» Mathematics: The Loss of Certainty

Mathematics: The Loss of Certainty Reviews

Customer Rating:
Summary: Dry, Boring and not that informative
Comment: There is a lot of history of the development of current mathematics and a lot of information of interest to mathematicians. Many of the concepts in this book will probably not be understood by the lay person (that is someone without adequate knowledge of the calculus) insofar as Kline provides lots of mathematical examples not well (or not at all) explained in lay person terms. This book has a lot of hype about it but is not at all worth the hype.

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Customer Rating:
Summary: Excellent survey of the history of mathematics
Comment: 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.

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Customer Rating:
Summary: Kline's uncertainty
Comment: 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.

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Customer Rating:
Summary: Great book by a great author
Comment: 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.
Italiano
Questo libro non intende essere un testo di matematica, ciò nonostante, offre un'ottima visione qualitativa dei temi che tratta - almeno se il lettore ha una competenza non nulla in matematica.
Non si dimentichi quello che Kant scrisse nel suo capolavoro "la critica della ragion pura", ovvero "molti libri sarebbero stati molto più chiari se non avessero voluto essere così chiari": ci sono argomenti che non possono essere spiegati in "termini troppo semplici".
Esistono molti testi divulgativi che non sono per nulli chiari per il lettore competente, e sembrano essere chiari per il lettore inadeguato: non è questo il caso del libro di Kline, che offre una lettura interessante per un lettore interessato.

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Customer Rating:
Summary: Mathematical Uncertainty
Comment: 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?"

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