The Mathematical Experience

PHILIP J. DAVIS & REUBEN HERSH

Preface

THE OLDEST MATHEMATICAL tablets we have date from 2400 B.C., but there is no reason to suppose that the urge to create and use mathematics is not coextensive with the whole of civilization. In four or five millennia a vast body of practices and concepts known as mathematics has emerged and has been linked in a variety of ways with our day-to-day life. What is the nature of mathematics? What is its meaning? What are its concerns? What is its methodology? How is it created? How is it used? How does it fit in with the varieties of human experience? What benefits flow from it? What harm? What importance can be ascribed to it?

These difficult questions are not made easier by the fact that the amount of material is so large and the amount of interlinking is so extensive that it is simply not possible for any one person to comprehend it all, let alone sum it up and compress the summary between the covers of an average-sized book. Lest we be cowed by this vast amount of material, let us think of mathematics in another way. Mathematics has been a human activity for thousands of years. To some small extent, everybody is a mathematician and does mathematics consciously. To buy at the market, to measure a strip of wall paper or to decorate a ceramic pot with a regular pattern is doing mathematics. Further, everybody is to some small extent a philosopher of mathematics. Let him only exclaim on occasion: "But figures can’t lie!" and he joins the ranks of Plato and of Lakatos.

In addition to the vast population that uses mathematics on a modest scale, there are a small number of people who are professional mathematicians. They practice mathematics, foster it, teach it, create it, and use it in a wide variety of situations. It should be possible to explain to nonprofessionals just what these people are doing, what they say they are doing, and why the rest of the world should support them at it. This, in brief, is the task we have set for ourselves. The book is not intended to present a systematic, self-contained discussion of a specific corpus of mathematical material, either recent or classical. It is intended rather to capture the inexhaustible variety presented by the mathematical experience. The major strands of our exposition will be the substance of mathematics, its history, its philosophy, and how mathematical knowledge is elicited. The book should be regarded not as a compression but rather as an impression. It is not a mathematics book; it is a book about mathematics. Inevitably it must contain some mathematics. Similarly, it is not a history or a philosophy book, but it will discuss mathematical history and philosophy. It follows that the reader must bring to it some slight prior knowledge of these things and a seed of interest to plant and water. The general reader with this background should have no difficulty in getting through the major portion of the book. But there are a number of places where we have brought in specialized material and directed our exposition to the professional who uses or produces mathematics. Here the reader may feel like a guest who has been invited to a family dinner. After polite general conversation, the family turns to narrow family concerns, its delights and its worries, and the guest is left up in the air, but fascinated. At such places the reader should judiciously and lightheartedly push on.

For the most part, the essays in this book can be read independently of each other.

Some comment is necessary about the use of the word "I" in a book written by two people. In some instances it will be obvious which of the authors wrote the "I." In any case, mistaken identity can lead to no great damage, for each author agrees, in a general way, with the opinions of his colleague.

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