To Infinity and Beyond

Rating:
Summary: To the limits of infinity
Review: Even as children we have a vague concept of infinity, thinking of it as the largest number; remember the familiar exchange of "I dare you!" "I double-dare!" "I dare you to infinity!" "I dare you to infinity plus one!" or some such thing. Even then, we realize to some extent that infinity is not truly the largest number because there is always something bigger.

Maor gives a brief history of the concept of infinity and how it fits into the worlds of art and science. This is a generally good book although there are a couple of errors (such as when he mixes up the concepts of whole numbers and integers). Maor is good at illustrating just how big infinity is without getting either overly technical or metaphysical (a problem with the last book I read on infinity, whose title I forget). Maor also shows how there are different sizes of infinity; it is often hard to conceive that there are more irrational numbers between 0 and .00001 then there are rational numbers along the whole number line.

With the exception of the couple of minor errors mentioned above, this is a good book. Infinity is a difficult concept to grasp, but with this book, you can do just that.

Rating:
Summary: To the limits of infinity
Review: Even as children we have a vague concept of infinity, thinking of it as the largest number; remember the familiar exchange of "I dare you!" "I double-dare!" "I dare you to infinity!" "I dare you to infinity plus one!" or some such thing. Even then, we realize to some extent that infinity is not truly the largest number because there is always something bigger.

Maor gives a brief history of the concept of infinity and how it fits into the worlds of art and science. This is a generally good book although there are a couple of errors (such as when he mixes up the concepts of whole numbers and integers). Maor is good at illustrating just how big infinity is without getting either overly technical or metaphysical (a problem with the last book I read on infinity, whose title I forget). Maor also shows how there are different sizes of infinity; it is often hard to conceive that there are more irrational numbers between 0 and .00001 then there are rational numbers along the whole number line.

With the exception of the couple of minor errors mentioned above, this is a good book. Infinity is a difficult concept to grasp, but with this book, you can do just that.

Rating:
Summary: The finest generally accessible math book I have seen.
Review: 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.

Rating:
Summary: The finest generally accessible math book I have seen.
Review: 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.

Rating:
Summary: The infinite is ubiquitous.
Review: Infinity is a rare topic for discussion. Most people don't know much about it because our lives revolve around the finite. We are concerned with our careers, relationships, and finances. Infinity is usually associated with religion, making it even less mundane for most people.

Infinity is discussed more often in scientific circles. Its abstractness makes it most suitable for mathematicians. Thus it is mathematics that best describes this concept.

Maor delves into the development of the secular conception of the infinite. He begins in ancient Greece. Although the Greeks, culminating in the life of Euclid, developed geometry, their mathematics were confined to practical matters. There were few studies into the abstract. The concept of limitlessness was rejected outright. It was the Moslems and Hindus who introduced into mathematics the concept of boundlessness.

Maor's book then focuses on the Western World's gradual acceptance of the infinite, culminating in the development of calculus.

This book is divided into three sections: mathematical infinity, geometric infinity and cosmological infinity. The first section is devoted to the history of the concept of the infinite and the development of number theory.

The section on geometric infinity explores patterns in artwork. Furthermore, the infinite was incorporated into geometry, leading to the development of non-Euclidean geometries such as projective geometry, hyperbolic geometry, and elliptical geometry. These mathematical breakthroughs led to scientific discoveries, culminating in the theory of relativity.

The third section, cosmological infinity, is devoted to the history of astronomy from ancient Greece to the twentieth century. The theme is humanity's discovery that the universe is a vast and maybe infinite expanse.

To Infinity and Beyond provides to the general reader the development of mathematics, of astronomy, and of the scientific method.

Rating:
Summary: Should appeal to both mathematicians and poets
Review: Maor has written a book for both mathematicians and poets. Since he is a mathematician himself there is, to be sure, plenty of math in Maor's book. But the book should also appeal to the aesthetic side of many readers (me included) by exploring human perspectives of infinity, such as how we try to relate to the concept at a personal level, and how different people have tried to capture the notion in art and prose.
The book is arranged in four parts, dealing with the mathematical concept of infinity (how it shows up in algebra, etc.), geometrical infinity, aesthetic infinity (both art and poetry) and cosmological infinity.
The section on mathematical infinity has the typical assortment of historical examples, beginning with examples like the runner's paradox made famous by Zeno. There are also examples of infinite series that converge, including examples of how ancient mathematicians invented infinite series for transcendental numbers like pi. There's a plethora of little tidbits found throughout this section in little mini chapters that are short essays, only a few pages long, that give surprisingly succinct, tantalizing, and often delicious examples of mathematical infinity. Reading this book I was struck by what good reading it makes for any student preparing to take a class in calculus.
Some of the author's most interesting material is the author's discussions about infinite series. I particularly enjoyed his examples how the associative property doesn't hold for infinite series (a non-intuitive fact that often comes as a surprise to many new students). Ordinarily, if you have a string of numbers that are connected by addition (x1+x2+x3+..+xn) for example, you can rearrange their order and get the same result. One of the strange things about infinity, though, is that rearranging the terms in an infinite series can result in the limit of the series changing from one number to another.
Of course no discussion about infinity would be complete without mentioning Cantor, which Maor does with particular clarity for first-time readers. Indeed, this is one of the things I like about Maor best - he's written a book that is fun to read, even if you already know most of the stuff. It's engaging and entertaining, and full of "ahh" and "ohhh" even when you find yourself reading about something you studied many years ago. At the same time this is a good introductory text for anyone (I'm thinking youngsters in high school) who wants to start exploring some of these mathematical concepts, and need a friendly introductory text. If you can manage first-year algebra you have the tools you need to follow what Maor is talking about, though be advised that he doesn't shirk when producing equations, though most of the math is relegated to the appendices.
The section on geometric infinity is punctuated by nice illustrations and those geometrical shapes that you may have heard about - the ones with things like finite volume but infinite surface area. This was one of those rare occasions where I found myself wishing Maor had gone a little further. Instead of simply showing how such objects exist in mathematics, he really should have explained the apparent "paradox" (it's not hard). Instead, he makes the example more of a "paradox" than it really is by mixing metaphors in talking about "painting" the surface. Of course mathematicians have one idea about painting a surface (mathematical paint has no thickness), but the beginning reader is likely to be mostly confused - too bad, since Maor clearly has the skill to explain the trick.
Maor's exploration of the infinite is (almost) infinite. He has a wonderful section on tiling, and some brilliant plates representing some of the best mathematical art that attempts to depict the nfinite. The section on cosmology and the infinite is a nice summary of the history of astronomy and how astronomers and cosmologists have vacillated over the years between a cosmos that is infinite, then finite and bounded. I thoroughly enjoyed reading this book. It is well written and both easy and fun to read. My only complaints are rather minor. Several times Maor treats infinity as a "big number" (it's not a number at all, and he makes that clear, but his terminology on this score isn't as consistent as it should be). And, he refers to mathematics as a science. Well, I suppose he's entitled to his opinion on that one,
though I imagine it will continue to be debated. Count me as one of those who puts mathematics in the "tools" category, separate from science.
The fact these inconsequential gripes are all I can find to complain about tells you what a really fine book this is. If you love mathematics, this book really needs to be in your library.

Rating:
Summary: A masterpiece of scholarship!
Review: 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!

Rating:
Summary: What do Nothingness and Infinity have in common?
Review: 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.

Rating:
Summary: The Infinite in Nature
Review: 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.

Rating:
Summary: The Infinite in Nature
Review: 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.