To Discover Mathematics
To Discover Mathematics
To relieve mathematics of the mystery surrounding it because its aims and methods are so seldom appreciated; to expose its utility, philosophy, and beauty; to accomplish these ends with elementary but significant material, available to all who, regardless of previous mathematical experience, are willing to cultivate the ability to reason; to present that material in interesting and stimulating style -- these are the purposes of this book.
The book, then, cannot be written in formal (and formidable) "textbook" style, and I have chosen to attempt an informal, sometimes chatty, exposition often verging on the narrative. At all times the reader is allowed to know the origin of the discussion and its goal. No phase of the subject is selected just because it is always in books on mathematics; but no important idea in elementary mathematics is dodged just because it is traditionally held to be too difficult for the uninitiated to understand. Tradition in mathematical presentation often erred, I am afraid, on the side of "compartmentalizing" material and grading it down to the presumed inability of the reader, or of adhering to rigid requirements of technical equipment which is many times merely a rote manipulation not fully understood. Calculus was not to be "taken" without full working knowledge of algebra, trigonometry, and analytic geometry, and these must be mastered in that order although the idea of calculus can be understood without any of the three. The price of prolonged apprenticeship is too severe, and to appreciate the aims of mathematics it is not necessary to pay the price. I have found many a student who has truly felt the nature and importance of non-Euclidean geometry but who had loathed his preparatory school memory course in Euclidean geometry; and many a student who was easily sunk in the progress of a boat in still water or up or down stream, but who delighted in notions about number, group, or limit.
Clearly, the aim of our exploration cannot be the cultivation of mathematics as a tool. We cannot insist on technical ability to begin with nor as an essential result of even a true comprehension of the essays. Emphasis must be on ideas rather than on technique. Yet, to be sure, the latter is in some degree necessary to the presentation of the former. Mathematics cannot be written without writing mathematics. The book properly correlates with those so-called "survey" courses now . . .