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on 3 January 2015
This is really a recreational mathematics book for mathematicians. It has a light-hearted approach but it deals with some very deep topics.. It is extremely well written and was a joy to read. I think it is aimed at the general lay-person and indeed the non-mathematician would get something from it but in the main you have to be a mathematician to get the most from it as it deals with some very fundamental features of the subject.
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5 of 7 people found the following review helpful
on 26 November 2007
Reuben Hersh's book fails to move on from the mistakes of previous philosophers. He is not the first to make them, and many of his mistakes are also the mistakes of those he argues against (Platonists) but it is nevertheless disappointing. First of all, he does not distinguish between Mathematics, in the sense of the human study of mathematical objects (statements, numbers, equations, etc.) and mathematics, in the sense of the items of study themselves (statements, numbers, equations, etc.). This terminology is my own, and not something I expect the reader to be familiar with. Nevertheless, there is a distinction here which ought to be made. Using this terminology, Mathematics is defined as the study of mathematics.

It seems that very often Hersh and people like him, and their opponents, are arguing solely for the reason that they have not made this distinction. But this simply defining of terms resolves the main disagreement. Mathematics is done by and created by people, mathematics is not. mathematics is discovered be people, Mathematics is not. For instance, algebraic topology is a fine human creation, a fine part of Mathematics. The other day, I discovered a remarkable fact. The system of equations x^2+y^2=z^2, y^2+z^2=t^2, has no positive integer solutions, and this is a piece of mathematics. I did not create that this was the case - I merely worked out that it was the case.

It is clear that Mathematics is a human discipline and would not "exist" without humans, like any other human discipline. This brings me on to my second point. In the first sentence of this paragraph, to say a human discipline "exists" is to say that there exist in the physical universe, intelligent lifeforms carrying out that discipline. Hersh argues with the Platonists over the question of whether mathematical objects, the usual example being numbers, would exist without humans. In the terminology I laid out earlier, this is the question of whether mathematics would exist without humans. But it is nonsensical to talk about the existence of mathematical objects. It is meaningless to say that the complex number i exists. It is meaningless to say that it does not exist. It is only meaningful to say it does exist in the complex numbers. It is only meaningful to say that it does not exist in the real numbers.

For this reason, the question of whether mathematics exists or not is a nonsensical one. Ergo, related questions such as where, how, why it exists or does not exist are nonsensical. Also the question of whether mathematics exists "timelessly" is nonsensical for not one reason, but two. Time is a physical concept, and simply does not apply to those things that make up mathematics because they are not physical things.

It may seem strange that such a hotly debated question could in fact be simply nonsense. But this is not really that strange. In fact, it provides an explanation for why these people keep on arguing and arguing and never make any progress.

Moving on, a favourite topic of Hersh's is uncertainty. One gets the feeling that if there were a kind of "uncertainty" breakfast cereal, made up of the respective letters of that word, it would be Hersh's favourite cereal. Maybe that's a weird thing to say but it's the only way I could think to describe how keen he is on uncertainty. He is a bit like Morris Kline in this respect. In love with uncertainty. He also seems to work mostly in differential equations and probability, both quite applied areas, but fortunately does not share Kline's anti-pure mathematics polemicisim.

Anyway, to get to the point, it is (here we go again) 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.

Hersh also has a topic of the left/rightness of mathematicians and philosophers. The less said about that the better. It is perfect for those who enjoy the meaningless and arbitrary.

One good thing about this book is that it emphasises Mathematics, the human study. Too often only mathematics is portrayed and this can lead to a view of Mathematics as inhuman, like mathematics. It needs to be emphasised that although change is not a notion that makes sense for mathematics, it makes perfect sense for Mathematics and happens all the time. It is a shame that Hersh cannot get this point across succinctly enough because he does not precisely define his terms.

The first duty of any philosopher is to make sense. Hersh, like those he argues against, neglects this duty. I give him two stars mainly out of sympathy, because he clearly went to a lot of effort to write this book.
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6 of 9 people found the following review helpful
on 23 August 1998
This book is another in the "Teachers First, Technically Proficient Last" genre. Hersh has written a Political Manifesto of the radical socialistic, liberal, "modern" teacher's misconception of what it means to teach. He is more interested in "cute" partitionings of (genuinely brilliant) philosophers of mathematics and mathematicians into "right" and "left" (the word "wing" left for the reader to insert), than he is in explaining mathematics (Really). Any discussions of mathematics are so trivial that he only makes a couple of basic errors (the most gross being employment of Platonic concepts); the rest is simply an expression of his apparent resentment at things being what they are rather than what he happens to want them to be. This whole book could have been written with one sentence; Mathematics is a social enterprise,and that's not even correct. Mathematics is a human enterprise (usually best done individually). The book will NOT explain what mathematics is really or otherwise. In its alleged field it is akin to Wilson's Consilience and Shapiro's Philosophy of Mathematics and even more unevenly written.
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on 16 August 2013
Excellent book by a renowned Mathematician. You do not need high powered mathematical background to follow the book. I recommend it.
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4 of 7 people found the following review helpful
on 28 May 1998
If you are very smart and enjoy thinking at the leading edge, this book can help you do that even better. Hersh, a mathematician and math teacher for 30 years, takes on the hard problems of what is knowledge and where does it come from, comparing the Platonist, formalist, structuralist, and humanist views. The work parallels what Kuhn has done for the philosophy of science. In particular he does a good job of showing why Platonism -- the commonly accepted view that math (read truth) exists a-priori and we merely "discover" it -- really does not hold up. Curiously, I found this deeply liberating, as it opens up much more breathing space for original thought, highlighting the role of the mathematician (thinker) as creator, as much as discoverer. "Must" reading for anyone who considers himself or herself at or near the genius bracket. I scaled my rating back to "9" because as the book acknowledges, it is the pioneer of a new genre (philosophy of mathematics) and hence it cannot deliver (on the first try) a complete answer. But what it does deliver is an extremely useful beginning.
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