The Art of the Infinite: The Pleasures of Mathematics

Rating:
Summary: As complex as math can be.
Review: As if math wasn't complex and confusing enough, a book with equally confusing english was written about it. With out bragging, I am fluent in mathmatics; I understand it as if it were my primary language. What I am not fluent in is English, and unfortunately this book was written only for the English elite. 1/4th of the time I understood half of the poetic correlations between mathmatics and philosophy described in this book, which, consequently happens to be 3/4ths of the context. Basically, if you understand mathematics as well as I do, but do not understand poetry and philosiphy well, do not read this book, you're well off where you are. If you do understand English, extreemly well, and want to know more about mathematics, read the book. But if you could care less about mathematics, or english, then don't even read this review.

Rating:
Summary: Is the Prose Delightfully or Excessively Rich?
Review: 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.

Rating:
Summary: Is the Prose Delightfully or Excessively Rich?
Review: 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.

Rating:
Summary: Infinite Delights?
Review: Here's human imagination at work. The flights of fancy the Kaplans show us are not about dragons and wizards, but about imaginary numbers, square roots, triangles, and infinite series.

I bought this book to mine for ideas to use in the notes I am writing to accompany the Third Edition of Geometry by Harold Jacobs, and I struck a rich lode. My professional interests made me look at material of a more technical nature, such as the proof of the theorem of Pappus. Pappus noticed that if you take six points A, B, and C on one side of an angle and a, b, and c on the other side of this angle and join each point to the two points labeled by *different* letters, then the three points of intersection of these six segments lie on a straight line. I knew this as a fact since my high school days, but it is not easy to give a proof that is reasonable at that level. The Kaplans have a beautiful explanation of this result, putting it in context and giving a gentle proof. Very nice indeed.

They have found just the right diagram or line of argument for many things I have seen before. Those of us who have suffered through the terrors of trigonometry will remember that there are some angle sum formulas, though we may not remember exactly what they are. The diagram at the top of page 187 tells you why these formulas are true and will make them unforgettable, if you decide to remember it. The path to this figure is made easy and natural in the book. What was new to me was the idea of adding a box around the tipped triangle --- suggested in the throw away line at the top of page 186. This gives us just what we need, neither too much nor too little.

One virtue of this book is that you can leaf through it and dive into the text wherever you see an interesting illustration or some idea you have been wondering about. The topics are mostly self-contained and there is always a nice story or bit of historical context to give you a sense of where you are and how this fits into the larger picture.

Buy this book, browse it, read it, and now and then get out your paper and pencil and puzzle through whatever tickles your fancy. This book is not just *about* mathematics, it gives you the real stuff.

Highly recommended.

Rating:
Summary: The proof of a.0 = 0 is incomplete.
Review: In the proof on page 40 of a.0 = 0,

Line 1: a.0 = a(1-1)
Line 2:.......= a - a
Line 3:.......= 0

since (1-1) is shorthand for 1+(-1), distributivity only yields

a(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.

Rating:
Summary: Interesting but could be written more clearly
Review: This book covers some very fine topics in math. It attempts to balance mathematical rigor with analogies and interesting historical points. The attempt however is not totally sucessful because the language used is too obscure. The mathematical topics discussed are complex enough by themselves and the additional obscure language makes them that much harder to understand. I would have vastly preferred the use of stright forward English for the discussion. The analogies and historical facts could have been presented separately alongside the main discussion. Nevertheless I enjoyed reading it and will recommend it to others as long as they have a good command of English and are willing to go along with the less than ideal presentation.

Rating:
Summary: The proof of a.0 = 0 is incomplete.
Review: 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 more complicated still, the final chapter has to do with Cantor's work. Common sense tells us there must be half as many even numbers as there are whole numbers, but Cantor showed that the infinity of both was equal. He showed that the infinite number of points in a line as long as your finger was equal to the infinite number in a line as long as from here to the Sun. In fact, the number of points on a line is equal to the number of points in a plane. And yet, some infinities are bigger than others. This is strange territory indeed, and requires some concentration to understand and enjoy, even with the Kaplan's literate, witty, and clear explanations. This is a fine introduction to different aspects of serious mathematics; true to its subtitle, it is a book full of pleasures.