Number: The Language of Science, The Masterpiece Science Edition [Hardcover]

Tobias Dantzig (Author), Joseph Mazur (Author), Barry Mazur (Author)

Book Description

Publication Date: March 10, 2005 | ISBN-10: 0131856278 | ISBN-13: 978-0131856271 | Edition: 2nd

Number is an eloquent, accessible tour de force that reveals how the concept of number evolved from prehistoric times through the twentieth century. Tobias Dantzig shows that the development of math—from the invention of counting to the discovery of infinity—is a profoundly human story that progressed by “trying and erring, by groping and stumbling.” He shows how commerce, war, and religion led to advances in math, and he recounts the stories of individuals whose breakthroughs expanded the concept of number and created the mathematics that we know today.

--This text refers to the Paperback edition.

Editorial Reviews

Review

Anyone interested in the history of numbers and mathematics should read this book. (Mario Livio, author of The Golden Ratio)

A classic . . . it deserves a place on the bookshelf of anyone interested in the history of thought. (Charles Seife, author of Zero and Decoding the Universe)

Beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. (Albert Einstein)

--This text refers to the Paperback edition.

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.

See all Editorial Reviews

Product Details

• Hardcover: 416 pages
• Publisher: Pi Press; 2nd edition (March 10, 2005)
• Language: English
• ISBN-10: 0131856278
• ISBN-13: 978-0131856271
• Product Dimensions: 8.2 x 5.8 x 1.6 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 4.6 out of 5 stars  See all reviews (16 customer reviews)
• Amazon Best Sellers Rank: #471,242 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

21 of 21 people found the following review helpful:

5.0 out of 5 stars Masterpiece Almost Forgotten, June 17, 2000

By 

Ary (Brazil) - See all my reviews

This review is from: Number. the Language of Science (Paperback)

This is a book hardly read in our times of "modern math" (we are living in a museum of great innovations!) and that shows the theory of numbers as a human activity, stressing the fundamental role of the intuition in the construction of the mathematics. It seems to me that the gradual forgetfulness of this kind of book is one of the important causes for the continuous decline in the number of interested (and interesting!) people in the field of mathematics. I recommend this reading. You'll find a lot of fun!

25 of 27 people found the following review helpful:

5.0 out of 5 stars Postmodern mathematics?, July 10, 2005

By 

Dennis Littrell (SoCal) - See all my reviews
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This review is from: Number: The Language of Science, The Masterpiece Science Edition (Hardcover)

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)

9 of 9 people found the following review helpful:

5.0 out of 5 stars Review of the 4th revised edition (not the new 2007 edition), August 26, 2007

By 

Richard Frost (San Diego, CA) - See all my reviews
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This review is from: Number. the Language of Science (Paperback)

I am a mathematics teacher and have used this book as either a required reading or suggested supplement for a variety of courses, including math history for liberal arts students, number theory for mathematics majors, etc.

The book (4th edition) is divided into Part I and Part II -- the latter comprising only the last 4th of the book. Any successful college student will find Part I informative, and at times wonderfully enlightening about the development of the concepts of number and measurement. This book was written for the armchair reader, so expect a reader-friendly style of writing. However, I have found that Part II can be quite challenging for liberal arts students -- and quite stimulating to those whose studies included a more rigorous tour of mathematics. Do not let this bother you! I think Part I is worth the price of the book on its own.

If you wish to learn more about the history of mathematics and mathematicians, you might wish to examine Notable Mathematicians: From Ancient Times to the Present edited by Robyn V. Young and Zoran Minderovic.