Mathematics: The Loss of Certainty (Galaxy Books)

by Morris Kline (Author)

30 customer reviews

ISBN-13: 978-0195030853

ISBN-10: 0195030850

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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

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From the Back Cover

Most intelligent people today still believe that mathematics is a body of unshakable truths about the physical world and that mathematical reasoning is exact and infallible. Mathematics: The Loss of Certainty refutes that myth.

About the Author

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

Product details

Series: Galaxy Books (Book 686)
Paperback: 384 pages
Publisher: Oxford University Press (June 17, 1982)
Language: English
ISBN-10: 0195030850
ISBN-13: 978-0195030853
Product Dimensions: 8 x 0.8 x 5.3 inches
Shipping Weight: 1.1 pounds (View shipping rates and policies)
Average Customer Review: 30 customer reviews
Amazon Best Sellers Rank: #446,693 in Books (See Top 100 in Books)
- #428 in Books > Science & Math > Mathematics > History
- #4928 in Books > Textbooks > Science & Mathematics > Mathematics

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Mathematics Unmakes the World

May 15, 2015

Format: PaperbackVerified Purchase

If you are interested in the origins of our current philosophical confusions, and you have a strong hold on the primacy of existence, this may be the book for you. Kline is basically a Kantian, but with a significant difference: his language is clear and his history thorough. Some of the history appears to be revisionist, i.e. intended to make a foundation for Kantian nihilism before Kant's time, but that being said, it is a valuable history.

Nowhere have I found such clear and stirring language about mathematics and the concept of the primacy of consciousness. If you are an Objectivist, you will read this carefully and with caution. Many contradictions are advocated: that a thing can be and not be, that the law of the excluded middle has been repealed, that identity is what we make it, that certainty is impossible and that we can be absolutely certain that certainty is impossible.

If I were to state this in mathematical terms, the book tells you that A = A is always false and A not = A is always true.

That being said, this book is useful because it presents a radically different picture of history of philosophy. I learned from it.

When I read Objectivist literature, it appeared to me that the Enlightenment was a period of unsullied glory, but according to Kline, it was more like a period of unsullied doubt. Advances in empirical science were accompanied by profound loss of certainty which the giants of science, philosophy and especially mathematics ignored at peril of their peace of mind. From Objectivism, I had the impression that Kant drove a crowbar into a tiny crack in the vast solid structure which we call philosophy, hit it with a sledgehammer, and the entire structure began to crumble into the nihilism we witness today. Kline tells a different story. According to him (and he is a Kantian), the structure was crumbling, the crack was miles wide and leagues deep. Kant just poured the cement of the unknowable over the ruins by cutting man off from his ability to know the "real" world because of the inadequacy of reason and the unreliability of his senses.

Whether Kline's characterization is correct and doubt was as widespread and as chronic before Kant as it has been after Kant, is not my point here. I really think he is exaggerating to make his case. However, the important accomplishment of this book is that he has thoroughly presented many of the mistakes of philosophers and scientists from Aristotle to the Enlightenment, which shows in exhaustive detail that Kant was not the thinker who originated nihilism, but the midwife who birthed it into a full-fledged philosophical anti-system.

Why then is this book important? Since we can know the world only by what exists, and we cannot know what does not exist, "Mathematics -- The Loss of Certainty" shows clearly that Kantianism and the current state of the world is a system based on non-existence, i.e. nothing. This book is a demonstration that the belief that a nothing is a something is the contradiction by which the world is self-destructing. As to how far the uncertainty has progressed, I give you Kline's closing sentences:

"Alfred North Whitehead once wrote, 'Let us grant that the pursuit of mathematics is a divine madness of the human spirit.' Madness, perhaps, but surely divine."

11 people found this helpful

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Mans most effective accomplishment is mathematics

June 12, 2018

Format: PaperbackVerified Purchase

In 1980 Morris Kline documented the history of mathematics from the Greeks onward.The style and content make it in an interesting read allowing him to inform his readership without baffling them. His choice of title indicates his preference for that short period in history when we humans thought we knew everything for sure and mathematicians were the trusted people of their time. His description of the infighting between the heroes of the past was informative but saddening for those who hope that academics would behave in a more mature way. Pure mathematicians had managed to run the show and Morris Kline thought this was a mistake as the understanding of nature is far from complete and applied mathematicians are needed in this quest. The book was written in 1980 and needs updating to reflect the 2018 mathematical universe.Overall an excellent source but needs a companion volume covering the last 40 years. His Thesis is taken from Henri Poincaré, "To foresee the future (of mathematics), the true method is to study its history and its present state."

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Point not taken, rating stays the same

December 14, 2009

Format: PaperbackVerified Purchase

I decided to write a review in 2006, after reading Walt Peterson's one, in which he says "My favorite mistake, and perhaps his [Kline's] most blatant blooper, is Kline's statement that the Loewenheim-Skolem Theorem implies Goedel's Incompleteness Theorem; he thinks that models with different cardinalities cannot satisfy the same sentences".

Then I was distracted by other matters, and it was only yesterday, when the book fell while I was rearranging others, that I remembered the issue.

Kline speaks about the L-S theorem in pp. 271-272 of the PB edition, and, in my opinion, nowhere does he say what Peterson accuses him of. The nearest he comes to it is in p. 273, where he writes: "The L-S theorem is not totally surprising. Gödel's incompleteness theorem does say that every [consistent] axiomatic system is incomplete. ... [six lines of text] ... . But the S-L theorem denies categoricalness in a far stronger or more radical way [than Gödel's]...". This is all. And, contrary to what Peterson says, in p. 272 Kline doesn't deny but on the contrary ASSERTS (using other words) that the L-S theorem means that models with different cardinalities CAN satisfy the same sentences: otherwise he wouldn't have mentioned it to reinforce his case about the loss of certainty. Did Peterson and I read the same edition of the book?

As for the rest, I readily admit that it may contain errors (that I, not being a professional mathematician, may have easily overlooked or not spotted). But Goethe was said -arguably- to be the last man who was able to know all that there was to know in his time. A mathematician nowadays can't possibly be at home in more than a small fragment of his chosen field. I remember the criticism levelled at Penrose by a logician in the number of the magazine "Sophia" totally dedicated to a discussion of his then new book "The Emperor's New Clothes". Or Fefferman's rebuttal, in "Psyché 2", of some inconsequential (for the sake of his central argument) mistakes Penrose had made in relation to Gödel's theorem in "Shadows of the Mind". Honest small mistakes are unavoidable in any single-person attempt to survey a large field; differences in interpretation even more so.

This said, I admit that Kline is maybe too inclined to slant the book toward his own point of view. But, as somebody -was it Nietzsche?- has said, impartiality is impossible to attain; the most that one can expect are honest biases, i.e. to interpret facts according to one's worldview (dishonest biases being suppressing known facts).

I think that overall Kline has done a splendid job: his book is eminently readable, concise, avoids major mistakes, and is one of the rare ones that, while written for laymen, is not dumbed down to them. So, four stars (not five, as he IS a little too opinionated for my taste in the last two chapters).

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