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on 25 February 2006
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.
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on 20 February 2009
I read this book last year, this is simply because, I said it is written by one of the most outstanding mathematical historian of the last century, Mr. Kline, but the only and an ultimatum example of his lost certainty of the absolute truth of mathematics is the Euclids parallel postulate. I have to say that Mr. Kline has repeated the same assumption in his other brilliant book MAthematics in the Westren culture.
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on 26 November 2007
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.
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