked vigorously anyone whose mathematics he disapproved.

His primary contributions were in the theory of equations and higher algebra. His major contributions were in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.

Kronecker led the opposition to Cantor's view of set theory and the foundations of mathematics. 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. The value of set theory, particularly in analysis and topology, made sure that it was accepted into mathematics despite initial opposition to the idea of the infinite.

Kronecker views were not put aside, however, and were 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.