Customer Reviews

Number. the Language of Science

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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!

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)

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.

I'm sending this book to my daughter, the doctor, who has expressed a desire to know something about the intellectual history of mathematics. I can't believe there is only one review! I have read this book three times; I may read it yet again before I die. Rather than list all its attributes, I suggest to the reader that s/he think of an attribute, and assume I gave I praise it to the limit of my ability!

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.

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.

I searched through many many many history of mathematics books and I finally found the ONE. Not just history, but primarily philosophy of mathematics. The brilliant thing about this book is that he tells the story around the most interesting about thing about mathematics: infinity. There are very VERY few people who can actually write a book that follows the natural wonder one goes through in their discovery. Dantzig can do this. Another of his books 'Aspects of Science' is also good. Just read the quote from Einstein on the cover.

Tobias Dantzig was the father of George Dantzig, the great operations Research scientist in 20th century.

Tobias was born in Russia, but went to France where he studied mathematics in Paris being taught there by Poincar¨¦. At this time Tobias met Anja who was at the Sorbonne at this time also studying mathematics. They married and emigrated to the United States, settling in Oregon. Tobias believed that his strong Russian accent would prevent him from obtaining jobs other than as a labourer, and at first his jobs included that of lumberjack, road builder and painter. It was into this very poor family that George was born.

Tobias and Anja chose names for their children hoping that these would influence their future careers. George was named "George Bernard" after George Bernard Shaw since his parents hoped their first child would become a writer. Similarly George's younger brother was named Henry after Henri Poincar¨¦, and he did indeed become a mathematician. Tobias was fortunate to gain the chance of reading for a Ph.D. in mathematics at the University of Indiana, while Anja obtained a Master's degree in French becoming a linguist at the Library of Congress in Washington D.C.

The family were now living in Washington D.C., and there George attended Powell Junior High School where his progress in mathematics was, at first, rather poor. Encouraged by his father, and determined to do well in mathematics and science, he soon began to obtain top marks in mathematics. This continued at Central High School where he became fascinated by geometry. By this time he was getting strong support from three people: an outstanding mathematics teacher at the High School, a school friend who would go on to become a professor of mathematics at Berkeley, and his father. George later wrote that his father:-

... gave me thousands of geometry problems while I was still in high school. ... the mental exercise required to solve them was the great gift from my father. The solving of thousands of problems during my high school days - at the time when my brain was growing - did more than anything else to develop my analytic power.

Tobias was working on his most famous work Number: the language of science in the late 1920s and George helped him. He later wrote:-

As a teenager, I prepared some of the figures that appeared in the book.

The book was published in 1930 and when it was reprinted in the 1970s a reviewer wrote:-

Since its first appearance nearly half a century ago the book has gone through a number of printings and has deservedly maintained its popularity.

Also, Albert Einstein is quoted saying: "This (Number the Language of Science) is beyond th most interesting book on the evolution of mathematics which has ever fallen into my hands"

Just finished reading the entire narrative of the book up to the appendicies. I learned so much about the connection of disparate branches of mathematics. Reading it again while attempting to work through some related problems is my next step.

Written by an author who died the year I was born, this book is still fresh, alive & compelling. I'd be interested if there is a sequel that incorporates new developements.

It was the Einstien quote that stimulated my impulse to buy the book.

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.