Introduction

The last century has been a golden age for mathematics. Many important, long-standing problems have been resolved, in large part because of the growing understanding of the complex interactions among the subfields of mathematics. As those relationships continue to expand and deepen,mathematics is beginning to reach out to explore interactions with other areas of science. These interactions, both within diverse areas of mathematics and between mathematics and other fields of science, have led to some great insights and to the broadening and deepening of the field of mathematics. I discuss some of these interactions and insights, describe a few mathematical achievements of the twentieth century, and pose some challenges and opportunities that await us in the twenty-first century.

The world of mathematics

In discussing our subject, we mathematicians face a dilemma. The most effective way to explain mathematics to general readers is to use metaphors, which entails a loss of precision and carries the risk of misunderstanding. On the other hand, advanced mathematical terms are incomprehensible tomost people—including other scientists. As my colleague David Mumford, former president of the International Mathematics Union, has said, “I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who … declare with an odd sort of pride that they are mathematically illiterate.”

Within the mathematical community, however, the use of a precise language is a distinct advantage. Because of its abstract nature and universality, mathematics knows neither linguistic nor political boundaries.