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Title: The Mathematical Experience by Phillip J. Davis, Reuben Hersh ISBN: 0-395-92968-7 Publisher: Houghton Mifflin Co Pub. Date: 14 January, 1999 Format: Paperback Volumes: 1 List Price(USD): $19.00

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Average Customer Rating: 4.92 (12 reviews)

Rating: 5
Summary: Philosophy, History and Myths of Mathematics
Comment: The Mathematical Experience by Philip J. Davis and Reuben Hersh
1981 Houghton Mifflin Company, Boston

Is all of pure mathematics a meaningless game? What are the contradictions that upset the very foundations of mathematics? If a can of tuna cost $1.05 how much does two cans of tuna cost (Pg. 71)? If you think you know the answer, don't be so sure. How old are the oldest mathematical tables? What is mathematics anyway, and why does it work? Can anyone prove that 1 + 1 = 2?
This is a book about the history and philosophy of mathematics. I'm certainly not a mathematician, and there are parts of the book I will never understand, yet the balance of it made the experience well worth while. The authors presented the material so that it is interesting and (mostly) easily understood. They have a creative way of making a difficult subject exciting. They do this by giving us insights into how mathematicians work and create. They live up to the title making mathematics a human experience by adding fascinating history. Frankly I was shocked when they pointing out how even mathematicians have made questionable assumptions and taken some basic "truths" on faith. They show the beauty of math in the "Aesthetic Component" chapter. Ultimately the question that comes up again and again is the question of whether or not we can really know anything about time and space independent of our own experience to make an adequate foundation for a complete system in mathematics. If you have ever wondered about the world of mathematics and the personalities involved you might consider this book. If you are a mathematics teacher you should read this book. If you are a mathematician you could find it quite unsettling.
It contains eight chapters, each one broken up into many subtitles so if you do get bogged down in the mathematics it isn't for long. There are 440 pages. I'd like to see a much more complete glossary for people like me who need it.

Rating: 5
Summary: Immerse yourself.
Comment: Back in the early 90's when I was an almost-penniless mathematics student I was standing in front of a bookshelf in my local bookstore and had to choose between this and Gödel, Escher, Bach. I chose this book and I still don't regret it. [I have also subsequently bought GEB :-)]
Driven by their obvious love of the subject, the authors do a credible job of tackling just what it is about mathematics that makes mathematicians love it so much, often to the bafflement of the rest of the world. A particular personal favourite is the series of four conversations between an "ideal mathematician" and, respectively, a University Public Information Officer, a philosophy student, a positive philosopher and a sceptical classicist.
I would recommend this book to students of mathematics at any level beyond the elementary, especially those with an interest in the foundations of their subject. The authors do however acknowledge that some parts of the book will seem alien to the layman.

Rating: 4
Summary: (probably) necessary, if not quite perfect.
Comment: As has been mentioned in the other reviews, this book takes the humanistic approach to mathematical philosophy, and the heuristic
approach to mathematical method. It does a very, very good job of presenting engaging and accessible accounts of many "advanced" topics, such as finite group theory and the forcing method. In a way, the ease with which they present these items might mislead the reader into taking them as much simpler or more superficial overall than they really are , but this is dealt with by a very liberal sprinkling of superlatives like "only a small handful of mathematicians understand X". Now this is the situation in pretty much every "popular math" book I've ever read (admittedly, not nearly enough), but here it helps to characterize my sole qualm with this book and the reviewers who praise it: overcompensation.
[can you tell yet that this is going to be another incredibly opinionated review?]
Basically, the situation is this:
The way math is presented to the general public is unsettlingly dogmatic. Sure, there's calculation, a little heuristics (mostly at around calculus level, if our average Tom or Mary can stand hanging around this long), but for the most part it's just "here's how it is: ..."
But *why*? And with this word must lie the beginnings of every mathematician's career. One simply cannot create mathematics, or even appreciate mathematics as a creative endeavor, without first digesting the fact that these amazing laws that we've been handed
and expected to just "believe and get on with it" have actually been created/discovered (to choose one is really just a matter of semantics) by real people just like you and me (assuming you're a complete weirdo who likes to make too many parenthetical remarks like me....).
And this is a great endorsement, to the intelligent general reader, of the above view. The only problem is that it overcompensates for the dogmatic status-quo. I probably would have just expected to take this with a pinch of salt (just as I expect my opinions to be taken), but apparently there's a good chance readers will come away with the unrealistic notion that mathematics can be studied just as well by studying the people who create it. I mean, sure - those budding math-ites who do this *will* have an advantage over those who don't (all other things being equal), but if you really want to *do* math (and this is where all the fun is!) you really have to get some serious problem-solving skills, and to learn anything of substance from within the last century you're going to end up having to read some very terse books indeed (*cough* Bourbaki).
This overcompensation also presents some philosophical difficulties. I completely agree that the "standard four" philosophies of math (formalism, intuitionism/constructivism, positivism, and platonism) leave something to be desired in that they neglect to account for the *huge* role played by society, and to varying degrees they neglect the role played by heuristic methods in both individual and social contexts. And I agree that any serious philosophy of math must take a *lot* of input from historical/biographical data. But one can go too far with the "social construct" idea of math, and this is done here. The "mere" fact that we are able to construct/discover/ the mathematics that we do and use it to interact with nature in the way we do is simply not trivial. I don't find it implausible that the authors might agree with this, but it's not a point emphasized enough here. You simply can't go out there and do whatever you want and expect it to work like mathematics or science. And a huge part of why both are the way they are today is because of increased emphasis on rigor. While the main advantages of this are increases in both precision and versatility of expression (that's right, rigor can *aid* creativity - just look at the work of Grothendieck!), there is something that has to be said about the objectivity of mathematics. It's true statements are really true, and in a way that largely generalizes our everyday notion of truth. But in many ways it's more - as one might overhear a mathematician say, it "has more structure". It's something in-between the trivial truth of grammatical rules (and other such stipulations) and scientific truth, which is a more faithful generalization of the everyday notion. It's difficult to define and relate all these notions of truth exactly, and that's just because they're not exact terms. In fact, most words aren't. This doesn't mean they're anything less than they were before - it just means that we've learned something new about them (an analogy due to Wittgenstein: solid materials are still solid, even though we now know that they're composed of discrete atoms connected together by force. We have simply learned something new about what it means for something to be solid).
Anyway, all this isn't explicitly negated in the book, which I'll say again is really great. Buy it, but think carefully about it. Philosophy is entirely about critical thought, even though mathematics isn't.

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