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Number: The Masterpiece Science Edition: The Language of Science Hardcover – Feb 2005

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Product details

  • Hardcover: 400 pages
  • Publisher: Prentice Hall; 1st New edition edition (Feb. 2005)
  • Language: English
  • ISBN-10: 0131856278
  • ISBN-13: 978-0131856271
  • Product Dimensions: 14.2 x 3.5 x 21.2 cm
  • Amazon Bestsellers Rank: 2,026,456 in Books (See Top 100 in Books)

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From the rudimentary mathematical abilities of prehistoric man to bizarre ideas at the edges of modern math, here is the story of mathematics through the history of its most central concept: number. Dantzig demonstrates that the evolution of numbers is inextricably linked with the history of human culture. 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 how the forces of history combine with human intuition to trigger revolutions in thought. Dantzig's exposition of the foundations and philosophy of math is accessible to all readers. He explores many of the most fascinating topics in math, such as the properties of numbers, the invention of zero, and infinity. First published in 1930, this book is, beyond doubt, the best book on the evolution of mathematics - now again in print.

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Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Amazon.com: 26 reviews
23 of 23 people found the following review helpful
Masterpiece Almost Forgotten 18 Jun. 2000
By Ary - Published on Amazon.com
Format: 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!
30 of 33 people found the following review helpful
Postmodern mathematics? 11 July 2005
By Dennis Littrell - Published on Amazon.com
Format: 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)
12 of 12 people found the following review helpful
A Human Story 1 April 2006
By Prahlad Vaidyanathan - Published on Amazon.com
Format: Hardcover
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.
9 of 9 people found the following review helpful
Review of the 4th revised edition (not the new 2007 edition) 26 Aug. 2007
By Richard Frost - Published on Amazon.com
Format: 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.
13 of 15 people found the following review helpful
An inferior edition 7 Dec. 2007
By Stewart A. Levin - Published on Amazon.com
Format: Paperback
This is a reprint of the author's 1954 fourth edition sandwiched between a new Foreword and Afterword. Neither the editor (Joseph Mazur) nor his brother (Barry Mazur, who wrote the Foreword) nor either of the advertised reviewers (Mario Livio or Charles Seife) apparently actually proofread the text as there are a distressing number of readily apparent typographic errors in the printing, both in the text and figures.
For a volume trumpeted on its title page as "The MASTERPIECE SCIENCE Edition" the many errors belie that mantle. In addition, the Afterword, which attempts to bring the reader up to date on relevant mathematical developments that occured after the fourth edition, fails to mention "undecidability" and the immense impact it has had on the issues discussed in the chapter entitled "The Anatomy of the Infinite."

Dantzig's Number continues to be accessible and generally insightful, but it is a shame that no one at Plume Books took due care and responsibility for its production.
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