To Infinity and Beyond: A Cultural History of the Infinite

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 in

I have read other books by Eli Maor. After "June 8, 2004", I had doubts about this one, but I wanted to clarify some Cantorian issues. Once I started this one, I could not put it down. It also answered my questions.Most, if not all of the material should be accessible to a motivated high school senior. It presents the history of infinity in a manner as fascinating as a mystery or adventure story (a true one, better than fiction); it reminds me of "Terrible Lizards" in that sense. Interspersed with the historical narrative, but easily separable, it contains good solid mathematics in a clear and concise fashion. Only the section on Bertrand Russell's paradoxes failed to satisfy.

Maor titles his book "a cultural study," but the cultural work domainates the second half of the book. The first half--which is more interesting than the second half--is a truly amazing analysis of just what the infinite is. Maor goes into detailed discussion of the nature of infinity in prime numbers, irrationals, rationals, and so on. The patterns, surprises, and mysteries of number fields are discussed with perfect clarity. Other issues involving infinity are mapped with equal precision and clarity for the beginner. The second half of the book involves studying the infinite in Escher's art, in geometric systems before and after Euclid, and in art, theology, science, singularities, and etc. Overall, for those interested in the mecahnics of nature, this book is not to be passed up!!! But be cautioned, this book is for beginners, for those only interested in grasping basic concepts of mathematics, not intense formulas that lead to singularities, for example. I am a graduate student in philosophy, so it served my purposes to the maximum level.

Maor is a great scholar! He's a professional mathematician with a deep knowldege of history of mathematics and astronomy and also a great writer. In addition, he has a deep love for music and culture. The book will give you a great sense of the diversity of mathematics. I strongly recomends all the four books by Maor!

Maor is thoroughly at home in the realm of mathematics, its history and the frequent detours into the lives of the men who have brought its secrets to light. To Infinity and Beyond is a lighter read than either e, the Story of a Number or Trignometric Delights (his two previous titles). However, this work is infinitely enlightening and exponentially chocked full of "aha's". Maor enriches the reader's understanding not only of mathematics but the culture in which it has flourished. An absorbing read.