To Infinity and Beyond: A Cultural History of the Infinite (Princeton Paperbacks) [Paperback]

I had just finsihed reading Maor's "biography" of e and looked up on Amazon to see if he had produced any other mathematical delights. I found this book on infinity - it is truly excellent. I had not expected "e" to be surpassed, but for me this book does it. What is really nice is that I can understand most of it - unlike Rudy Rucker's book on the same subject which I'd really like to fully understand but don't. Having read Maor's "infinity" I'll return to give Rucker's another go.

It should be noted that this book was written in 1987, several years before Maor's book: "e- the Story of a Number" - [the computer rendered image on the cover may belie the age of the book]. As I write this I am a second year mathematics undergraduate; I read "To Infinity and Beyond" during the summer. I was first introduced to Cantor's levels of infinity in this book, (extending countability and uncountability to aliph values, and that in effect bijective functions could map one set to another providing it has the same aliph value. First learning this principle felt like an eery truth of reality of high subtlety and high information. All of the great ideas have the eery convincingness that overcomes initial non-intuitivity... either non-intuitivity or that people before missed all together that which was sprawled in front of them). There was a rewarding explanation of the conception of non-Euclidean geometry, which to my shock but not to my surprise, was yet another fundamental idea that came from the mind of Gauss. Tragedy is touched upon as the tale of the Italian scientist, (he probably would have been called 'Philosopher' back then), named Bruno was burnt to the stake for his beatiful imagination in conjecturing that the universe is truly infinite and that there are infinitely many other worlds of creatures like us. The final chapter fills the reader with inspiration as Maor explains our place in the universe and ends with the comment, (not verbatim): "Perhaps the greatest question remaining to human enquiry is whether we are alone."

Israeli mathematician Eli Maor's beautiful book came out in 1987 and has remained in print ever since. The reason is simple: it is authoritative yet accessible. There are numerous graphs, drawings and equations; but the focus, as the subtitle expresses it, is on the cultural history of the infinite.

The book is divided into four parts for four types of infinity: mathematical, geometric, aesthetic, and cosmological. The highlight of mathematic infinity has to be Georg Cantor's discovery and demonstration in the 19th century that there are hierarchies of infinity--that is, that some infinities are larger than others! Cantor's proof is most amazing and indeed one of the great triumphs of mathematics. What I found fascinating about geometric infinity is tessellation, which is the art and science of laying geometric patterns on a surface, such as squares, triangles, circles, etc. Probably the best known and most delightful expression of aesthetic infinity is in the work of M. C. Escher. Maor includes a number of Escher's drawings and paintings including five pages of color plates in the middle of the book. As for cosmological infinity, well, physicists and cosmologists shy away from infinity, of course, but it is impossible to think about the cosmos without having our notions tinged with the infinite. After all, it is hard to escape from the idea that the universe came from nothing or has always been. If it's always been, then that is infinity; and if there was once nothing, for how long was there nothing?

Maor adorns the text with numerous quotes about the infinite from scientists, mathematicians, artists, and others. William Blake's beautiful

To see a world in a grain of sand
And heaven in a wild flower,
Hold infinity in the palm of your hand
And eternity in an hour.

appears on pages 95 and 137. Perhaps the quote I like best for its simplicity is this very ancient one from Anaxagoras: "There is no smallest among the small and no largest among the large; but always something still smaller and something still larger." (p. 2)

Which brings me to two ideas about infinity. First, as Maor informs us, infinity is not a number, but an idea. The second is the strange disconnect that exists between the idea of infinity in physics and in mathematics. Again as Maor notes, in mathematics the idea of infinity is right there inescapably at the very beginning since there is no end to the integers. "One, two, three--infinity" so said George Gamow, and so it is unavoidably true. But in physics there still exists something like a horror of infinity so much so that should an infinity come up in the equations, that is considered a sure sign that something is wrong! Indeed, if I am reading the frustrating history of string theory correctly, it would appear that physicists are more comfortable with notions of upwards of 11 dimensions than they are with infinities.

The problem I think is that, although the mind of humanity cannot avoid the idea of infinity, in the physical world about us there is no proof of anything infinite. The grains of sand can be (in theory) counted. So too can the stars--well, maybe. Contrary to what is often thought, physicists insist that energy and matter, time and space do have a limit to their divisibility--Planck's limits. But I am guessing that even the carefully construed quanta of modern physics may prove to be divisible in ways at present incomprehensible to humankind. It wasn't so many years ago that it was thought that nothing existed beyond the Big Bang universe, or at least it was not considered "scientific" to speculate on such matters. Now we see eminent scientists speaking of a possible infinity of parallel universes, worlds (forever?) beyond our ken.

Maor presents an appendix in which Euclid's proof of the infinitude of prime numbers is given along with proofs that the square root of the number 2 is irrational and that there are only five regular solids. Included are technical discussions of seven other topics. Clearly this is a book that has appeal for both the professional mathematician and the layperson alike. It is a beautiful and fascinating piece of work.