Customer Reviews

To Infinity and Beyond: A Cultural History of the Infinite

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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 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.

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.

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.

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.

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!

This 235 page book attempts to place the concept of Infinity within the cultural realm. To accomplish this, the author has to first establish what infinity means and then to show how it was used in such divers arenas as art and astronomy. Therefore, the book is divided into sections. In the first section, therefore, we focus on the arithmetic meaning of infinity. This is an excellent explanation of the concept and with the stretching of the information in to Calculus serves as a good way of introducing people to why Calculus was developed. Certainly this is a more informative way of looking at it than what is typically taught in the normal high school math curricula!

From arithmetic we move to geometry and there are introduced to the way that the concept of inifinity allows the mathematician to create interesting geometrical constructs including such things as non-euclidean geomtries (plural!!!). This part of the book can be a bit dense and even the inclusion of a practical example of the creation of Mercator projections of the world's map do not help much.

Next we move to the realm of art. Here the author expresses his admiration for the work of the Dutch artist Escher and uses several of his prints as examples. These prints are great and fun illustrations of how one moves from infinity to center stage and back to infinity and as an admirer of Escher's works myself, it's fun to read about it in such glowing terms. However, since Escher himself claimed to not understand any of the mathematics behind his inventions, it is somewhat puzzling why the two are interpolated here.

Finally, we move on to deal with astronomy. Since astronomy is the science that deals with very large numbers and concepts, I suppose that is appropriate. But, at this point the book moves away from the mathematics and becomes a somewhat straight-forward recounting of astronomical history. This is interesting but it is not clear to me how the concept of inifinity really applied despite the somewhat tortuous attempts the author makes.

From very large distances we go to the infintesimal when the author spends one, two page chapter on the atom. This is clearly an attempt to be all inclusive and does not work - in my opinion. We are already past the 200 page mark when this happens and one has to ask why other topics deserve such long descriptions, but sub-atomic physics gets only a paragraph or two?

In any case, this was an interesting survey of various topics that seem to be connected through the concept of infinity. It will probably not teach you too much, but will also illuminate some dark recesses of the world's thoughts, so it is probably worth a quick read.

I have liked everything written by Maor and this does not fail either. Very interesting weave of math and art with some new angles on old methods of looking things. A keeper.

This book should be accessible to middle school children, there is nothing about the subject matter or the proofs in the book that is beyond their mathematical reach. So I bought this book for my kids who are very bright in mathematics. Unfortunately, the writing is horrible. Why would you write a popular science book and target it at readers in the middle of college?

I ran the first six paragraphs (1000 words, 6Kb) of the introduction of this book through the AT&T Bell Labs "Writers Workbench" to support my thesis that the writing in this book is needlessly complex. Here is an objective report on the reading grade level, using 4 very-widely accepted formulas for reading grade-level :

readability grades:
(Kincaid) 13.8 (auto) 14.0 (Coleman-Liau) 10.3 (Flesch) 13.8 (44.9)
sentence info:
no. sent 36 no. wds 981
av sent leng 27.2 av word leng 4.62
no. questions 2 no. imperatives 0
no. nonfunc wds 537 54.7% av leng 6.11
short sent (<22) 39% (14) long sent (>37) 17% (6)
longest sent 76 wds at sent 30; shortest sent 7 wds at sent 16
...

So by 3 out of 4 measures, the book is appropriate for somebody in 14th grade, i.e. a sophomore in college. What a total waste !!

It's the last time I EVER purchase something from Birkhauser Academic books of Boston, the original publisher. Birkhauser takes all the leftovers that Addison Wesley of Massachusetts won't publish. The Birkhauser books are usually overly technical for the subject matter they present, whereas Addison Wesley books are a paragon of simplicity. Even though the current edition is published by Princeton University Press, the original version was published by Birkhauser, and that should have been my tip-off that this is not a well-written book.

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.