An Appreciation of Kronecker

Abstract

I am especially honored by the invitation to speak at this annual joint session of the mathematics section and the history section of the Academy because my predecessor last December was Andr6 Weil. As you well know, Weil said nothing last year about the man who is my subject tonight, but, as you probably do not know, he might well have. In fact, at the International Congress of Mathematicians in Cambridge, Massachusetts, in 1950, he gave an invited address which could have borne the title of my talk tonight, "An Appreciation of Kronecker". I will return to the content of Wefts Cambridge address later in my talk. First, I had better begin with the basic facts about Kronecker, since I imagine that even many of the mathematicians among you know little more about him than that he gave his name to the "Kronecker delta". Leopold Kronecker was born in 1823 in the Silesian town of Liegnitz--now in Poland, then in Prussia-into a wealthy Jewish family. The source and the extent of the family weal th is unknown to me. Frobenius's memorial speech on Kronecker, which is, I believe, the source all others have used for biographical informat ion about Kronecker, refers to Kronecker's father as a "businessman", but he also mentions banking and land in connection with the family wealth. Wherever it came from, there seems to have been plenty of it. Kronecker's father took great care with the boy's education, in the early years hiring a tutor for him and then sending him to the gymnasium in Liegnitz. The boy was an avid and gifted student. At the gymnasium he had the good fortune to have as his mathematics teacher the young E. E. Kummer, who later won world renown not only as a creative mathematician but also as an outstanding teacher. From his time with Kummer at the gymnasium forward, Kronecker appears to have been committed to mathematics, although he had a lifelong interest in many other subjects--philosophy, the classical languages, and history among them. After Liegnitz, he studied primarily at the University of Berlin, although he also spent semesters at the Universities of Bonn and Breslau, the latter while Kummer was there. To give mathematicians an idea of mathematics in Berlin at this time, it will suffice to say that Dirichlet, another great teacher and great mathematician, was at his zenith then, and in Berlin, and that both Jacobi and Eisenstein were there, too. Kronecker received the doctorate from Berlin in 1845 at the age of 22. Then he did a strange thing. He left Berlin and its lively mathematical environment to return to Silesia and manage various family properties. He appears to have been mathematically active during this time, but he published nothing. Frobenius says that he maintained an energetic scientific correspondence with Kummer and that he said in later years that the period of isolation in Silesia had had the fortunate consequence " that he was able to mature slowly." Frobenius goes on to say, "For his colleagues, however, it