The Art of the Infinite: The Pleasures of Mathematics

Fifty years ago, if you were to randomly select a book from the mathematics section in the library, it most likely would have been uniformly colored grey, or some other neutral/dark hue, with a drab but utilitarian title in the language of professional mathematicians. Well, things certainly have changed. The standard grey hardbacks have given way to covers filled with color, while the utilitarian titles - boring in their simplicity - have given way to poetry and hyperbole that would make a thespian blush. In days past the Kaplan's book would have been called "introduction to number theory. Now, it's called "The art of the infinite." I'd have called it "number theory set to poetry, with story problems." I selected this book because I thought it might have something to do with infinity. After leafing through it, though, it was immediately apparent that it covers lots more than just the "infinite." I can imagine conversations between the Kaplans and their publisher. Publishers are fond of telling science/mathematics authors that most people won't buy a book with lots of equations, and that they needed to make the cover snazzier by including a catch word like "infinite," or something like that. Robert and Ellen Kaplan have written what turns out to be a first-rate book, showing that it's possible to make number theory understandable and very interesting. It's particularly fun the way they make frequent use of mental or mathematical "experiments," to tune "intuition" as a means for solving mathematical problems. While this style may offend or at least annoy pure mathematicians, others will see in their examples key insights into how the human mind works through mathematical problems, and how learn. The Kaplans are both accomplished mathematicians, but they are also excellent teachers. The authors used geometry and pictures to show how to construct the counting numbers, the set of integers (positive and negative), the rational numbers, the real numbers, and finally complex numbers. The interesting thing about this book is that the reader learns all this stuff while having fun with some of the most interesting mathematical asides you can imagine. Yes, infinity does enter into the book. Again, the Kaplans do a masterful job of describing the mathematics of sets. It's a common misconception that infinity is a number - many (most?) people don't understand that it's a quality of sets. You will, though, after reading this book. The book is chuck full of diagrams, and plenty of equations, too. It's an easy book to understand (for the most part) but it's not for intellectual slouches, either. Mostly, I found the explanations to be clear and understandable, with the exception of the chapter that deals with perspective. I was able to glean new concepts from the chapter, but I think I would have been lost, had I not already understood the subject fairly well before I read the book. When you get to the end, don't stop reading. Th

This is an excellent tome... entertaining. Written with whit and charm, it gives one pause for thought and contains a lovely subtle humor... which is too bad for the authors as this dooms the book to wide rejection from those who are still in need of redrafting their sixth grade expositions on 'Where The Red Fern Grows'... too bad, too bad. Now, will those of you who are playing in the match this afternoon move your clothes down onto the lower peg immediately after lunch, before you write your letter home, if you're not getting your hair cut, unless you've got a younger brother who is going out this weekend as the guest of another boy, in which case, collect his note before lunch, put it in your letter after you've had your hair cut, and make sure he moves your clothes down onto the lower peg for you... ok?

As you can read from other reviews, this book rates 5 stars for its excellent description and illustration of many fascinating topics in mathematics. Not all readers, in contrast, will appreciate the authors' most unusual prose style. At times they can't seem to write a sentence without a metaphor, and often a startling or even madcap one. Allusions, philosophical insights, snatches of poetry and unusual quotations, verbs that wriggle or hop--they are all crammed together. So at times the mathematics seems a good deal easier to handle than the prose.I was at first tempted just to dismiss this style as mere overwriting, but as I read further I started to see that it nicely fit the remarkable turns of thoughts of the master mathematicians as they tested their brains on the challenges of number and space. The more-than-quirky prose, including its philosophical and quasi-religious asides, definitely adds to the interest and instructiveness of the book, I finally decided.This book is, as you can imagine, far more absorbing than the school math most of us were subjected to. Five stars.

In the proof on page 40 of a.0 = 0, Line 1: a.0 = a(1-1) Line 2:.......= a - aLine 3:.......= 0since (1-1) is shorthand for 1+(-1), distributivity only yieldsa(1-1) = a[1+(-1)] = a.1+a(-1) so that going from Line 1 to Line 2 implicitly assumes that a(-1) is equal to -a, which has not been previously established from the axioms.

We all take our pleasures where we find them, and everyone is different, with different sources to draw upon. It will seem peculiar to many people that others could take pleasure in mathematics. Children usually learn to be bored or frightened by math, but there isn't any reason for this, other than incompetent teaching. As an attempt at remedy, husband and wife team Robert and Ellen Kaplan in 1994 began the Math Circle, Saturday morning sessions for kids who just wanted to find out more about mathematics. (The sessions were changed to Sunday morning when soccer practice interfered). Some kids (especially those who were pushed into the classes by their parents) dropped out, but some have come back, year after year, and the Kaplans have found that posing questions, inviting conjectures, asking for examples, and even suggesting ways towards proofs can be something children can enjoy. Mathematicians have been telling us for centuries about the beauty of the objects and systems that they have explored. The Math Circle seems to have taught math in a way to at least some kids who have caught the spirit of the quest for mathematical beauty. In _The Art of the Infinite: The Pleasures of Mathematics_ (Oxford University Press), the Kaplans have put some of those lessons into book form, concentrating on infinities of various kinds. This is a book for adults, or kids who hanker to think about math like adults ought to, but it is full of a sense of play.As you might expect, things start simple and get very complicated, and this is true right off in the first chapter, considering more and more complicated numbers. The Natural Numbers are introduced with patterns, as if you had stones to position on a table. 1, 3, 6, and 10 stones make pleasing equilateral triangles, and 1, 4, 9, and 16 make pleasing squares. We move from these to zero and negative numbers: "Certainly zero and the negatives have all the marks of human artifice: deftness, ambiguity, understatement." Are these numbers invented or discovered? The profundity of this question is plumbed throughout the book. Rationals, irrationals, and finally the complex numbers are all included. As the numbers mount up, the irregularity and regularity of the primes is considered, one of the most fruitful arenas of number theory. Euclid had to make an assumption about the infinite, his famous fifth postulate; but it is only an assumption; assuming that parallel lines meet eventually produces also a worthy geometry that tells us much about how the Einsteinian universe works. But there is no need to look into these strange worlds to find wonders; before leaving Euclid's terra firma, we are reintroduced to the triangle, and are presented with some astonishing revelations of secret points within and around the simple three sides that will remind you that no matter how simple things look, or even how simple things are, everything is more complicated than you can imagine.And if you want your infinities