Customer Reviews

The Art of the Infinite: The Pleasures of Mathematics

5 star:		(6)
4 star:		(4)
3 star:		(2)
2 star:		(1)
1 star:		(0)

 

 

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.

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.

The mathematics in this book is clear and absolutely delightful - reminiscent of high school math. The derivations, proofs, figures and equations are all very clear and the words immediately associated with them are very useful complements. The problem arises when we are in-between the mathematical expositions, i.e., where historical and miscellaneous other snipets are presented; these would normally be pleasant diversions and would make the book even more interesting. But here, this is not the case. The prose is rather obscure, complex and cryptic and tends towards the quasi-poetic, quasi-philosophical and quasi-parabolic all at once. This is most unfortunate for a math book where simplicity and clarity of expression are paramount. Had the historical and other digressions been written clearly and in plain English, then this book, in my opinion, would have easily been 5-star material. But as it is, the math is worth an easy 5 stars, the prose an unfortanate 1 star for an average of 3 stars.

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.

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.

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. The Appendix has some of the most interesting and worthwhile reading in the book.

I've criticized other authors for being too poetic with their math books. The Kaplans do it a lot, but they manage to do it in a way that doesn't interfere with understanding the key mathematical concepts.

Needless to say, I highly recommend this book for anyone who enjoys mathematics, and wants to brush up on what numbers mean, how we invented them, and how to have fund with counting - and a whole lot of other stuff.

This book trys to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don't. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don't know any and don't have a natural talent for it, you will find it very dense or impossible. Being somewhere in the middle I skimmed thru most of it and slowed down when it got interesting. If you have only a little time I would suggest the last chapter 'The Abyss` about Georg Cantor and transfinite arithmetic.

At points they wax philosophical and ask the perennial question: is math is out there in the world or in here in our heads. Why not ask this about art or music or literature or computer programs or philosophy itself? In a very general way math must come from the same place that words and ideas and images come from---our brain evolved to make them and they must in many ways(every way?) reflect the structure of our brains, which reside in our dna which was shaped by natural selection which was shaped by the geology of the earth and the structure of our universe which comes from particle physics which comes from the laws of nature which are just there.

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.

1. On page 98, the authors mis-define what they call a tower, defining it as x raised to the x-th power, then that raised to the x-th power, and so on and so on. In other words, if f(n) is defined to be the value of the tower after n steps, then:

f(n+1) = f(n)^x, for every positive integer n.

However, with that definition, and x= sqrt(2), the tower's terms quickly diverge - A spread sheet shows that f(20) is greater than 10 to the 108-th power. Instead, they should have defined the tower by the rule:

g(n+1) = x^g(n), for every positive integer n.

With that definition, and x = sqrt(2) the series {g(n)} converges to the number 2, as they claim. Indeed, a spread sheet shows that g(20) is approximately 1.999586. Also, with this new definition, the book's proof is legitimate, because the proof used the rule:

x^y = y, where y is the limiting value of the tower as n approaches infinity. That equation is not true with the book's definition of y, because it uses f(n) instead of g(n).

The heart of the matter is that the operation of taking successive powers is not "associative", even when the sequence is finite. For example,

(3^3)^3 = 27^3 = 19683, and that is not equal to

3^(3^3) = 3^27 = 7.6256E+12. Associativity fails!

2. The first term in the equation at the top of page 97 should be 1 - 1/2,
not (1 - 1/2)/2.

George Monser

Like a good mathematical proof, the work is clearly written, insightful and elegant. The book gives a history of how infinity is understood by mathematicians. The last chapter on one of my mathematical heroes, Cantor, is worth the price alone, in my opinion. It gives the clearest version of Cantor's diagonalization proof I have yet encountered. Diagonalization proofs since then have played a very important role in modern math in the works of Godel and Turing, e.g. Not only does the book accomplish this expository feat but it does so while interweaving the math with a very touching biographical sketch of Cantor showing the beauty and depth of his thoughts, his personal struggle with his colleagues and his struggle with trying to prove the impossible (and possibly the most interesting question in all of mathematics, the Continuum Hypothesis). We now know that this problem he struggled with throughout much of his adult life and which brought him misery and maybe even to the brink of madness is neither provable nor disprovable (independent) within mathematics (thanks to the work of Kurt Godel and Paul Cohen). The previous chapters are likewise entertaining and informative (for me at least).

I only wish that the book went further into contemporary times and gave an exposition of the work done by mathematicians such as Cohen and Hugh Woodin on the Continuum Hypothesis. But that is probably asking too much of the authors as this is a very specialized area and outside of their expertise.