Customer Reviews

Mathematics: The Loss of Certainty (Galaxy Books)

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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.

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!

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.

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.

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.

A delightful and important book for all math enthusiasts. A must read for budding mathematicians.

This book authoritatively chronicles the gradual realization that mathematics is not the exploration of hard edged objective reality or the discovery of universal certainties, but is more akin to music or story telling or any of a number of very human activities.

Kline is no sideline popularizer bent on de-throwning our intellectual heros - he speaks knowledgeably from within the discipline of mathematics, revealing the evolution of mathematical thought from "If this is real, why are there so many paradoxes and seeming inconsistencies?" to "If this is just something people do, why is it so damned powerful?"

One reviewer said, ``First, Barbosa attacks Morris Kline (he's got some nerve doing that) for Prof. Kline's supposed lack of understanding of mathematics. This frivolous insult is so ridiculous that it isn't necessary to discuss it further.'' I won't claim that Kline doesn't understand mathematics, but it is quite clear from this book that he does not understand logic. I looked up reviews in the professional literature by logicians and found they made the same point.

Kline makes many technical errors in his account of the foundational debates in the early twentieth century. My favorite mistake, and perhaps his 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. (For non-logicians: they can and do; Kline's alleged implication is wrong.) His account of the history of mathematics is not as bad.

Kline was an applied mathematician, and in his last two chapters informs us in very strong terms that applied mathematics is good and true, but pure mathematics is not. He urges mathematicians to abandon the study of analysis, topology, functional analysis, etc., and devote themselves to the problems of science.

The book is lively and entertaining, if not entirely reliable.

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).

This is a look at the history of mathematics through the lens of logical justification. The "loss of certainty" referred to in the subtitle is a consequence of the pursuit of rigor in both mathematics and logic. As mathematics developed, judgments of what constitutes logical justification, and hence of what constitutes a mathematical proof, reached a severity that pushed the requirements of justification to the very conceptual and structural foundations of mathematics. Attempts to prove that mathematics, taken as a whole, is logically consistent and that every true mathematical statement is within the reach of mathematical proof, that it is consistent and complete, encountered troubling difficulties.

Jumping into the second half of the book:

The question of whether Euclid's parallel postulate was a necessary axiom or whether it could be mathematically derived from the others had resulted in the discovery of non-euclidean geometries and thus the logical independence of the euclidean parallel postulate from the others. With this discovery, geometry as a subject of mathematical study became dissociated from the physical world. With refinements in the demands of rigor and of what constitutes a defensible axiomatic structure, geometry was restructured by Hilbert and in the process was emptied of all conventional geometric meaning. Questions of consistency having arisen in the study of the independence of the parallel postulate, the urgent question of the consistency of non-euclidean geometries was resolved through proving their consistency to be certain, relative to the unproven consistency of the now more rigorously axiomatized euclidean geometry. Euclidean geometry itself was translated rigorously into algebraic form, thus pushing the question of its logical consistency into that of algebra and, further, into arithmetic and number theory.

Cantor's investigations of infinity and the paradoxes that arose in thinking about sets, and Zermelo's controversial axiom of choice raised questions of how to locate arithmetic, infinity, and mathematical proof within the boundaries of logical justification. Procedures such as Peano's method of building numbers from the integers upward upon the foundation of five axioms weren't sufficient to assuage critical doubts, since infinite sets and the axiom of choice were implicit in the system. The axiom of choice was discovered also to be implicit in valuable and highly regarded areas of advanced mathematics. Not only was the question unsettled of where to find a logical and fully explicit basis of number, but with the discovery of the hitherto unknown use of the axiom of choice in the higher reaches of mathematics, the goal of finding a logical justification for mathematics seemed absolutely essential.

At the same time, logic itself was undergoing development into a more rigorous form. In fact, it was looking as if this new formalized, and thus more mathematically structured logic could fulfill the need for a foundation and rigorous justification of mathematics. Mathematics, it was suggested, was simply a further extension of logic and inseparable in principle from it. The program ran afoul, however, in controversial details of its explication within Russell and Whitehead's Principia Mathematica, specifically in the axioms and methods applied to avoid troubles arising from imprecise uses of infinity. The Principia system didn't look in its foundations to be pure logic. Aside from these complaints, the question arises that if mathematics is simply logic, and logic is a process within the human mind, then what is the logical justification of using mathematics to describe the physical world?

Pushing mathematics further into the mind and detaching it from logic, Brouwer held that mathematics is a human construct arising from the mind's intuition, independently of any apprehension of the external world, and that logic is aligned with language, wholly separate from intuition, which precedes all language, and therefore logic has no independent authority over what constitutes mathematics. Mathematics does not exist outside the mind, so only that which the mind intuits can be considered legitimate mathematics. Any mathematical assertion of mathematical existence is legitimate only when an intuitively clear procedure is given to construct or display that about which the assertion is made. The natural numbers, addition, multiplication, and mathematical induction are intuitively clear, and from this, infinite sets can be intuitively justified; but there are far reaching differences because of the primacy of intuition over logic. Although logic is aligned with language and thus suspect of intuitive accuracy, there is an intuitive basis for a portion of logic. This intuitively clear portion of logic does not, however, include the law of excluded middle: it may be the case that both p and not-p are true in this intuitively justified portion of logic, especially when it comes to thinking about infinite sets.

Hilbert took the view that although logic and mathematics are logically distinct, the approach taken in the Principia of laying down an axiomatic basis presented in rigorously formalized mathematical language is the key to proving the logical consistency and completeness of mathematics; but unlike in the Principia, because mathematics would not be presented as an extension of logic, the axiomatic basis does not need to be restricted only to axioms for logic. The system could be examined through the syntax of its formalized mathematical language, and using undisputed logical principles, the question of its logical consistency and completeness could be investigated. Complaints arose that Hilbert intended to empty mathematics of all mathematical meaning. Hilbert countered that he had been misunderstood: it was only during the investigation of the logical status of the system that the syntax of the system would hold prominence. The mathematical meaning was still retrievable, just as it was in an axiomatic presentation of geometry.

Zermelo and Fraenkel took a fourth approach. The Principia had included sets within its system, but sets were presented as arising within the purview of logic. Zermelo first, and later Fraenkel with amendments to Zermelo's system, derived set theory from its own axioms. These axioms were designed to recover what was desirable in Cantor's non-axiomatic set theory and to avoid the problems that had arisen through thinking of sets, and especially infinite sets, in an imprecise way. From this axiomatically based theory of sets, it was claimed, all of mathematics could be derived. If set theory is consistent and complete, then so is mathematics.

Everyone believed that once the elements of number were given a logically secure basis, all the rest of mathematics could be derived from the substructure of arithmetic. The work of Gödel, and later Cohen's work adding to Gödel's on Zermelo's axiom of choice and the continuum hypothesis of Cantor, brought this belief to ruin. Systems such as those of Principia or the kind Hilbert proposed were impaired when Gödel proved any formal system rich enough to provide arithmetic could not yield every mathematical truth of the system using the undisputed logical principles Hilbert advocated, and thus that the formal system was necessarily incomplete; and moreover, by logical implication, the consistency of the system could not be proved. Then, in the realm of set theory, in which Zermelo and Fraenkel had included the axiom of choice because of its essential role in many valuable proofs and mathematical concepts, it turned out that the axiom of choice was logically independent of the other axioms, just as in euclidean geometry the axiom of parallel lines was not derivable from the others. So just as there could be logically justified and thus mathematically legitimate non-euclidean geometries, there could be theories of sets without the axiom of choice with as much mathematical legitimacy as ZF set theory, upon which the superstructure of mathematics was being built. Other results, concerning the continuum hypothesis and the cardinality of models of ZF set theory, added to the realization that ZF is only one kind of mathematically legitimate set theory. Because ZF was touted, and seemed to be working well, as a foundation for mathematics, these discoveries subverted the idea that mathematics was a coherent unity, organized around the core idea of a set. The certainty of mathematics was lost.

Kline concludes the book with three non-historical chapters, where he discusses the relationship of mathematics to natural science and the value of science in determining the reliability and trustworthiness of mathematics. He argues pointedly against the eminence of pure mathematics and is concerned that mathematicians, in avoiding scientific applications, are loosing touch with what historically gave mathematics its vitality.

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.