Mathematics: The Loss of Certainty (Galaxy Books) [Paperback]

Morris Kline (Author)

Book Description

Publication Date: June 17, 1982 | Series: Galaxy Books (Book 686)

This work stresses the illogical manner in which mathematics has developed, the question of applied mathematics as against 'pure' mathematics, and the challenges to the consistency of mathematics' logical structure that have occurred in the twentieth century.

Editorial Reviews

About the Author

Morris Kline is Professor Emeritus at the Courant Institute of Mathematical Sciences, New York University.

Product Details

• Paperback: 384 pages
• Publisher: Oxford University Press, USA (June 17, 1982)
• Language: English
• ISBN-10: 0195030850
• ISBN-13: 978-0195030853
• Product Dimensions: 8.1 x 5.3 x 0.7 inches
• Shipping Weight: 7.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (20 customer reviews)
• Amazon Best Sellers Rank: #454,455 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

41 of 41 people found the following review helpful:

5.0 out of 5 stars Critical review of history and foundations of mathematics, August 31, 2000

By 

Juergen Kahrs (Bremen, Germany) - See all my reviews
(REAL NAME)   

This review is from: Mathematics: The Loss of Certainty (Galaxy Books) (Paperback)

This is one of the few affordable books about the history of mathematics. Others are the books by Howard Eves (ISBN 0-486-69609-X), by Courant and Robbins (ISBN 0-19-510519-2), and by Dunham (ISBN 0-14-014739-X).

Kline starts his review with the old Greeks, goes on with medieval mathematics and emphasizes the influential movement of rigorization in the 19th century. Unlike most authors, he does not stop his review in the early 20th century with Hilbert, Russel and Brouwer. Kline goes much further and explains the importance of Gödel, Skolem, Bourbaki and even Cohen, whose name I had not heard before.

It is one of the unique features of Kline's style how he manages to develop the sequence of ideas and approaches while constantly telling anecdotes. Some people even think that this book is just a collection of anecdotes and funny stories about mathematicians. But don't be misled by such comments. Although Kline illustrates his arguments so vividly, he is always on track. He often starts a paragraph by explaining something in detail, followed by a more intuitive point of view and finally tells an anecdote about exactly the point to be made.

In contrast to most other books about the history of mathematics, this book does not try to please the reader by telling him what a perfect body of knowledge mathematics is. Kline is really serious about the title "Mathematics - The Loss of Certainty". Throughout the whole book Kline explains the relation between mathematics and the other sciences (mostly astronomy and physics). While mathematics strived to reveal some truth about nature when it was young, it is today an isolated and fragmented discipline. Kline leaves no doubt that he dislikes the current situation of mathematicians ignoring the other sciences and playing with arbitrary formalisms.

Comparison of this book with Eves' reveals interesting details. Eves seems to like geometry much more than algebra and therefore talks much more about Euclid and Hilbert than about Gauss, Hamilton, and Gödel.

Compared to the book by Courant and Robbins, this book is completely different. Kline presents the background knowledge about the history of ideas while Courant and Robbins present remarkable theorems, methods and tasks. Maybe you understand the latter much better after reading the former.

The cover of the book is annoying. There is a pile of digits, accompanied by this sentence:

"A thinker who understands numbers better than anyone since Euclid delivers a ringing indictment of modern mathematics. Omni"

What a stupid comment. It misses the point completely. This book is not primarily about numbers and Euclid did most of his work on geometry, not numbers.

The book ends with a well chosen bibliography and a very reliable index. All in all one of my favourite books on the history of mathematics.

35 of 36 people found the following review helpful:

5.0 out of 5 stars A great book on the nature of mathematics!, January 5, 2001

By 

Sando Anoff (Holmdel, NJ United States) - See all my reviews

This review is from: Mathematics: The Loss of Certainty (Galaxy Books) (Paperback)

I wouldn't normally write a review of any book, but this book is really good (read the other reviews if you don't believe me), and I felt I had to write something. I highly recommend it for anyone who has ever wondered about the nature of mathematics.

I have always been fascinated by mathematics, but doubts started creeping into my mind about it when I was taught about the calculus, and all of a sudden, I began to question whether this was reality I was being taught, or just some convenient invention. After all, zero divided by zero doesn't make sense, and the idea of the "ultimate limit" seemed to be a trick, or dangerously close the Infinite, which is isn't much easier to swallow either.... Many years of engineering didn't make me feel any more comfortable, although clearly, it worked!

On reading this book, to my surprise (and somewhat to my consolation), I realized that even the great Newton and Leibniz did not justify their thoughts on this in a totally logical way, even though they helped to invent it.

Which makes you wonder...why does the physical world seem to follow mathematical patterns (or does it really...)? And did the thinkers justify their "laws" of mathematics and establish them beyond any doubt? Did "constructive intuition", whatever that might be, play the most important role in the creation of mathematics?

You may not get all the answers to these questions in this book (you won't get it in any other book this side of the universe), but you will certainly get a very thorough, deep and entertaining discussion these and many other questions you may not even have thought of. It is almost like being in a room with all these historical figures and listening to them arguing it out!

Best part is, the book is quite cheap! You'll like it!

25 of 26 people found the following review helpful:

5.0 out of 5 stars One of the most valuable books on the history of mathematics, August 19, 2000

By 

Mark Poyser (Los Angeles) - See all my reviews

This review is from: Mathematics: The Loss of Certainty (Galaxy Books) (Paperback)

I have read this book about twenty times. Besides being an entertaining review of the development of mathematics, it also touches on perhaps the most sensitive topic of all: Is mathematics describing something Real?

Klein establishes that for most of its history, mathematics was developed without a serious examination of foundational issues. Not only that, but things were invented when needed (infinitesimals, sqrt of -1), unexpected crises popped up (non-euclidean geometry), special pleading was invoked (theory of types in Principia), and wild and woolly ideas appeared (Cantor). One is forced to painfully conclude that as much as we would like mathematics to be Real in some way, in the end it is just a highly rigorous language with a mild empirical foundation. It has great powers of application - but only 'when applicable' [!]

Probably the most entertaining portion of the book is when the three schools (Logical, Intuitionist, Formalists) get into a tussle at the beginning of the 20th century. It reads like a theological debate - which it probably was. When extremely intelligent people (Russell, Browder, Hilbert) disagree, you know something has gone wrong at a deep level of understanding. Klein celebrates Godel's theorems as a triumph for the 'loss of certainty' - a view this reader does not share (the mapping of arithmetic to meta seems invalid) - but other than that, the author has done an excellent job of showing how the efficacy of mathematics have blinded many from its shaky foundations.

At the end of the book you will have an appreciation for mathematics as a useful tool, for the difficulties surmounted in its development, but also for the fragility of its claim to represent Truth. Anyone who has majored in mathematics at college and mastered it - though with a nagging feeling that they were only manipulating symbols on paper - will enjoy Klein's work.