To Infinity and Beyond: A Cultural History of the Infinite

by Eli Maor

Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement. "Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates."--Los Angeles Times "[Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama."--Choice "Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics."--Science

Genres: Mathematics, Science, Nonfiction, History, Art, Education, Stem

304 pages, Paperback

First published January 28, 1986

Reviews

[Dennis Littrell · Rating 5 out of 5]

Splendid exploration 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 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--the Planck 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.

--Dennis Littrell, author of “The World Is Not as We Think It Is”

[Anthony O'Connor · Rating 3 out of 5]

Pretty good book in some ways. Could have been spectacular.

The author looks at how concepts of infinity arise naturally when looking at numbers and geometries. He looks at how it is depicted in visual arts and music. There are some great Escher pictures.
He points out some of the peculiar properties of infinity. Culminating in Cantor's so called 'naive set theory'. And then finally has a look at the infinite in astronomy and cosmology.
Full of interesting detail, beautifully written, and entertaining. Though for the most part very elementary. But with a few nice mathematical elucidations.

It falls short in four annoying ways.

Firstly the author keeps digressing into extensive explanations not relevant to the topic at hand. This is not the place for a history of math and science. Keep on topic.
Secondly his description of the Big Bang contains the usual error. Primordial localized body!!! Please.
Thirdly with sets he goes as far as Cantor and then stops. And even the appendix on set theory after Cantor is brief and inadequate. No mention of transfinite ordinals or large cardinals. Surely this is a must in a cultural history of the infinite to 1991 ( when published). If you want to stop it at 1900 you should say so in the title. And Large Cardinals are the deepest and most exciting thing we know about 'infinity'.
Fourthly his 'philosophy of math' - on the significance of the independence of AC and CH, and inevitably Godel's theorems - which he should have just left out is just so much simplistic nonsense. Stick to your lane.

So what should have been, and started out as, a beautiful book was ultimately a disappointment.

[Laura Buckby · Rating 5 out of 5]

This book is exactly what I needed for my thesis "What is the importance of infinity in mathematics"
It includes little quotes on infinity too, I might use a few of those as well!

Brilliant.

Now for everyone who doesn't like mathematic (probably most of you) i dont want to bore you with a review; so have this as a little thanks for putting up with me,

[Remo · Rating 4 out of 5]

Esta obra nos cuenta las distintas visiones y tratos que ha tenido por nuestra parte el concepto del infinito a lo largo de la historia. Cantor, Gödel, Escher, Bach... son muchos los autores tratados y muchos los datos y detalles: Teoría de grupos, sucesiones, series... La bibliografía es de las que da ganas de leer más. Muy buena, igual que el libro.

[Eliot Long · Rating 3 out of 5]

It's a fine book for some general knowledge as it covers some pretty random subjects, from theology to M.C. Escher to space and math, but if you're looking for a more in-depth look at a subject, I'd consider another book; depends on your intent going into the book. However, even for a general knowledge book, I thought that there were somewhat useful sections (like the sections on Cantor, infinite series, some functions, or astronomy) but also some that seemed out of place in this book. Some portions also tended to drag on in my opinion, such as the art section. Overall fine, but there are better books out there that cover the same topics.

Edit: I will say, the appendix does provide some in-depth explanations/proofs, mainly on the math, but not really on topics beyond the chapter contents. Also, the quotes that Maor includes are a nice addition, I'll give him credit for that ;)

[Adrian Manea · Rating 3 out of 5]

As some other readers have pointed out, it's a good book that could have been great.

Some chapters are really enjoyable and partly unknown to me, the poetic excerpts and quotes are much appreciated, but some subjects are too roughly sketched I think. Yes, it's a book of popular science and (scientific) culture and as such, most likely its purpose is mostly to spark curiosities and invite readers to a more thorough study. And it does that quite well.

But my preference at least would have been a bigger book with perhaps less chapters but lengthier.

A good read, nevertheless.

[Robert Woodman · Rating 3 out of 5]

Don't read this book expecting to be entertained. It IS entertaining in places, but it is meant to be informative and explanatory. Eli Maor is a well-known historian of mathematics. In this book, he tackles the history of infinity as a concept in math, science, and eventually culture. I somewhat enjoyed reading this book, but it was a bit tedious in places, leading me to give it a score of 3/5 instead of something higher.

[Bill Ardis · Rating 3 out of 5]

Good discussion on infinity, some parts I found more interesting than others. My biggest interest was the mathematical history (part 1) and while it is brief, it was a good starting place. Parts 2 & 3, were ok, did not have as much interest in them, did like part 4.

[Paige McLoughlin · Rating 4 out of 5]

Intro to infinity from the Greeks and Kabbala to astronomy to cosmology to math theorems Cantorism and paradoxes of infinity and history of this key concept. I have seen most of it in other places but others may not have.

[Caroline · Rating 4 out of 5]

For a math book, it actually held my interest. A lot of the book is about art and the study of space, as well as the history behind it. It was pretty cool.

[John Badger · Rating 5 out of 5]

Well written, quite diverse, and covers a lot of commonly known material now as well as less well known stuff. Made me appreciate eschers art

[Tim · Rating 4 out of 5]

In organization and character this earlier work of Maor's foreshadows the better edited Princeton published "e: The Story of a Number." It's worth the aside to note that this work by comparison with that later volume demonstrates the sincerity of the rich thanks authors extend editors in prefaces. A top-notch editor clarifies an author's voice the way heat clarifies butter leaving no thermal trace in the final product. That said, this is a finely felt, thoughtfully designed, and beautiful work that largely carries through in clarity an exploration of mathematical, geometric, aesthetic, and cosmological infinity. It is a work replete with the words of others as well as Maor's own. The wide range of thinkers and artists who have reached for the infinite and the indivisible is itself a striking pattern that appears in these pages. Indeed, in an important way this work is about the intellectual and emotional kinship we may all find in reaching both within and beyond ourselves to the extremities of rational thought, aesthetic vision, and the physical world. In that kinship Maor opens obliquely as well the portal on the infinite that Kant and many others found in the moral law. Concern with Truth and Beauty carry his account of infinity, as one might expect of a mathematician. But his frequent allusions to various intrusions of infinity into Judaism, Islam, and Christianity offer at least a cultural lens on the Good as well. In the end this is a work uncertain of conclusion but convinced of enjoyment in the endless avenues of inquiry open to an active mind.

[Ann · Rating 4 out of 5]

This book is a bit of a challenge if you are rusty of math but Maor does a good job of presenting the material with interesting footnotes (mostly historical in nature) and proofs in the appendix. He addresses infinity in several categories: mathematical, geometry, art, and cosmology. The first two sections are basic background for the latter two. There is a whole chapter on M. C. Escher. This is an accessible and interesting treatment of the topic. Highly recommended for those interested in infinity and the historical, philosophical, and cultural connections.

[Tiffany · Rating 4 out of 5]

A discussion of the mathematics/discoveries of infinity, and how the concept of infinity has affected religion, art, music, and science. Being me, I especially found the math and art portions interesting, especially because there are a whole lot of references (and even a chapter) to M.C. Escher.

A high enough 3 stars that I gave it 4 stars, although it might have really been more like 3.75 stars.

[Marcia · Rating 5 out of 5]

A very readable exposition on history of the concept of infinity. Excellent for the layman and still satisfying for the expert.

[Chris Lawnsby · Rating 4 out of 5]

Pretty interesting book about infinity. Less interesting than "Fermat's Enigma" but more interesting than "The Golden Ratio." I'd recommend it only if you are interested in numbers.

[Kevin]

Will still has this, I want it back.

[Michael · Rating 4 out of 5]

An excellent historical overview of the concept of infinity.

[Kate · Rating 4 out of 5]

georg cantor is cool.

[Zac · Rating 4 out of 5]

a decently written book that still holds MOSTLY true considering it was written in 1987. good concepts, well explained.

[Richard · Rating 3 out of 5]

History of infinity. Okay, tails off at the end.

[Dave Nichols · Rating 3 out of 5]

Pretty good for a rather boring subject. Lots of common stuff, a few trinkets here and there. There were a few areas I pretty much just skimmed through.