This final chapter contains three survey articles on mathematics, dating from 1900, 1951, and 2000, as well as a brief and subjective account of the Second International Congress of Mathematicians, held in Paris in August 1900.
George Bruce Halsted was one of the American delegates to the Congress and wrote a report for the Monthly shortly after he returned. The major part of the paper deals with his reactions to Hilbert's famous address on the problems of mathematics, an address that set the agenda for twentieth-century work in mathematics. But the other talk that particularly interested Halsted was one on Japanese mathematics, by Rikitaro Fujisawa (1861–1933). Fujisawa's conclusions as to the Japanese independent discovery of both zero and the square root of – 1 are not accepted today.
In a report written for the beginning of the twentieth century, G. A. Miller discusses some “new fields” of mathematics, fields that seemed to him to be particularly fertile. Among the important areas currently under active investigation, Miller picked the arithmetization of analysis, the development of set theory, and the study of groups as particularly worthy of further attention. He also noted that practical applications of mathematics were important; in particular, he was impressed with the discovery of a linkage that would construct a straight line.
A half-century later, Hermann Weyl discussed the mathematics of the first half of the twentieth century.