Mathematics: The Loss of Certainty

by Morris Kline

Book Overview

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... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 1435136063
• ISBN13: 9781435136069
• Release Date: January 2011
• Publisher: Fall River Press Edition
• Length: 448 Pages
• Weight: 1.70 lbs.
• Dimensions: 9.2" x 1.3" x 6.3"

Customer Reviews

engaging intellectual history in the domain of mathematics

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 un

A great book on the nature of mathematics!

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!

One of the most valuable books on the history of mathematics

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.

An outstanding account of the history of mathematics

Kline is brilliant. He writes an outstanding account of the history of mathematics and its effect on humanity. He includes great examples and a multitude of quotes. It gets a bit slow in the middle but starts and finishes strong. It is an inspiration to all who appreciate math. Readers should have taken at least through calculus.

The drama of the unfolding of mathematical thought

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.