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The great evolution of the physical, engineering, and social sciences during the past half century has cast mathematics in a role quite different from its familiar one of a powerful but essentially passive instrument for computing answers. In fact that view of mathematics was never a correct one, but it has had a strong influence in determining the standard undergraduate mathematics curriculum. Its inadequacy is becoming increasingly apparent with the growing recognition that mathematics is at the very heart of many modern scientific theories—not merely as a calculating device, but much more fundamentally as the sole language in which the theories can be expressed. Thus mathematics plays an organic and creative part in science, as a limitless source of concepts which provide fruitful new ways of representing natural phenomena.
The view of mathematics as a calculating device, and the traditional pre-eminence of analytic geometry and calculus in the undergraduate mathematics curriculum, can be traced in part to the dominant influence of classical physics, especially mechanics, and to the almost ineradicable prejudice in favor of expressing the laws of Nature in the form of simple mechanical analogies. This billiard-ball conception of Nature still persists, but its limitations have been known for a long time modern science cannot confine itself to that naive conception, and that part of mathematics with which it is linked—though utterly indispensable—is nonetheless inadequate for science. An ability to deal fluently with abstract systems has become a necessity.
Yet algebra, the mathematics of abstract systems par excellence, has been commonly neglected in undergraduate curricula indeed it is often omitted almost entirely from the mathematical education of science students, greatly to the detriment of their understanding of mathematics. The aim of this book is to acquaint students in the physical, engineering, and social sciences with the most important algebraic structures and with the mathematician’s way of discussing them. The book contains material which is not usually given before the junior or senior year of college, and much of the subject matter covered here is not generally presented to science students at all. It may therefore seem surprising that the book was designed to enable the student to begin his study of algebra at a very early stage of his undergraduate career. A preliminary mimeographed version was used with gratifying success in a course given by the authors at Johns Hopkins University in 1960-1961. Among those who took the first half of the course were freshmen who contemplated more than one term of mathematics in college. (Entry of freshmen into the second half of the course was restricted to those in the top third of their class who studied linear algebra t simultaneously with a course in calculus.) Although the material was of a level usually considered rather advanced, it was our experience that it was understood and learned as readily by the freshmen as by those students who had taken the standard analytic geometry and calculus courses. The explanation for this appears to be that the lesser mathematical experience of the freshmen was more than compensated by their freedom from stubborn misconceptions and by their stimulation upon encountering something that was not just a prolongation of high school. Before the first term was over, we were able to communicate with them in the precise and lucid language of mathematics