Einstein called this the most interesting book on the evolution of mathematics he had ever read.

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Product Details

• Amazon Sales Rank: #869843 in Books
• Published on: 2005-03-10
• Original language: English
• Number of items: 1
• Binding: Hardcover
• 416 pages

From the Back Cover

"It is the aim of this book to...present the evolution of number as the profoundly human story which it is."

—Tobias Dantzig

"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."

—Albert Einstein

"Tobias Dantzig's Number: The Language of Science is one of the truly great classics of mathematical exposition, perhaps the most lucid history of the number concept ever written. Its republication should be a cause for celebration by every scientifically minded person, regardless of his or her mathematical background."

—Eli Maor, author of e: The Story of a Number and To Infinity and Beyond

"Tobias Dantzig's Number is a classic. A fascinating account of the evolution of mathematics, it deserves a place on the bookshelf of anyone who is interested in the history of thought."

—Charles Seife, author of Zero and Alpha and Omega

"A classic! Anyone interested in the history of numbers and mathematics should read this book."

—Mario Livio, author of The Golden Ratio

From the rudimentary mathematical abilities of prehistoric man to the counterintuitive and bizarre ideas at the edges of modern math, this masterpiece of science writing tells the story of mathematics through the history of its most central concept: number.

Dantzig succeeds in his aim to reveal a human story, and in making that story accessible to the non-expert. In his friendly and welcoming style, he shows how math developed from basic faculties present in us all, beginning with our "number sense"—the ability to discern that an object has been added to or removed from a small collection of objects without counting. The subsequent evolution of the concept of number is inextricably linked with the history of human culture, as Dantzig demonstrates. He shows how advances in math were spurred by the demands of growing commerce in the ancient world; how the pure speculation of philosophers and religious mystics contributed to our understanding of numbers; how the exchange of ideas between cultures in times of war and imperial conquest fueled advances in knowledge; and, ultimately, how the forces of history combine with human intuition to trigger revolutions in thought.

Sweeping in scope, Number is an open doorway into the world of math. Dantzig explains the foundations of mathematics with ease, and eloquently explores deeper philosophical questions that arise along the way. He describes the properties of all kinds of numbers—integers, primes, irrationals, transcendentals, and more. He explains the significance of zero, and shows that its invention had revolutionary consequences for arithmetic. He shows how the invention of symbols for use in algebra—a radical departure from tradition at the time—ushered in a new era of math; how arithmetic and geometry reflect each other; and how calculus uses infinity to model the continuity of space and time.

With a new afterword, notes section, and bibliography written by math professor and author Joseph Mazur, and a new foreword by mathematician Barry Mazur, the Masterpiece Science edition of Number—which was first published in 1930—is the first update of Dantzig's classic work in over fifty years. It is a story that ranges from the dawn of man to the genius of history's greatest mathematicians, vividly revealing how the pursuit of knowledge transcends the rise and fall of civilizations.

© Copyright Pearson Education. All rights reserved.

About the Author

Tobias Dantzig was born in Latvia in 1884. As a young man, he was caught distributing anti-Tsar propaganda and fled to Paris, where he studied under Henri Poincaré. He moved to the United States in 1910 and took a job as a lumberjack in the forests of Oregon. He received his Ph.D. in mathematics from Indiana University in 1916, and taught at Johns Hopkins, Columbia University, and the University of Maryland. He died in 1956.

Joseph Mazur is Professor of Mathematics at Marlboro College where he has taught a wide range of classes in all areas of mathematics, its history and philosophy. He is the author of Euclid in the Rainforest: Discovering Universal Truth in Logic and Math.

Barry Mazur is the Gerhard Gade University Professor at Harvard University. He teaches and does research in mathematics. He is the author of Imagining Numbers (especially the square root of minus fifteen).

© Copyright Pearson Education. All rights reserved.

Excerpt. © Reprinted by permission. All rights reserved.

A 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 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

© Copyright Pearson Education. All rights reserved.

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Customer Reviews

Postmodern mathematics?
Einstein called this "the most interesting book on the evolution of mathematics which has ever fallen into my hands."

Number was first published in 1930 with the fourth edition coming out in 1954. This is a republication of that fourth edition (Dantzig died in 1956) edited by Joseph Mazur with a foreword by Barry Mazur. It is an eminently readable book like something from the pages of that fascinating four-volume work The World of Mathematics (1956) edited by James R. Newman in that it is aimed at mathematicians and the educated lay public alike.

Part history, part mathematics and part philosophy, Number is the story of how we humans got from "one, two...many" to various levels of infinity. Strange to say it is also about reality. Here is Dantzig's concluding statement from page 341 in Appendix D: "...modern science differs from its classical predecessor: it has recognized the anthropomorphic origin and nature of human knowledge. Be it determinism or rationality, empiricism or the mathematical method, it has recognized that man is the measure of all things, and that there is no other measure."

Or more pointedly from a couple of pages earlier: "Man's confident belief in the absolute validity of the two methods [mathematics and experiment] has been found to be of an anthropomorphic origin; both have been found to rest on articles of faith."

These are inescapably the statements of a postmodernist. I was surprised to read them in a book on the theory of numbers, and even more surprised to realize that if mathematics is a distinctly human language, it is entirely possible that beings from distant worlds may speak an entirely different language; and therefore our attempts to use what many consider the "universal" language of mathematics to communicate with them may be in vain.

And this thought makes me wonder. Is the concept "two," for example, (as opposed to the number "2") really just a human construction? Would not intelligent life anywhere be able to make a distinction, just as we have, between, say, two things and three things? And if so, would they not be able to count? And would not then the entire edifice of mathematics (or at least most of it) follow?

I wonder if Dantzig was not in contradiction with himself on this point because earlier he writes (p. 252) "...any measuring device, however simple and natural it may appear to us, implies the whole apparatus of the arithmetic of real numbers: behind any scientific instrument there is the master-instrument, arithmetic, without which the special device can neither be used nor even conceived." Does this not imply that measurements (by any beings) and therefore numbers have an existence outside of the human mind and do not rest on "articles of faith"?

As to the numbers themselves (putting philosophy aside) we learn that the two biggest bugaboos in the history of number are zero and infinity. It took a long, long time for humans, as Dantzig relates, to accept the idea of zero as a number. Today zero is also a place-holder. But what does it mean to say that there are zero pink elephants dancing about my living room? I can see one cow in the yard, or two or three, but I cannot see zero cows in the yard.

Of course, today it is easy to see that zero is a number that is less than one and greater than minus one. I have one cow and I sell that one cow. Now I have zero cows. (Curiously, note that the plural noun "cows" is grammatically required.) However, the imperfect fit within the entire structure of mathematics that zero has achieved may be appreciated by realizing that every other number can be a denominator; that is, three over one equals three, three over two equals 1.5, etc., but what does three over zero equal?

It is a convention of mathematics to say that division by zero is "undefined." There is no other number about which the same can be said.

I used to think when I was young that infinity was the proper answer to division by zero. For Dantzig this is clearly not correct because to him infinity is not a number at all but a part of the process. He writes, "the concept of infinity has been woven into the very fabric of our generalized number concept." He adds, "The domain of natural numbers rested on the assumption that the operation of adding one can be repeated indefinitely, and it was expressly stipulated that never shall the ultra-ultimate step of this process be itself regarded as a number." Of course he is talking about "natural" numbers. He notes in the next sentence that in the generalization to "real" numbers, "the limits of these processes" were "admitted...as bona fide numbers." (p. 245) In other words, part of the process became a number itself!

The culmination of Dantzig's argument here is that infinity itself is a construction of the human mind and exists nowhere (that we can prove) outside of the human mind. He believes that the basis for our belief in the existence of infinity comes from our (erroneous) conception of time as a continuum. Dantzig notes that Planck time and indeed all aspects of the world are to be seen in terms of discrete quanta and not continuous streams.

Ultimately, Dantzig gives this sweeping advice to the scientist: "...he will be wise to wonder what role his mind has played in...[a] 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." (p. 242)

A Human Story
The striking facts about Danzig's book are :

1. It does not claim to be a 'popular' science book. At the outset, he warns the reader ".. it is not written for those who are afflicted with an incurable horror of the symbol". In doing so, I think he has gained more readership, simply because noone likes to be patronised, and most 'popular' science books are extremely patronising.

2. He makes it a point to explain to the reader that mathematics is not something that was made by the Hand of God. He clearly explains the mistakes made by some of the most eminent mathematicians, and thus brings out the 'human' element in the evolution of mathematics very beautifully.

3. He interweaves his philosophy with that of the history of math, and thus makes it eminently readable.

So deserves its "classic" status
NLS is, in a word, masterful. It is a fascinating and penetrating introduction to the "language of science". After laying a foundation of first principles (what is a number, what does it mean to count), Dantzig goes on to construct a veritable cathedral of mathematics. As the reader climbs ever higher, the mathematics Dantzig describes grows increasingly abstruse, but the exposition remains lucid and compelling throughout. Dantzig is a terrific guide -- an exceedingly good writer and a very deep thinker. Many of the concepts developed in NLS are treated ad nauseum in the popular mathematics literature, but nowhere as clearly. After reading the opening pages of NLS, I was impressed with the writing (very literary in style) but was skeptical that I would learn much from this slim volume. How very wrong I was. Time and again Dantzig clarified concepts and connections that have long eluded my full grasp. NLS is a superb book, and a fascinating read. This new edition is a useful improvement on its predecessors -- with excellent endnotes and bibliography, and a well-considered division of the book into text and appendices.