Reviews from Amazon

Reviews elsewhere on the web:

Morris Kline

Mathematics, The loss of certainty

Mathematics has an air of being the most secure form of knowledge. In Mathematics, The Loss of Certainty, however, Morris Kline shows that this is not necessarily deserved. He shows how, rather than mathematics being an obvious progression of knowledge, in fact many ideas in the subject were strongly resisted when first introduced, and even when accepted often rested on insecure foundations. He explains how Euclidean geometry turned out not be as 'obviously true' as people thought, how calculus was based on the shaky ground of infinitesimals, and how grudgingly imaginary numbers came to be accepted as a valid way to do calculations.

I felt that Kline tends to overdo the uncertainty of mathematics, in particular at the start of the book, making it look like each surprising result implied that the subject was in total disarray. The book improves as it goes on though. The later parts look at the development of Cantor's set theory and the axiomatisation of mathematics leading up to Göel's incompleteness theorem and beyond. This is certainly worth reading for those with an interest in the philosophy of mathematics, especially since the book requires very little previous mathematical experience on the part of the reader.

Amazon.com info

Paperback 384 pages
ISBN: 0195030850
Salesrank: 141520
Weight:0.45 lbs

Published: 1982 Oxford University Press, USA

Amazon price $17.96

Marketplace:New from $12.95:Used from $7.38

Buy from Amazon.com

Amazon.co.uk info

Paperback 384 pages
ISBN: 0195030850
Salesrank: 165041
Weight:0.45 lbs

Published: 1983 Oxford University Press Inc, USA

Amazon price £7.91

Marketplace:New from £7.90:Used from £6.84

Buy from Amazon.co.uk

Amazon.ca info

Paperback 384 pages
ISBN: 0195030850
Salesrank: 236173
Weight:0.45 lbs

Published: 1982 Oxford University Press

Marketplace:New from CDN$ 13.51:Used from CDN$ 7.58

Buy from Amazon.ca

Product 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.

Product 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.

Dry, Boring and not that informative **
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.

Dry, Boring and not that informative **

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.

Excellent survey of the history of mathematics *****
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.

Excellent survey of the history of mathematics *****

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.

Kline's uncertainty **
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.

Kline's uncertainty **

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.

Great book by a great author *****
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.

Great book by a great author *****

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.

Mathematical Uncertainty *****
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?"

Mathematical Uncertainty *****

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?"

Ultimately disappointing ***
Kline's book traces the development of mathematics from the ancient Greeks to the present time. It tries, ultimately, to develop a philosophy of mathematics through presenting the history of mathematics, displaying the sometimes enormous struggle mathematicians had in trying to come to terms with new concepts, and trying to develop rigour. For anyone who believes in the eternal verities of mathematics, this book is a salutary reminder that, even if such exist, mankind's ability to intuit them is severely limited. Although the subject matter is extremely interesting, Kline's style makes the book somewhat irritating, and I believe that the book is considerably the poorer for it. I am broadly sympathetic to the position outlined by Kline, but I fear that neither his history nor his philosophy of mathematics receive much support from this work. It is undoubtedly the case that the separation of applied and pure mathematics has weakened both, that the "Greats" (Gauss, Poincare, etc.) worked equally easily in both areas, and so on. It is interesting to note, for example, the tremendous impetus imparted to areas of pure mathematics in recent years by developments in physics (e.g., read Peter Woit's book, "Not even wrong", which supports this view, although, paradoxically, Woit's view, rightly or wrongly, is that areas such as superstring theory are not physics at all). But the potentially interesting philosophical implications for mathematics of this dialogue are, I fear, lost in Kline's rather heavy-handed, perhaps evangelical extolling of the virtues of applied mathematics. I am compelled to write a word or two about the review of Kline's book by D. Thomas (entitled: Tries to prove a nonsensical point). I think D. Thomas must have been reading another book! It is simply not true that Kline's book bases its entire argument for "uncertainty" on Godel's well-known incomplete theorems. Indeed, the discussion of Godel and concomitant issues occupies a relatively small fraction of the volume, and those hoping for a "Godelian" slant on mathematics will be sorely disappointed. Kline's arguments for "uncertainty", whether right or not, extend back into the history of mathematics much further than Godel, and indeed much further than the late nineteenth century/early twentieth century programme to develop the foundations of mathematics. Alas, D. Thomas's polemics miss the target by a light-year.

Ultimately disappointing ***

Kline's book traces the development of mathematics
from the ancient Greeks to the present time. It tries,
ultimately, to develop a philosophy of mathematics through
presenting the history of mathematics, displaying the
sometimes enormous struggle mathematicians had in trying to
come to terms with new concepts, and trying to develop
rigour. For anyone who believes in the eternal verities
of mathematics, this book is a salutary reminder that, even
if such exist, mankind's ability to intuit them is severely
limited.

Although the subject matter is extremely interesting, Kline's
style makes the book somewhat irritating, and I believe that
the book is considerably the poorer for it. I am broadly
sympathetic to the position outlined by Kline, but I fear that
neither his history nor his philosophy of mathematics receive
much support from this work. It is undoubtedly the case that
the separation of applied and pure mathematics has weakened both,
that the "Greats" (Gauss, Poincare, etc.) worked equally easily
in both areas, and so on. It is interesting to note, for example,
the tremendous impetus imparted to areas of pure mathematics in
recent years by developments in physics (e.g., read Peter Woit's
book, "Not even wrong", which supports this view, although,
paradoxically, Woit's view, rightly or wrongly, is that areas such
as superstring theory are not physics at all). But the potentially
interesting philosophical implications for mathematics of this
dialogue are, I fear, lost in Kline's rather heavy-handed, perhaps
evangelical extolling of the virtues of applied mathematics.

I am compelled to write a word or two about the review of Kline's
book by D. Thomas (entitled: Tries to prove a nonsensical point).
I think D. Thomas must have been reading another book! It is simply
not true that Kline's book bases its entire argument for "uncertainty"
on Godel's well-known incomplete theorems. Indeed, the discussion of
Godel and concomitant issues occupies a relatively small fraction of
the volume, and those hoping for a "Godelian" slant on
mathematics will be sorely disappointed. Kline's arguments for
"uncertainty", whether right or not, extend back into the history
of mathematics much further than Godel, and indeed much further than
the late nineteenth century/early twentieth century programme to
develop the foundations of mathematics. Alas, D. Thomas's polemics
miss the target by a light-year.

Tries to prove a nonsensical point *
The writer of this book, the late Morris Kline, is an anti-pure maths polemicist. His main goal is to show that uncertainty reigns in mathematics. Let me denote by Mathematics the study of mathematical objects such as statements, integers, functions and so forth, and let me denote by mathematics the mathematical objects themselves. Morris Kline does not make this distinction clear so it is not always clear what he is talking about. Anyway, to get to the point, it is nonsensical to talk of mathematical statements as certain or uncertain. Certainty is not a property of statements, it is a property of people, an emotion. The same goes for uncertainty. And one cannot dictate to someone their emotions. Therefore any talk of certainty of mathematics makes no sense. Also talk of the certainty of Mathematics makes no sense. There may be a statement in combinatorics of which a combinatorialist is certain, and a statement in commutative algebra of which an algebraist is certain, and they may be mutually uncertain of each other's statements due to lack of expertise. One can only talk of the certainty of an individual mathematician with respect to a particular statement or list of statements, and not of the certainty of 10,000 mathematicians all at once. As such, the main point that Morris Kline is trying to prove, that Mathematics or mathematics is uncertain, makes absolutely no sense. Obviously, he fails to prove it, as one cannot prove nonsense. His basis for the claim he makes about uncertainty is Godel's theorem, which more or less says that there is no general algorithm for determining whether a given arithmetical statement is true or not. Equivalently, for any consistent list of axioms there is a true arithmetical statement which is true but cannot be proven using only those axioms. But this doesn't stop one from being certain that a given list of axioms is true, or being certain that a given list of axioms is consistent, and doesn't stop one from being certain of a truth derived from a certain list of axioms. Godel's theorem simply does not imply what Kline is suggesting. Kline brings up the parallel postulate in support of what he is trying to say. But the apparent support is really an illusion. All the non-euclidean drama means is that there are possible geometries that violate the parallel postulate. That is, there are many possible geometries. But within each geometry, things are perfectly clear, not hazy and uncertain as Kline wants to put across. Then there is the continuum hypothesis in set theory, and the axiom of choice in set theory. There are set theoretical statements which appear to be neither true nor false, and this might seem shocking, but these statements do not touch most of mathematics. In fact one can prove that large areas of mathematics, all mathematics developed before Cantor, and that which logically follows from it and similar investigations, are unchanged whether one assumes the axiom of choice (or the continuum hypothesis) to be true, or assumes it to be false. In other words, why should I care about an isolated pathological statement of set theory? However, the Godel sentence in Godel's theorem is not a pathological statment of set theory, but a statement of arithmetic. If we were unable to prove or dispose of some such statements, even in principle, that might be a cause for panic. But this is not necessarily the case, and Godel's theorem does not imply that it is so. Kline also makes some technical mistakes on logic, which are annoying. It is also ironic that the front cover comment hails Morris Kline as the guy who understands numbers better than anyone since Euclid. Morris Kline presumably had quite a limited knowledge of numbers seeing as he was not a number theorist but an applied mathematician, and he never published a single paper on the subject. In other parts Kline discusses various fallacious proofs or reasons for declaring a theorem to be true. However numerous these may be, it is not clear that they have any philosophical significance. Yes people have sometimes declared things to be true for fallacious reasons, and sometimes have been wrong, so what? Kline describes Godel's theorems as disastrous, but they simply were not. They were disasters for Bertrand Russell and people who shared his views, but a lot of people did not. Godel's views and his theorem tie in just fine with people like Poincare. Outside the field of mathematical logic, mathematicians carried on as normal. Godel's theorem did not and does not indicate that anyone should do anything otherwise, and this is a point Kline fails to address (probably because it flies in the face of what his agenda is). Godel's work and Cohen's work on the axiom of choice and the continuum hypothesis merits the title "disaster" a little more, but nevertheless they are outlandish statements of set theory, one might even call them meaningless, and they have no relevance to "normal" mathematics, being disastrous only for set theorists and point-set topologists.

Tries to prove a nonsensical point *

The writer of this book, the late Morris Kline, is an anti-pure maths polemicist. His main goal is to show that uncertainty reigns in mathematics. Let me denote by Mathematics the study of mathematical objects such as statements, integers, functions and so forth, and let me denote by mathematics the mathematical objects themselves. Morris Kline does not make this distinction clear so it is not always clear what he is talking about.

Anyway, to get to the point, it is nonsensical to talk of mathematical statements as certain or uncertain. Certainty is not a property of statements, it is a property of people, an emotion. The same goes for uncertainty. And one cannot dictate to someone their emotions. Therefore any talk of certainty of mathematics makes no sense. Also talk of the certainty of Mathematics makes no sense. There may be a statement in combinatorics of which a combinatorialist is certain, and a statement in commutative algebra of which an algebraist is certain, and they may be mutually uncertain of each other's statements due to lack of expertise. One can only talk of the certainty of an individual mathematician with respect to a particular statement or list of statements, and not of the certainty of 10,000 mathematicians all at once.

As such, the main point that Morris Kline is trying to prove, that Mathematics or mathematics is uncertain, makes absolutely no sense. Obviously, he fails to prove it, as one cannot prove nonsense. His basis for the claim he makes about uncertainty is Godel's theorem, which more or less says that there is no general algorithm for determining whether a given arithmetical statement is true or not. Equivalently, for any consistent list of axioms there is a true arithmetical statement which is true but cannot be proven using only those axioms. But this doesn't stop one from being certain that a given list of axioms is true, or being certain that a given list of axioms is consistent, and doesn't stop one from being certain of a truth derived from a certain list of axioms. Godel's theorem simply does not imply what Kline is suggesting.

Kline brings up the parallel postulate in support of what he is trying to say. But the apparent support is really an illusion. All the non-euclidean drama means is that there are possible geometries that violate the parallel postulate. That is, there are many possible geometries. But within each geometry, things are perfectly clear, not hazy and uncertain as Kline wants to put across. Then there is the continuum hypothesis in set theory, and the axiom of choice in set theory. There are set theoretical statements which appear to be neither true nor false, and this might seem shocking, but these statements do not touch most of mathematics. In fact one can prove that large areas of mathematics, all mathematics developed before Cantor, and that which logically follows from it and similar investigations, are unchanged whether one assumes the axiom of choice (or the continuum hypothesis) to be true, or assumes it to be false. In other words, why should I care about an isolated pathological statement of set theory?

However, the Godel sentence in Godel's theorem is not a pathological statment of set theory, but a statement of arithmetic. If we were unable to prove or dispose of some such statements, even in principle, that might be a cause for panic. But this is not necessarily the case, and *Godel's theorem does not imply that it is so*.

Kline also makes some technical mistakes on logic, which are annoying.

It is also ironic that the front cover comment hails Morris Kline as the guy who understands numbers better than anyone since Euclid. Morris Kline presumably had quite a limited knowledge of numbers seeing as he was not a number theorist but an applied mathematician, and he never published a single paper on the subject.

In other parts Kline discusses various fallacious proofs or reasons for declaring a theorem to be true. However numerous these may be, it is not clear that they have any philosophical significance. Yes people have sometimes declared things to be true for fallacious reasons, and sometimes have been wrong, so what?

Kline describes Godel's theorems as disastrous, but they simply were not. They were disasters for Bertrand Russell and people who shared his views, but a lot of people did not. Godel's views and his theorem tie in just fine with people like Poincare. Outside the field of mathematical logic, mathematicians carried on as normal. Godel's theorem did not and does not indicate that anyone should do anything otherwise, and this is a point Kline fails to address (probably because it flies in the face of what his agenda is). Godel's work and Cohen's work on the axiom of choice and the continuum hypothesis merits the title "disaster" a little more, but nevertheless they are outlandish statements of set theory, one might even call them meaningless, and they have no relevance to "normal" mathematics, being disastrous only for set theorists and point-set topologists.

A compehensive review of the state of Mathematics today. *****
This book, like all those written by Morris Kline, is well worth reading by anyone interested in mathematics since it deals in a non-technical manner with history of the rise of mathematics and what Kline sees as its present decline.

A compehensive review of the state of Mathematics today. *****

This book, like all those written by Morris Kline, is well worth reading by anyone interested in mathematics since it deals in a non-technical manner with history of the rise of mathematics and what Kline sees as its present decline.

What is certainty, and how is it lost? ****
Clearly Morris Kline is an historical master, and his retelling of the story of mathematic is lush and rewarding. Not having had math of any sort since high school, I found the story riveting and confirming of many inchoate intuitions, especially with regard to the rather counter-intuitive status of irrational numbers, negative numbers, complex numbers, and infinitesimals, etc. However, having some training in epistemology, the book was somewhat less convincing in demonstrating its grandiose claims of the sort that (paraphrasing) "there is no truth in mathematics," or that there is no "justification for calculus" or "no factual evidence that supports the calculus," etc. His notions of truth, justification, evidence, and certainty seem entirely too dependent upon the rather limited portrait of certainty to be gotten from ancient Greek ideals of unassailable first principles and the deductions gotten from them. Epistemology itself has an evolving story that must be taken into account, and the epistemic notions (i.e. truth, evidence, etc.) require elaboration since they are central to Kline's evaluations of mathematics at every point. Since we are not ancient Greeks beholden to this limited epistemic ideal, there is only a "loss of certainty" for us to the extent that we adhere blindly to self-evident first principles and deduction as the only norms that could confer epistemic values like "certainty." Kline is persuasive in his arguments that alternate algebras and geometries are possible and useful, and one can hardly doubt that such alternatives lend themselves to a degree of modesty and potential relativity in mathematical claims to knowledge. Once one begins to admit other epistemic norms (like adequacy to empirical reality/experience or applicability to future problems, etc.) into the picture, one wonders if all the alternate algebras and geometries remain on equal footing. In any event, a degree of relativity in mathematical description and expression is not incompatible with a modest realism in mathematics. Nor are self-evidence and deduction the only norms for rationality, justification, evidence, warrant, or certainty. So long as one is cautious about the epistemic premises upon which the "loss of certainty" is predicated by Kline, the book is a great read!

What is certainty, and how is it lost? ****

Clearly Morris Kline is an historical master, and his retelling of the story of mathematic is lush and rewarding. Not having had math of any sort since high school, I found the story riveting and confirming of many inchoate intuitions, especially with regard to the rather counter-intuitive status of irrational numbers, negative numbers, complex numbers, and infinitesimals, etc. However, having some training in epistemology, the book was somewhat less convincing in demonstrating its grandiose claims of the sort that (paraphrasing) "there is no truth in mathematics," or that there is no "justification for calculus" or "no factual evidence that supports the calculus," etc. His notions of truth, justification, evidence, and certainty seem entirely too dependent upon the rather limited portrait of certainty to be gotten from ancient Greek ideals of unassailable first principles and the deductions gotten from them. Epistemology itself has an evolving story that must be taken into account, and the epistemic notions (i.e. truth, evidence, etc.) require elaboration since they are central to Kline's evaluations of mathematics at every point. Since we are not ancient Greeks beholden to this limited epistemic ideal, there is only a "loss of certainty" for us to the extent that we adhere blindly to self-evident first principles and deduction as the only norms that could confer epistemic values like "certainty." Kline is persuasive in his arguments that alternate algebras and geometries are possible and useful, and one can hardly doubt that such alternatives lend themselves to a degree of modesty and potential relativity in mathematical claims to knowledge. Once one begins to admit other epistemic norms (like adequacy to empirical reality/experience or applicability to future problems, etc.) into the picture, one wonders if all the alternate algebras and geometries remain on equal footing. In any event, a degree of relativity in mathematical description and expression is not incompatible with a modest realism in mathematics. Nor are self-evidence and deduction the only norms for rationality, justification, evidence, warrant, or certainty. So long as one is cautious about the epistemic premises upon which the "loss of certainty" is predicated by Kline, the book is a great read!

engaging intellectual history in the domain of mathematics *****
Dedicated to Elizabeth Ruffle, M.Sci. Mathematics, on her Day of Liberation, July 15, 2003 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. P.S. Say you are Mistress, not Master, in Science.

engaging intellectual history in the domain of mathematics *****

Dedicated to Elizabeth Ruffle, M.Sci. Mathematics, on her Day of Liberation, July 15, 2003

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.

P.S. Say you are Mistress, not Master, in Science.

Everyone else should be convinced by Morris Kline's book *****
I want to start by saying that I agree with all of the positive reviews of Morris Kline's book (from what I can tell, all but one person gave this book high marks). Morris Kline is indeed a great mathematician as well as a great writer/expositer of his chosen field. This book -- which explores the philosophical ramifications of mathematics via its history and argues for the essentially uncertain nature of mathematics -- is definitely a great book for math fans out there. What has motivated my review is the one negative review by Sr. Barbosa of Puerto Rico (or so he claims). It is a sad but true fact that people who give their opinions on the web do not always give fair and reasonable opinions and/or are motivated by ulterior motives. Sr. Barbosa's review seems to fall into that category. 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. The negative review further contends that mathematics really is not uncertain. Sr. Barbosa, in that line of thought, also says that Kurt Godel didn't really believe in his own famous theorem! (Or at least that's the only way one can interpret Sr. Barbosa's statements.) Even layman that are familiar with popular works on mathematics -- Godel, Escher, Bach, Godel's Proof, etc. -- realize that mathematics as a formal, axiomatic system has been PROVEN (for all time) to be incomplete and inconsistent ... i.e., "uncertain." These ideas have been further amplified by the works of Alan Turing and Alonzo Church (the Halting Problem) as well as Gregory Chaitin (Algorithmic Information Theory -- along with Andrei Kolmogorov and Raymond Solomonoff). In fact, Chaitin has proven that the natural number system -- ie, the counting numbers (1,2,3,...) -- is itself random (i.e., uncertain). If that was not enough evidence in favor of Morris Kline (and contra Sr. Barbosa), then consider quantum physics and chaos theory. Both of those fields add further fuel to the idea that nature itself is uncertain. If nature is uncertain, then why shouldn't math (which often elegantly represents nature) be uncertain? Sr. Barbosa winds up looking foolish for arguing that Copernicus and other great thinkers of physics can be used to support Sr. Barbosa's views. On the contrary, physics seems to support Morris Kline. In short, Morris Kline's book does a valuable service by looking at how mathematics has hisorically developed in an uncertain manner in order to further highlight the uncertainity in mathematics that has been logically PROVEN by others. Shame on Sr. Barbosa and others who constantly write misleading, unfair, and irrational reviews of books that can lead customers astray and unfairly malign quality work.

Everyone else should be convinced by Morris Kline's book *****

I want to start by saying that I agree with all of the positive reviews of Morris Kline's book (from what I can tell, all but one person gave this book high marks). Morris Kline is indeed a great mathematician as well as a great writer/expositer of his chosen field. This book -- which explores the philosophical ramifications of mathematics via its history and argues for the essentially uncertain nature of mathematics -- is definitely a great book for math fans out there.

What has motivated my review is the one negative review by Sr. Barbosa of Puerto Rico (or so he claims). It is a sad but true fact that people who give their opinions on the web do not always give fair and reasonable opinions and/or are motivated by ulterior motives. Sr. Barbosa's review seems to fall into that category.

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.

The negative review further contends that mathematics really is not uncertain. Sr. Barbosa, in that line of thought, also says that Kurt Godel didn't really believe in his own famous theorem! (Or at least that's the only way one can interpret Sr. Barbosa's statements.)

Even layman that are familiar with popular works on mathematics -- *Godel, Escher, Bach*, *Godel's Proof*, etc. -- realize that mathematics as a formal, axiomatic system has been PROVEN (for all time) to be incomplete and inconsistent ... i.e., "uncertain." These ideas have been further amplified by the works of Alan Turing and Alonzo Church (the Halting Problem) as well as Gregory Chaitin (Algorithmic Information Theory -- along with Andrei Kolmogorov and Raymond Solomonoff). In fact, Chaitin has proven that the natural number system -- ie, the counting numbers (1,2,3,...) -- is itself random (i.e., uncertain).

If that was not enough evidence in favor of Morris Kline (and contra Sr. Barbosa), then consider quantum physics and chaos theory. Both of those fields add further fuel to the idea that nature itself is uncertain. If nature is uncertain, then why shouldn't math (which often elegantly represents nature) be uncertain? Sr. Barbosa winds up looking foolish for arguing that Copernicus and other great thinkers of physics can be used to support Sr. Barbosa's views. On the contrary, physics seems to support Morris Kline.

In short, Morris Kline's book does a valuable service by looking at how mathematics has hisorically developed in an uncertain manner in order to further highlight the uncertainity in mathematics that has been logically PROVEN by others. Shame on Sr. Barbosa and others who constantly write misleading, unfair, and irrational reviews of books that can lead customers astray and unfairly malign quality work.

Did not Convince Me ***
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.

Did not Convince Me ***

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.

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!

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!