Tài liệu Number the language of science book tobias dantzig

Tobias Dantzig NUMBER The Language of Science Edited by Joseph Mazur Foreword by Barry Mazur The Masterpiece Science Edition Pi Press New York PI PRESS An imprint of Pearson Education, Inc. 1185 Avenue of the Americas, New York, New York 10036 Foreword, Notes, Afterword and Further Readings © 2005 by Pearson Education, Inc.© 1930, 1933, 1939, and 1954 by the Macmillan Company This edition is a republication of the 4th edition of Number, originally published by Scribner, an Imprint of Simon & Schuster Inc. Pi Press offers discounts for bulk purchases. For more information, please contact U.S. Corporate and Government Sales, 1-800-382-3419, [email protected]. For sales outside the U.S., please contact International Sales at [email protected]. Company and product names mentioned herein are the trademarks or registered trademarks of their respective owners. Printed in the United States of America First Printing: March, 2005 Library of Congress Number: 2004113654 Pi Press books are listed at www.pipress.net. ISBN 0-13-185627-8 Pearson Education LTD. Pearson Education Australia PTY, Limited. Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia, Ltd. Pearson Education Canada, Ltd. Pearson Educatión de Mexico, S.A. de C.V. Pearson Education—Japan Pearson Education Malaysia, Pte. Ltd. Contents Foreword vii Editor's Note xiv Preface to the Fourth Edition xv Preface to the First Edition 1. Fingerprints xvii 1 2. The Empty Column 19 3. Number-lore 37 4. The Last Number 59 5. 79 Symbols 6. The Unutterable 103 7. This Flowing World 125 8. The Art of Becoming 145 9. Filling the Gaps 171 10. The Domain of Number 187 11. The Anatomy of the Infinite 215 12. 239 The Two Realities Contents vi Appendix A. On the Recording of Numbers 261 Appendix B. Topics in Integers 277 Appendix C. On Roots and Radicals 303 Appendix D. On Principles and Arguments 327 Afterword 343 Notes 351 Further Readings 373 Index 385 Foreword T he book you hold in your hands is a many-stranded meditation on Number, and is an ode to the beauties of mathematics. This classic is about the evolution of the Number concept. Yes: Number has had, and will continue to have, an evolution. How did Number begin? We can only speculate. Did Number make its initial entry into language as an adjective? Three cows, three days, three miles. Imagine the exhilaration you would feel if you were the first human to be struck with the startling thought that a unifying thread binds “three cows” to “three days,” and that it may be worthwhile to deal with their common three-ness. This, if it ever occurred to a single person at a single time, would have been a monumental leap forward, for the disembodied concept of three-ness, the noun three, embraces far more than cows or days. It would also have set the stage for the comparison to be made between, say, one day and three days, thinking of the latter duration as triple the former, ushering in yet another view of three, in its role in the activity of tripling; three embodied, if you wish, in the verb to triple. Or perhaps Number emerged from some other route: a form of incantation, for example, as in the children’s rhyme “One, two, buckle my shoe….” However it began, this story is still going on, and Number, humble Number, is showing itself ever more central to our understanding of what is. The early Pythagoreans must be dancing in their caves. viii NUMBER If I were someone who had a yen to learn about math, but never had the time to do so, and if I found myself marooned on that proverbial “desert island,” the one book I would hope to have along is, to be honest, a good swimming manual. But the second book might very well be this one. For Dantzig accomplishes these essential tasks of scientific exposition: to assume his readers have no more than a general educated background; to give a clear and vivid account of material most essential to the story being told; to tell an important story; and—the task most rarely achieved of all— to explain ideas and not merely allude to them. One of the beautiful strands in the story of Number is the manner in which the concept changed as mathematicians expanded the republic of numbers: from the counting numbers 1, 2, 3,… to the realm that includes negative numbers, and zero … –3, –2, –1, 0, +1, +2, +3, … and then to fractions, real numbers, complex numbers, and, via a different mode of colonization, to infinity and the hierarchy of infinities. Dantzig brings out the motivation for each of these augmentations: There is indeed a unity that ties these separate steps into a single narrative. In the midst of his discussion of the expansion of the number concept, Dantzig quotes Louis XIV. When asked what the guiding principle was of his international policy, Louis XIV answered, “Annexation! One can always find a clever lawyer to vindicate the act.” But Dantzig himself does not relegate anything to legal counsel. He offers intimate glimpses of mathematical birth pangs, while constantly focusing on the vital question that hovers over this story: What does it mean for a mathematical object to exist? Dantzig, in his comment about the emergence of complex numbers muses that “For centuries [the concept of complex numbers] figured as a sort of mystic bond between reason and imagination.” He quotes Leibniz to convey this turmoil of the intellect: Foreword ix “[T]he Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.” (212) Dantzig also tells us of his own early moments of perplexity: “I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.” (254) The interplay between algebra and geometry is one of the grand themes of mathematics. The magic of high school analytic geometry that allows you to describe geometrically intriguing curves by simple algebraic formulas and tease out hidden properties of geometry by solving simple equations has flowered—in modern mathematics—into a powerful intermingling of algebraic and geometric intuitions, each fortifying the other. René Descartes proclaimed: “I would borrow the best of geometry and of algebra and correct all the faults of the one by the other.” The contemporary mathematician Sir Michael Atiyah, in comparing the glories of geometric intuition with the extraordinary efficacy of algebraic methods, wrote recently: x NUMBER “Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. (Atiyah, Sir Michael. Special Article: Mathematics in the 20th Century. Page 7. Bulletin of the London Mathematical Society, 34 (2002) 1–15.)” It takes Dantzig’s delicacy to tell of the millennia-long courtship between arithmetic and geometry without smoothing out the Faustian edges of this love story. In Euclid’s Elements of Geometry, we encounter Euclid’s definition of a line: “Definition 2. A line is breadthless length.” Nowadays, we have other perspectives on that staple of plane geometry, the straight line. We have the number line, represented as a horizontal straight line extended infinitely in both directions on which all numbers—positive, negative, whole, fractional, or irrational—have their position. Also, to picture time variation, we call upon that crude model, the timeline, again represented as a horizontal straight line extended infinitely in both directions, to stand for the profound, ever-baffling, ever-moving frame of past/present/futures that we think we live in. The story of how these different conceptions of straight line negotiate with each other is yet another strand of Dantzig’s tale. Dantzig truly comes into his own in his discussion of the relationship between time and mathematics. He contrasts Cantor’s theory, where infinite processes abound, a theory that he maintains is “frankly dynamic,” with the theory of Dedekind, which he refers to as “static.” Nowhere in Dedekind’s definition of real number, says Dantzig, does Dedekind even “use the word infinite explicitly, or such words as tend, grow, beyond measure, converge, limit, less than any assignable quantity, or other substitutes.” Foreword xi At this point, reading Dantzig’s account, we seem to have come to a resting place, for Dantzig writes: “So it seems at first glance that here [in Dedekind’s formulation of real numbers] we have finally achieved a complete emancipation of the number concept from the yoke of time.” (182) To be sure, this “complete emancipation” hardly holds up to Dantzig’s second glance, and the eternal issues regarding time and its mathematical representation, regarding the continuum and its relationship to physical time, or to our lived time—problems we have been made aware of since Zeno—remain constant companions to the account of the evolution of number you will read in this book. Dantzig asks: To what extent does the world, the scientific world, enter crucially as an influence on the mathematical world, and vice versa? “The man of science will acts as if this world were an absolute whole controlled by laws independent of his own thoughts or act; but whenever he discovers a law of striking simplicity or one of sweeping universality or one which points to a perfect harmony in the cosmos, he will be wise to wonder what role his mind has played in the discovery, and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity, or is but a reflection of his own mind.” (242) Dantzig writes: “The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and of delight!” (240) xii NUMBER This bears some resemblance in tone to the famous essay of the physicist Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” but Dantzig goes on, by offering us his highly personal notions of subjective reality and objective reality. Objective reality, according to Dantzig, is an impressively large receptacle including all the data that humanity has acquired (e.g., through the application of scientific instruments). He adopts Poincaré’s definition of objective reality, “what is common to many thinking beings and could be common to all,” to set the stage for his analysis of the relationship between Number and objective truth. Now, in at least one of Immanuel Kant’s reconfigurations of those two mighty words subject and object, a dominating role is played by Kant’s delicate concept of the sensus communis. This sensus communis is an inner “general voice,” somehow constructed within each of us, that gives us our expectations of how the rest of humanity will judge things. The objective reality of Poincaré and Dantzig seems to require, similarly, a kind of inner voice, a faculty residing in us, telling us something about the rest of humanity: The Poincaré-Dantzig objective reality is a fundamentally subjective consensus of what is commonly held, or what could be held, to be objective. This view already alerts us to an underlying circularity lurking behind many discussions regarding objectivity and number, and, in particular behind the sentiments of the essay of Wigner. Dantzig treads around this lightly. My brother Joe and I gave our father, Abe, a copy of Number: The Language of Science as a gift when he was in his early 70s. Abe had no mathematical education beyond high school, but retained an ardent love for the algebra he learned there. Once, when we were quite young, Abe imparted some of the marvels of algebra to us: “I’ll tell you a secret,” he began, in a conspiratorial voice. He proceeded to tell us how, by making use of the magic power of the cipher X, we could find that number which when you double it and Foreword xiii add one to it you get 11. I was quite a literal-minded kid and really thought of X as our family’s secret, until I was disabused of this attribution in some math class a few years later. Our gift of Dantzig’s book to Abe was an astounding hit. He worked through it, blackening the margins with notes, computations, exegeses; he read it over and over again. He engaged with numbers in the spirit of this book; he tested his own variants of the Goldbach Conjecture and called them his Goldbach Variations. He was, in a word, enraptured. But none of this is surprising, for Dantzig’s book captures both soul and intellect; it is one of the few great popular expository classics of mathematics truly accessible to everyone. —Barry Mazur Editor’s Note to the Masterpiece Science Edition T he text of this edition of Number is based on the fourth edition, which was published in 1954. A new foreword, afterword, endnotes section, and annotated bibliography are included in this edition, and the original illustrations have been redrawn. The fourth edition was divided into two parts. Part 1, “Evolution of the Number Concept,” comprised the 12 chapters that make up the text of this edition. Part 2, “Problems Old and New,” was more technical and dealt with specific concepts in depth. Both parts have been retained in this edition, only Part 2 is now set off from the text as appendixes, and the “part” label has been dropped from both sections. In Part 2, Dantzig’s writing became less descriptive and more symbolic, dealing less with ideas and more with methods, permitting him to present technical detail in a more concise form. Here, there seemed to be no need for endnotes or further commentaries. One might expect that a half-century of advancement in mathematics would force some changes to a section called “Problems Old and New,” but the title is misleading; the problems of this section are not old or new, but are a collection of classic ideas chosen by Dantzig to show how mathematics is done. In the previous editions of Number, sections were numbered within chapters. Because this numbering scheme served no function other than to indicate a break in thought from the previous paragraphs, the section numbers were deleted and replaced by a single line space. Preface to the Fourth Edition quarter of the century ago, when this book was first written, I had grounds to regard the work as a pioneering effort, inasmuch as the evolution of the number concept— though a subject of lively discussion among professional mathematicians, logicians and philosophers—had not yet been presented to the general public as a cultural issue. Indeed, it was by no means certain at the time that there were enough lay readers interested in such issues to justify the publication of the book. The reception accorded to the work both here and abroad, and the numerous books on the same general theme which have followed in its wake have dispelled these doubts. The existence of a sizable body of readers who are concerned with the cultural aspects of mathematics and of the sciences which lean on mathematics is today a matter of record. It is a stimulating experience for an author in the autumn of life to learn that the sustained demand for his first literary effort has warranted a new edition, and it was in this spirit that I approached the revision of the book. But as the work progressed, I became increasingly aware of the prodigious changes that have taken place since the last edition of the book appeared. The advances in technology, the spread of the statistical method, the advent of electronics, the emergence of nuclear physics, and, above all, the growing importance of automatic computors— have swelled beyond all expectation the ranks of people who live on the fringes of mathematical activity; and, at the same time, raised the general level of mathematical education. Thus was I A xvi NUMBER confronted not only with a vastly increased audience, but with a far more sophisticated and exacting audience than the one I had addressed twenty odd years earlier. These sobering reflections had a decisive influence on the plan of this new edition. As to the extent I was able to meet the challenge of these changing times—it is for the reader to judge. Except for a few passages which were brought up to date, the Evolution of the Number Concept, Part One of the present edition, is a verbatim reproduction of the original text. By contrast, Part Two—Problems, Old and New—is, for all intents and purposes, a new book. Furthermore, while Part One deals largely with concepts and ideas. Still, Part Two should not be construed as a commentary on the original text, but as an integrated story of the development of method and argument in the field of number. One could infer from this that the four chapters of Problems, Old and New are more technical in character than the original twelve, and such is indeed the case. On the other hand, quite a few topics of general interest were included among the subjects treated, and a reader skilled in the art of “skipping” could readily circumvent the more technical sections without straying off the main trail. Tobias Dantzig Pacific Palisades California September 1, 1953 Preface to the First Edition his book deals with ideas, not with methods. All irrelevant technicalities have been studiously avoided, and to understand the issues involved no other mathematical equipment is required than that offered in the average highschool curriculum. But though this book does not presuppose on the part of the reader a mathematical education, it presupposes something just as rare: a capacity for absorbing and appraising ideas. Furthermore, while this book avoids the technical aspects of the subject, it is not written for those who are afflicted with an incurable horror of the symbol, nor for those who are inherently form-blind. This is a book on mathematics: it deals with symbol and form and with the ideas which are back of the symbol or of the form. The author holds that our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a fine mind. It is the aim of this book to restore this cultural content and present the evolution of number as the profoundly human story which it is. This is not a book on the history of the subject. Yet the historical method has been freely used to bring out the rôle intuition has played in the evolution of mathematical concepts. And so the story of number is here unfolded as a historical pageant of ideas, linked with the men who created these ideas and with the epochs which produced the men. T xviii NUMBER Can the fundamental issues of the science of number be presented without bringing in the whole intricate apparatus of the science? This book is the author’s declaration of faith that it can be done. They who read shall judge! Tobias Dantzig Washington, D.C. May 3, 1930 CHAPTER 1 Fingerprints Ten cycles of the moon the Roman year comprised: This number then was held in high esteem, Because, perhaps, on fingers we are wont to count, Or that a woman in twice five months brings forth, Or else that numbers wax till ten they reach And then from one begin their rhythm anew. —Ovid, Fasti, III. M an, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense. This faculty permits him to recognize that something has changed in a small collection when, without his direct knowledge, an object has been removed from or added to the collection. Number sense should not be confused with counting, which is probably of a much later vintage, and involves, as we shall see, a rather intricate mental process. Counting, so far as we know, is an attribute exclusively human, whereas some brute species seem to possess a rudimentary number sense akin to our own. At least, such is the opinion of competent observers of animal behavior, and the theory is supported by a weighty mass of evidence. Many birds, for instance, possess such a number sense. If a nest contains four eggs one can safely be taken, but when two are removed the bird generally deserts. In some unaccountable way the bird can distinguish two from three. But this faculty is by no 1