Mathematics: A Cultural Approach

Mathematics: A Cultural Approach

Mathematics: A Cultural Approach

Mathematics: A Cultural Approach

Excerpt

This text attempts to show what mathematics is, how mathematics has developed from man's efforts to understand and master nature, what the mathematical approach to real problems can accomplish, and the extent to which mathematics has molded our civilization and our culture. Though the essence of the material presented is mathematics, through motivations, applications, and implications the subject is shown to be intimately related to physical science, philosophy, logic, religion, literature, the social sciences, music, painting, and other arts.

The book is primarily intended for one-year terminal courses addressed to liberal arts students. It should serve in the training of teachers of elementary mathematics and of secondary-school teachers in nonmathematical subjects. It may also be helpful in giving high-school students some perspective on mathematics and in illustrating the subject's relevance to other fields. Indeed even the prospective specialist would do well to learn something about his subject before deciding to devote his life to it.

The content of this book ranges over elementary mathematics with the emphasis on ideas. No techniques are taught for the sake of techniques because technique per se is worthless knowledge and is not utilized later in life by the nonprofessional. The proofs of trigonometric identities are not important, but the fact that trigonometry has given man his understanding of the heavens is significant. Let us cease teaching scales to students who do not intend to play mathematical sonatas. In general, mathematical literacy, by which I mean understanding, is worth far more than technical proficiency. Hence there are very few exercises calling for drill, and the selection of exercises is, in general, based on the principle that a few, well chosen and thoroughly explored, are worth more than a thousandfold hastily done and poorly understood. To avoid mechanical imitation of stock procedures, any examples which may serve as a guide to the exercises are made an integral part of the text in the hope that students will be encouraged to read it, a practice by no means common among students of mathematics.

The material can be handled in various ways without loss of continuity. The first two chapters are straightforward reading matter, which, if desired, can be left to the student. The technical material can be started at several places. For students who have had recent training in arithmetic, algebra, and plane geometry, a course could begin with Chapter 7 or the latter part of Chapter 6. The chapters on philosophy, religion, literature, and the deductive approach to the social sciences can be left for student reading if time does not permit their inclusion or if the teacher believes that the material is best treated as reading assignments. Many of the mathematical chapters, such as the ones on projective geometry, non-Euclidean geometry, arithmetics and their algebras, and the chapters on trigonometric functions are logically independent. The material in the two chapters on the calculus is not used in the subsequent text.

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