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Mathematics: The Loss of Certainty [Hardcover]

Morris Kline
3.0 out of 5 stars  See all reviews (3 customer reviews)

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Product details

  • Hardcover: 374 pages
  • Publisher: Oxford University Press Inc (Dec 1980)
  • Language: English
  • ISBN-10: 019502754X
  • ISBN-13: 978-0195027549
  • Product Dimensions: 3.2 x 15.9 x 23.5 cm
  • Average Customer Review: 3.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 816,879 in Books (See Top 100 in Books)

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Refuting the accepted belief that mathematics is exact and infallible, the author examines the development of conflicting concepts of mathematics and their implications for the physical, applied, social, and computer sciences.

About the Author

Morris Kline is Professor Emeritus at the Courant Institute of Mathematical Sciences, New York University. --This text refers to the Paperback edition.

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Any civilization worthy of the appelation has sought truths. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Customer Reviews

3.0 out of 5 stars
3.0 out of 5 stars
Most Helpful Customer Reviews
4 of 7 people found the following review helpful
This book, like all those written by Morris Kline, is well worth reading by anyone interested in mathematics since it deals in a non-technical manner with history of the rise of mathematics and what Kline sees as its present decline.
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2 of 4 people found the following review helpful
I read this book last year, this is simply because, I said it is written by one of the most outstanding mathematical historian of the last century, Mr. Kline, but the only and an ultimatum example of his lost certainty of the absolute truth of mathematics is the Euclids parallel postulate. I have to say that Mr. Kline has repeated the same assumption in his other brilliant book MAthematics in the Westren culture.
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6 of 11 people found the following review helpful
1.0 out of 5 stars Tries to prove a nonsensical point 26 Nov 2007
The writer of this book, the late Morris Kline, is an anti-pure maths polemicist. His main goal is to show that uncertainty reigns in mathematics. Let me denote by Mathematics the study of mathematical objects such as statements, integers, functions and so forth, and let me denote by mathematics the mathematical objects themselves. Morris Kline does not make this distinction clear so it is not always clear what he is talking about.

Anyway, to get to the point, it is nonsensical to talk of mathematical statements as certain or uncertain. Certainty is not a property of statements, it is a property of people, an emotion. The same goes for uncertainty. And one cannot dictate to someone their emotions. Therefore any talk of certainty of mathematics makes no sense. Also talk of the certainty of Mathematics makes no sense. There may be a statement in combinatorics of which a combinatorialist is certain, and a statement in commutative algebra of which an algebraist is certain, and they may be mutually uncertain of each other's statements due to lack of expertise. One can only talk of the certainty of an individual mathematician with respect to a particular statement or list of statements, and not of the certainty of 10,000 mathematicians all at once.

As such, the main point that Morris Kline is trying to prove, that Mathematics or mathematics is uncertain, makes absolutely no sense. Obviously, he fails to prove it, as one cannot prove nonsense. His basis for the claim he makes about uncertainty is Godel's theorem, which more or less says that there is no general algorithm for determining whether a given arithmetical statement is true or not. Equivalently, for any consistent list of axioms there is a true arithmetical statement which is true but cannot be proven using only those axioms.
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Most Helpful Customer Reviews on (beta) 4.5 out of 5 stars  29 reviews
52 of 54 people found the following review helpful
5.0 out of 5 stars Critical review of history and foundations of mathematics 31 Aug 2000
By Juergen Kahrs - Published on
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.
36 of 38 people found the following review helpful
5.0 out of 5 stars A great book on the nature of mathematics! 5 Jan 2001
By Sando Anoff - Published on
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!
33 of 35 people found the following review helpful
5.0 out of 5 stars One of the most valuable books on the history of mathematics 19 Aug 2000
By Mark Poyser - Published on
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.
20 of 23 people found the following review helpful
5.0 out of 5 stars engaging intellectual history in the domain of mathematics 14 July 2003
By los desaparecidos - Published on
Morris Kline, Professor Emeritus of Mathematics at New York University, offers us with this book a superb popular intellectual history in the domain of mathematics focusing on a single theme, the search for the perfection of truth in mathematical formalism. The outcome of this quest is described in its essence on page 257:

"The science which in 1800, despite the failings in its logical development, was hailed as the perfect science, the science which establishes its conclusions by infallible, unquestionable reasoning, the science whose conclusions are not only infallible but truths about our universe and, as some would maintain, truths in any possible universe, had not only lost its claim to truth but was now besmirched by the conflict of foundational schools and assertions about correct principles of reasoning."

Kline informs us that the current state of the science is that in which in true postmodern fashion several schools somewhat peacefully coexist--among them, Russell's logicism, Brouwer's intuitionism, Hilbert's formalism, and Bourbaki's set theory--in apparent abandonment of the nineteenth-century goal of achieving the perfection of truth in formal mathematical structures. In this coliseum of competing paradigms, the tipping point that engenders the status quo of peaceful coexistence is, of course, Kurt Godel, who in 1931 with his Incompleteness Theorem of almost cultic fame showed that any mathematical system will necessarily be incomplete because there will always exist a true statement within the system that cannot be proven within the system.

Despite this Babel, Kline believes that mathematics is gifted with the intellectual wherewithal to fruitfully pursue even the farthest and most abstruse reaches of abstraction because in this quest it is always assured the boon of the Holy Grail by virtue of the touchstone of empiricism. He concludes on the last page:

"Mathematics has been our most effective link with the world of sense perceptions and though it is discomfiting to have to grant that its foundations are not secure, it is still the most precious jewel of the human mind and must be treasured and husbanded."

In Scripture the counterpart of this outlook might be, "Test everything; retain what is good" (1 Thessalonians 5:21), while in common proverbs it would be, "The proof of the pudding is in the eating."

Although the book is written as a popular intellectual history and therefore is accessible to every educated reader, I believe that the extent to which readers would appreciate various historical portions of this book would depend on their formal mathematical preparation. From the time of Euclid's Elements to Newton's Principia Mathematica, sufficient for a deep appreciation on the reader's part is a high school background in mathematics. Beginning with Newton's fluxions and Leibniz's differentials and ending with nineteenth-century efforts to place algebra on formal footing, a finer understanding of the book requires the undergraduate-level background in mathematics that is usually obtained by scientists and engineers. Starting in the late eighteenth-century with Gauss' investigation of non-Euclidean geometry until twentieth-century disputes concerning mathematical philosophy, the discussion is probably more accessible to trained mathematicians or logicians.

Here and there I picked up interesting trivia, such as the historical fact that algebra, unlike geometry, was not initially developed as a formal system but rather as a tool of analysis, or that the intellectual enterprise to cast mathematics as a complete, consistent formal system really began in the second decade of the nineteenth century.

For lovers of mathematics, I recommend this book as engaging diversion in intellectual history. Read it on vacation.
9 of 9 people found the following review helpful
5.0 out of 5 stars The drama of the unfolding of mathematical thought 4 Aug 1997
By A Customer - Published on
The heart of this book is a great narrative about
the development of mathematical thought from
Euclid's time to the modern time. Though the book
asks the question of why mathematics "works" in
applied disciplines despite the fact that its
theoretical underpinnings have repeatedly been
revealed to have substantial gaps, at bottom it
works best as a great story, wonderfully
researched and coherently told, of how, and to
whom, the major mathematical lightbulbs turned
on. All of the familiar names -- men like
Gauss, LeGendre, Russell, Cantor, Liebniz,
Cauchy, Godel -- play roles. If you would like
an overview of mathematical thought, to step
back and see the big picture and understand what
the big issues have been, read this book. You
do not need more than a conceptual understanding
of certain advanced math concepts (e.g., calculus,
trigonometry, set theory) to enjoy this book.
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