INTRODUCTION TO FINITE MATHEMATICS

By Profs. John G. Kemeny, J.Laurie Snell, and Gerald L. Thompson, allof the Mathematics Department. New York:Prentice-Hall, 1957. 572 pp. $5.00.

College mathematics has traditionally been analytic geometry and the calculus; and these austere subjects have been symbols of the unchanging and unchangeable. It was good to suffer these things, knowing that future generations would suffer in exactly the same way.

It is true that as much as forty years ago other mathematical disciplines were timidly making their way: group theory, Boolean algebra, probability. But these were not for undergraduates who meant business. These were pretty-pretties, playgrounds in research for second-class scholars who couldn't do important things in the older and purer fields.

Then these new fields blossomed amazingly. The workers are no longer timid. They find ideas and methods in their fields which seem to them to be of as much value as the ideas and methods of the old fields. These ideas are new, and they have an appeal because they are new - the older mathematics is so terribly old. They want undergraduates to see these ideas. More than that, these pioneers challenge the validity of parts of the traditional. They tell the old guard that most of analytic geometry is itself a pretty-pretty, a luxury which simply can't be afforded. The calculus, they admit, is still the Rock of Gibraltar, but the long, tortuous, dimly lighted paths hewn out of solid rock, should be replaced by escalators and funiculars. The important thing is to reach the top. From there you can take off for a flight to the Pyrenees, then to the Alps, and beyond the Alps lies Italy. (If the reader loses the analogy, think nothing of it; one recognizes the enthusiasm of the flight pilot.)

The book Introduction to Finite Mathematics is not a revolutionary book. It is simply the first intelligent, adequate attempt to bring the modern developments of logic, probability, and matrix algebra to the freshman-sophomore level. A few half-hearted attempts had been made; this is the first one that clicks. And it clicks because three men, each competent in his own field, were able to select intelligently, and fuse sensitively.

The selection of topics and the applications were made with the interests of students in the social and biological sciences in mind. Yet this new mathematics is also becoming increasingly important for chemists, physicists, and engineers. Perhaps only experience can tell whether one book will fit the varied needs of mathematicians, chemists, engineers, behavioral scientists, and uncommitted liberal arts students. The reviewer would hope that it can. Experience seems to show that a college of liberal arts makes its greatest contributions precisely where it rises superior to minor variations. And it would be a rash man who would venture to predict exactly what mathematics would be "useful" to a biochemist twenty years from now.

Thus while this is not a revolutionary book, it is a concrete symbol of a revolutionary trend. Here at Dartmouth a course in finite mathematics will be inserted bodily into the analytic geometry - calculus - differential equations sequence. This is not going to be done because three members of the Department wrote a book; they wrote the book so that it could be done. This is real pioneering, and the reviewer is delighted to be able to voice his opinion that they have done a firstrate job, and because this has been done, Dartmouth is assured the leading role in a new and significant development of college mathematics.