Product Description

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

Product Details

• Amazon Sales Rank: #237101 in Books
• Published on: 1991-07-09
• Number of items: 1
• Binding: Paperback
• 304 pages

Review
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 )

Fascinating and enjoyable . . . [P]laces the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics.
(Science )

Review
Fascinating and enjoyable . . . [P]laces the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics.

Customer Reviews

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--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.

Infinity vs. indefinity
Apparently Mr Eli Maor somehow missed the 20th century French educated mathematician and metaphysician René Guénon's book, translated from the French original, The Metaphysical Principles of the Infinitesimal Calculus (Guenon, Rene. Works.) in which this author succinctly clarifies some basic questions of mathematical vocabulary, thus resolving apparent theoretical difficulties of calculus.

One of Guénon's central themes is the difference between (mathematical) indefinity, often inappropriately referred to as "infinity" by e.g. Mr Maor, and absolute or metaphysical infinity, a quite different notion.

May this reviewer cordially suggest that Mr Eli Maor study Guénon's book and revise the next edition of his own?

Incidentally, the set of "irrational" numbers and similar notions are an indefinity.

Math and its influence on culture
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.