Mathematics: The Loss of Certainty (Galaxy Books) (Paperback)
by Morris Kline (Author)
Editorial Reviews
Book Description
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.
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; Reprint edition (June 17, 1982)
Language: English
engaging intellectual history in the domain of mathematics, July 14, 2003
Reviewer:
los desaparecidos (Makati City, Philippines) - See all my reviews
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.
30 of 55 people found the following review helpful:
Did not Convince Me, April 17, 2002
Reviewer:
Pedro Rosario (Río Piedras, PR USA) - See all my reviews
I wish to point out first the positive aspects of the book. First of all, it should be noted that Morris Kline is one of the greatest mathematicians and now discusses a very important philosophical issue that is pertinent today.
Kline shows a great insight concerning the history of the development of mathematics, a recount of the problems that different mathematicians had throughout history, the way they pretended to solve the problem, their logical and illogical reasons for doing so. He at least defends himself very well looking to history to prove how uncertain mathematics is.
However, his book lives up according to a fallacy. Let's say that somebody thinks that certainty depends on a property "F" characteristic of some "a" mathematical system. Then the fact that up to that point it was believed by many people that F(a), then mathematics was certain, while when they discovered that it was not the case that F(a) then certainty of mathematics can no longer be established. An analogy with science will make clear the fallacy. Galileo insisted that the certainty of science on the universe depended greatly on the fact that the planets and stars moved in perfect circular orbits; Kepler on the other hand proved that the planets move in eliptical orbits. It would be an exaggeration to think, that the certainty of science is lost just because planets move in eliptical orbits.
Another problem is that he states that mathematics is also uncertain because the irrational reasons to admit certain mathematical entities or axioms. However, the *validity* of the axioms is what is at stake in mathematics, not the subjective reasons that somebody had to admit them. An analogy again with science can show this second fallacy. Some of the reasons Copernicus admited that the Sun was the center and not the Earth, was because the Sun was the noblest star, and because it would restore the perfection of the circles in which planets revolve, because it had been lost in the Ptolemaic geocentric view of the universe. Do these reason should really dismiss the validity of Copernicus' theory? No. The same happens with mathematics. The illogical reasons that somebody might have to discover something, is irrelevant concerning the validity and certainty of mathematics.
Also, there is the fallacy that because that there is a development of mathematics in one area that seems to be unorthodox at some moment, might compromise the certainty of mathematics. For example, he uses the development of "strange" algebras or "strange" geometries as examples of this. Non-Euclidean geometry doesn't invalidate Euclidean geometry, as Morris seems to suggest, nor does imply the loss of certainty of Euclidean geometry. It only means that Euclidean geometry is one of infinite possible mathematical spaces. Certainty is guaranteed in each one of them.
Also, he seems to use the word "disaster" concerning Godel's theorems. But it was a "disaster" only to *some* philosophical schools. Godel's theorems doesn't seem at all to imply the uncertainty of mathematics, since Godel himself believed in its certainty during his entire life. In fact, Platonist propoposals such as Husserl's, though Edmund Husserl posited the completeness of mathematics, his main philosophy of mathematics is supported *even after* Godel's discovery. The only thing refuted in his philosophy is the completeness of mathematics, but not his mathematical realism, nor his account of mathemathical certainty. Interestingly, Husserl is never mentioned in the book (just as many philosophers of mathematics ignore his philosophy).
Though the book is certainly instructive and Morris shows his knowledge of history of mathematics, due to these fallacies, he never proves his case.
Customer Reviews
Average Customer Review:
Write an online review and share your thoughts with other customers.
1 of 1 people found the following review helpful:
Excellent survey of the history of mathematics , February 9, 2007
Reviewer:
Michael Emmett Brady "mandmbrady" (Bellflower, California ,United States) - See all my reviews
Kline demonstates, in a clear and detailed fashion ,that the pursuit of " pure " mathematics(the set theoretical, real analysis approach),as opposed to the applied mathematics useful to scientific discovery ( the differential and integral calculus plus ordinary and partial differential equations),leads to a dead end as far as scientific discovery is concerned. This is well illustrated in his discussion of the rise of the Nicholas Bourbaki school that has come to dominate mathematics(pp.256-257)since the mid -1930's and its impact on the social sciences.
The field of economics is an excellent example of Kline's point. Economists are notorious for trying to copy the latest technical developments that occur in mathematics, statistics, physics, biology, etc., irrespective of whether or not such techniques will yield useful knowledge which economists can use to analyze the events/historical processes occurring in the real world so that they can explain and predict why and when these events/processes will occur/reoccur.The best examples of the non or anti-scientific approach of the economics profession are the (a) Arrow-Debreu-Hahn general equilibrium approach based on various fixed point theorems,(b)the Subjective Expected Utility approach of Ramsey-De Finetti-Savage ,and(c)the universal belief of econometricians in the applicability of multiple regression and correlation analysis based on a least squares approach which requires the assumption of normality. It is not surprising that no econometrician in the 20th century ever did a basic goodness of fit test on their time series data to check to see whether or not the assumption of normality was sound. It took a Benoit Mandelbrot to demonstrate that the assumption of normality did not stand up.
The result has been that the economists simply are incapable of dealing with phenomena in the real world. Their pursuit of the latest fad or gimmick or technique to copy leads to the type of comment made by Robert Lucas,Jr.,the main founder of the rationalist expectationist school, that his theory can't deal with uncertainty, but only risk which must be represented by the standard deviation of a normal probability distribution. It is unfortunate that Lucas never did any goodness of fit test on business cycle time series data before constructing a theory that is only applicable if business cycles can be represented by multivariate normal probability distributions.
Kline's approach to the nature of mathematical discovery is very similar to that of J M Keynes and R Carnap-"The recognition that intuition plays a fundamental role in securing mathematical truths and that proof plays only a supporting role suggests that ...mathematics has turned full circle.The subject started on an intuitive and empirical basis...the efforts to pursue rigor... have led to an impasse..."(p.319).It can easily be observed that all of the three economist approaches mentioned above have ended in an impasse also.
3 of 6 people found the following review helpful:
Kline's uncertainty, June 2, 2006
Reviewer:
Walt Peterson (Ames, Iowa) - See all my reviews
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.
1 of 3 people found the following review helpful:
Great book by a great author, February 23, 2005
Reviewer:
G. A. Meles "Sirtor" (Italy) - See all my reviews
English:
This book isn't meant to be a mathematics book, still it offers a very good qualitative view of the problems it describes - at least as long as the reader has a not-zero competence in mathematics.
Don't forget what Kant wrote, in the introduction of his masterpiece "Critique of Pure Reason" i.e. "that many a book would have been much clearer if it had not made such an effort to be clear": there are topics that can't be explained in "too simple words".
There are a lot of divulging books that are not clear for competent reader and seem to be clear for inadequate readers: this is not the case of Kline books, which provides a interesting reading for an interested reader.
3 of 4 people found the following review helpful:
Mathematical Uncertainty, December 23, 2004
Reviewer:
Jefferson D. Bronfeld "always_reading" (Binghamton, New York USA) - See all my reviews
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?"
Mathematics: The Loss of Certainty. by Morris Kline. Oxford. 366 pp. $19.95.
Professor Kline recounts a series of ``shocks'', ``disasters'' and ``shattering'' experiences leading to a ``loss of certainty'' in mathematics. However, he doesn't mean that the astronaut should mistrust the computations that tell him that firing the rocket in the prescribed direction for the prescribed number of seconds will get him to the moon.
The ancient Greeks were ``shocked'' to discover that the side and diagonal of a square could not be integer multiples of a common length. This spoiled their plan to found all mathematics on that of whole numbers. Nineteenth century mathematics was ``shattered'' by the discovery of non-Euclidean geometry (violating Euclid's axiom that there is exactly one parallel to a line through an external point), which showed that Euclidean geometry isn't based on self-evident axioms about physical space (as most people believed). Nor is it a necessary way of thinking about the world (as Kant had said).
Once detached from physics, mathematics developed on the basis of the theory of sets, at first informal and then increasingly axiomatized, culminating in formalisms so well described that proofs can be checked by computer. However, Gottlob Frege's plausible axioms led to Bertrand Russell's surprising paradox of the the set of all sets that are not members of themselves. (Is it a member of itself?). L.E.J. Brouwer reacted with a doctrine that only constructive mathematical objects should be allowed (making for a picky and ugly mathematics), whereas David Hilbert proposed to prove mathematics consistent by showing that starting from the axioms and following the rules could never lead to contradiction. In 1931 Kurt Goedel showed that Hilbert's program cannot be carried out, and this was another surprise.
However, Hilbert's program and Tarski's work led to metamathematics, which studies mathematical theories as mathematical objects. This replaced many of the disputes about the foundations of mathematics by the peaceful study of the structure of the different approaches.
Professor Kline's presentation of these and other surprises as shocks that made mathematicians lose confidence in the certainty and in the future of mathematics seems overdrawn. While the consistency of even arithmetic cannot be proved, most mathematicians seem to believe (with Goedel) that mathematical truth exists and that present mathematics is true. No mathematician expects an inconsistency to be found in set theory, and our confidence in this is greater than our confidence in any part of physics.
Mathematics: The Loss of Certainty
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. Morris Kline points, out that today there is not one universally accepted concept of mathematics - in fact, there are many conflicting ones.
Yet the effectiveness of mathematics in describing and exploring physical and social phenomena continues to expand. Indeed, mathematical activity is flourishing as never before, with the rapidly growing interest in computers and the current search for quantitative relationships in the social and biological sciences. "Are we performing miracles with imperfect tools?" Kline asks.