Rating: 
Summary: A survey on exactly that.
Review: Along with Ivars Peterson's books on math, I suppose this has changed my life, too.I was going to study history. Math? Who cared about math? Math was for those science-types. I had an image of mathematicians as bespectacled, socially-inept, hunch-shouldered gnomes who lived in universities and ventured out of their burrows for--well, maybe they didn't venture out at all. The joke's on me. I'm a math major now. This book is one of the reasons. I've always loved history: the march of events, the ebb and flow of cause and effect and unexpected accident. I didn't realize that math, too, had a history, an ebb and flow. If I'd ever thought about it, I would have realized that an angel didn't come down from the heavens bearing The Big Book of Math, complete with proofs. But that's what it seemed like, until I read about the almost architectural building of theorem upon theorem, idea upon idea. Math wasn't a Big Book; it evolved and grew. Grows still, I should say. Did numbers exist? Well, of course they existed. Wait a second. What *is* a number anyway? How *does* one exist? Would they exist if there were no people? And so I learned that math, too, has its philosophies. Most of all, I learned that mathematicians were and are people, not gnomes in burrows who have nothing to do with the rest of the world. That math is important for more than the homework assignments that plagued my high school evening hours. That math is worth studying. If you could convey this to heaven knows how many disgruntled and frustrated math students around the world, I wonder if they might like the subject better. I sure did.
Rating: 
Summary: (probably) necessary, if not quite perfect.
Review: 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/<verb> 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.
Rating:
Summary: Immerse yourself.
Review: 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:
Summary: Immerse yourself.
Review: 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:
Summary: rare.
Review: It was about five years ago. Physics suddenly seemed fascinating but I was struggling with math. My tutor suggested two books for me. One of them was this book. I cannot say this book was particularly helpful but it gave me a good sense of what mathematics is: its people, culture, history, and philosophy. Quite unlike E.T. Bell's Men of Mathematics, this book does not contain romantically presented stories of some math heros. And unlike some popular math books by Ian Stewart, it does not attempt to explain (rather unsuccessfully) some esoteric theories. It is just as the title suggest--what a mathematical experience can be. A book of this kind is rare. P.S. Now, some five years later, I am not sure if mathematical knowledge maintains a separte existence as Plato had thought, and as the authors believe. (Ref. Plato, Phaedo)
Rating:
Summary: Math is People
Review: Math, like any subject can be studied from the point of view of conceptual constructions or from from that of the people that did the constructing. This is absolutely a people first book. If this were used in high school math, the world would have far, far fewer mathphobes. If you're a mathphobe, the cure is in these pages.
Rating:
Summary: One of the best books about math.
Review: Some books are of such depth that it is impossible to completely digest all that they contain even after multiple re-readings. Many achieve this through their level of technicality, or through sheer obscurity. The true gems are those that achieve it through clear intelligible discussion of deep concepts. Books like this point outside of themselves, leading one to whole new conceptual worlds. They force new connections to be made in the reader's brain. I reserve my highest recommendation for books of this type, and "The Mathematical Experience" is certainly one of them. Popular books such as Ivars Peterson's "Mathematical Mystery Tour" and Keith Devlin's "Mathematics: The Science of Patterns" excel at giving the non-mathematician a glimpse into the world of modern mathematics, and an appreciation of the beauty and interest found therein. Depending on the level of sophistication of the reader, some popular math books are more appealing than others, in as much as they convey more or less actual mathematical knowledge. However I would venture to guess that these works hold little interest for real mathematicians, being much too shallow in their description of modern problems, even outside the specialized field of the reader. Davis and Hersch on the other hand should strike a chord with most practicing professionals, as well as with the lay audience. As the authors state in the introduction, the layman reader may at times "feel like a guest who has been invited to a family dinner. After polite general conversation, the family turns to narrow family concerns, its delights and its worries, and the guest is left up in the air, but fascinated." We receive the same service of exposure to intriguing mathematical ideas as in other popular books, but we also get healthy doses of philosophy and history. We get glimpses of truly mind-boggling (or mind-expanding ... the authors would perhaps say that bogglification is a primary path to expansion), mathematical concepts such as the Frechet ultrafilter, the truly huge integer known as a moser, or Weiss's restatement of the Chinese Remainder Theorem which is so abstract and generalized as to defy the understanding of all but a handful of practicing mathematicians. The book tackles problems of mathematical experience which are tough because they fall into the realm of philosophy: the meaning of proof, the goal of abstraction and generalization, the existence of mathematical objects and structures, and the necessary interplay between natural and formal language, or between algorithmic and dialectic processes. What is amazing is that Davis and Hersch make these ideas not only accessible to an intelligent layman, but also interesting and vital, without (I presume), losing the interest of real mathematicians. Rather than a zoo of mathematical curiosities, the book is an anthology of essays about the practice of mathematics, with illustrations ranging from the elementary to the extraordinarily deep. I suspect that the questions "What is mathematics?" and "What does a mathematician actually do?" are rather off-putting to the majority of professionals in the field. But "The Mathematical Experience" asks these questions, and rather than giving a terse answer, takes them very seriously and fearlessly analyses them from a variety of stances. Of course, the authors don't presume to give definitive answers. They do, however, provide much food for thought ... so much that the reader is likely to come away from the book transformed. If you are not a mathematician, but a curious layman, "The Mathematical Experience" is the best place to go after you've read William Dunham, Ivars Peterson, Keith Devlin, Ian Stewart, or others like them. If you are a student of mathematics, or a student or practitioner of any other science, you'll do yourself a great favor by reading this book. If you are a mathematician who generally dislikes and avoids pop-math writing, give this a try. You may be very pleasantly surprised.
Rating:
Summary: Informative and engaging
Review: The authors deal with various important aspects of mathematics and about practising mathematics. They also deal with the philosophy of mathematics. By and large, they do it engagingly. Specifically, they tackle why mathematics seems to 'work'; how a mathematician actually goes about doing mathematics; they offer some light treatment of a few mathematical topics, and they illustrate mathematical thinking as well.
This book is best read by students thinking about choosing mathematics as a career, or even just as a field of study. Although, any layperson will come off with a greater appreciation of what mathematics is, and what mathematicians do.
Rating:
Summary: The Perfect Mix of Mathematics, Philosophy and History
Review: The best book I have read of it's type. Seemlessly incorporating Mathematics, Philosophy and History. Makes one want to really read everything in the Bibliography.
Rating:
Summary: Philosophy, History and Myths of Mathematics
Review: 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.
|
