Mathematics and the Search for Knowledge
This book is not an account of mathematical analysis, and is not intended to supply the reader with a survey of the various branches and achievements of the subject. In this respect it is notably different from books that attempt to do just this, such as the fascinating What Is Mathematics? by Richard Courant and Herbert Robbins (first published in 1941). As Professor Kline makes explicit, his object is to make clear the important part mathematics has played in the investigation of physical nature. The book is thus primarily a history of the development of physical knowledge, from the time of the ancient Greeks to the revolutionary discoveries of the present century.
Dr. Kline writes appreciatively and with enthusiasm about the advancement of knowledge gained with the help of mathematical analyses of natural phenomena; and except for the chapter on quantum theory his exposition is highly readable and informative, and generally illuminating. He makes evident that the Ptolemaic theory of planetary motion was a great triumph of mathematical analysis—a triumph that was repeated in the heliocentric theory of Copernicus and Kepler, in the mechanics and gravitational theory of Newton, in Maxwell’s fusion of electrical and magnetic forces into a comprehensive theory, in Einstein’s special and general theories of relativity, and in the revolutionary ideas employed in exploring atomic and subatomic processes.
Kline’s account of most of these developments in science before the early years of the twentieth century can be understood by readers with only moderate familiarity with elementary geometry and algebra. For example, he explains Newton’s unification of terrestrial and celestial mechanics with the help of some calculations that require no more mathematics than a high school student normally possesses. He gives an engrossing account of the development of non-Euclidean geometries (geometries with parallel postulates different from Euclid’s), and so prepares the reader to understand the geometrization of physics that characterizes relativity theory. To be sure, to appreciate Maxwell’s electromagnetic theory some familiarity with the differential calculus is necessary; and the structure of quantum theory involves more advanced mathematics, which Kline merely sketches in general terms without attempting to describe the actual content of the relevant theoretical equations. Kline has succeeded remarkably well in conveying to the layman the power, the penetration, and the indispensability of mathematical analysis in exploring physical nature.
Kline is far less successful in his explanation of the role of mathematics in quantum theory. His chapter on the subject describes the hierarchy of structures constituting the nature of matter and gives the reader a good idea of the small dimensions characterizing the basic components of physical reality. But the reader is also likely to be overwhelmed by the details presented in his account of subatomic structures, and may fail to obtain a satisfactory grasp of the pattern of analysis. Perhaps no one can do a better job in this respect than Kline has done—certainly not without a much more leisurely paced explanation of the details than can be given in a relatively brief chapter.…
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