The Mathematical Experience (Paperback)

~ Phillip J. Davis (Author), (Author) "A naive definition, adequate for the dictionary and for an initial understanding, is that mathematics is the science of quantity and space..." (more)
Key Phrases: inner issues, mathematical reality, mathematical experience, Further Readings, Selected Topics, Nonstandard Analysis (more...)

Editorial Reviews

Amazon.com Review

We tend to think of mathematics as uniquely rigorous, and of mathematicians as supremely smart. In his introduction to The Mathematical Experience, Gian-Carlo Rota notes that instead, "a mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof ... is more often than not a way of making sure that our minds are not playing tricks." Philip Davis and Reuben Hersh discuss everything from the nature of proof to the Euclid myth, and mathematical aesthetics to non-Cantorian set theory. They make a convincing case for the idea that mathematics is not about eternal reality, but comprises "true facts about imaginary objects" and belongs among the human sciences. --This text refers to an out of print or unavailable edition of this title.

Review

"A true gem, one of the masterpieces of our age." -- -- AMERICAN MONTHLY

"A true gem, one of the masterpieces of our age." -- Review

See all Editorial Reviews

Product Details

• Paperback: 464 pages
• Publisher: Mariner Books (January 14, 1999)
• Language: English
• ISBN-10: 0395929687
• ISBN-13: 978-0395929681
• Product Dimensions: 9.1 x 6.3 x 1.3 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (17 customer reviews)
• Amazon.com Sales Rank: #239,894 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

43 of 45 people found the following review helpful:

5.0 out of 5 stars A survey on exactly that., January 22, 2000

By	Yoon Ha Lee (Ithaca, NY United States) - See all my reviews

This review is from: The Mathematical Experience, Study Edition (Hardcover)

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.

18 of 18 people found the following review helpful:

5.0 out of 5 stars Excellent dialog on the development of mathematics.., February 12, 2001

By	Kersi Von Zerububbel "Kersi" (San Diego, CA USA) - See all my reviews

This book was a sheer joy to read and digest. The authors skilfully comingle history, mathematics, philosophy, and biography. The result is a truly fantastic voyage into the meaning and gist of discovery and conjecture. In chapter after chapter important ideas like Fourier analysis, Non-Cantorian Set Theory, and Objects and Structures are scrutinized in a very interesting manner.

The deeper you go into the book the more will you revel in the sheer majesty and scope of the topics. I had to read the chapter on Inner Issues twice to really get everything out of the text. Topics such as Teaching and Learning are very insightful and full of little hidden gems.

If you are prepared to expend some effort and if you wish to know what mathematics "really is like", grab this book. I am sure this will become a permanent treasure in your library and you will peruse it often long into the night.

21 of 23 people found the following review helpful:

5.0 out of 5 stars One of the best books about math., December 1, 2000

By	Shard (Birmingham, AL United States) - See all my reviews

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.