Part VII
Special Topics
The ordering of material that we have used up to now—first, by cultures and, within each culture, roughly chronological—becomes useless after the beginning of the eighteenth century. From that point on, there is essentially only one mathematical culture, a world-wide one, with a broad consensus as to methods, although some specialties are more concentrated in one geographical area than another. As for chronology, so much mathematics has been produced every year, and mathematics has been advancing along so many broad fronts, that a chapter devoted to a single decade in the eighteenth century or a single year in the twentieth would be prodigiously long. As the time period grew shorter and the chapters grew longer, all perspective would be lost. For that reason, this final part of the history, except for Chapter 35, which discusses women mathematicians in the late nineteenth and early twentieth centuries, consists of chapters, each of which is devoted to the development of a single subject area.
In an effort to convey as much mathematics as possible in this book, we have slighted some other questions of sociological and political interest, such as the increasing democratization of mathematics that accompanied the increase in prosperity after the industrial revolution, its opening up to people from working-class backgrounds. Especially important in that democratization was the gradual involvement of women in the mathematical world. We shall devote the first chapter in this final part to that subject. For lack of space, we are forced to omit other interesting subjects, such as the influence of the Nazi and Communist regimes on mathematics in Germany and the Soviet Union and the impact of the Cold War on mathematical research in the United States.
We ended our narrative of the development of different mathematical subjects at different points. We left the story of both algebra and geometry at the point they had reached around the beginning of the seventeenth century, and we left the story of calculus and its outgrowths in the nineteenth century. Certain prominent parts of mathematics, such as probability, mathematical logic, set theory, and modern number theory have hardly been mentioned at all. While the enormous literature generated by these subjects in the modern era makes the task of summarizing them nearly impossible, we can at least make a grand sweep of each of them to provide some measure of completeness to our coverage of the world of mathematics. These last eleven chapters will fill in some of these gaps. These chapters, much more than those that have preceded, are written in the style that we called heritage in Chapter 1. That is, they aim to show how certain familiar features of modern mathematics arose rather than to describe objectively what mathematical life was like in the past.