Customer Reviews

A Brief History of Infinity: The Quest to Think the Unthinkable

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Perhaps I should state what I think should have been made a little more obvious. Clegg's "A Brief History of Infinity" is not a mathematics book. It is definitely a history book. In fact, it outlines the history of man's struggle to come to grips with the exceedingly complex and devilishly bewildering concept of infinity. Of necessity, of course, it touches on matters mathematical but the meat of this book is the history.

A BRIEF HISTORY OF INFINITY delves into man's contemplation of matters infinite from the earliest days of its discussion by Greek philosophers, to St Augustine's theological musings of creation, to Leibniz and Newton battling over bragging rights for the creation of calculus, to Cantor's transfinite numbers and even to the implications of infinity in quantum physics.

Having noted that the book is more focused on history than mathematics, it's definitely worth pointing out that the mathematics would still be daunting for a complete neophyte. That said, my hope was for somewhat more mathematics and a little less of the historical background. For example, I found the section on Leibniz and Newton's battles with Bishop Berkley over infinitesimals quite dreary and plodding.

But, the misunderstanding as to the exact nature of the book can probably be laid more my doorstep. A more careful examination of previous reviews and the marketing info on the book cover would have better informed me as to what I was stepping into.

Recommended.

Paul Weiss

A Brief History of Infinity - The Quest to Think the Unthinkable by Brian Clegg is a good introduction for the layman to the historical development of the concept of infinity. Clegg's book is a good book, but not an exceptional one. I found the first half to be less engaging, as perhaps the sections on Greek mathematics and the historical development of calculus were too familiar. The second half is much better, especially the discussion of Cantor's transfinite numbers. On balance, A Brief History of Infinity rates four stars and should appeal to a wide audience.

Clegg touches on a wide range of topics, often exploring unexpected connections and tangential matters. Sometimes his side trips are diverting and enjoyable, but at other times I found these digressions to be overly distracting. (The discussions on quantum physics seemed unnecessary; there so many good books for the layman available.) But to be fair, infinity is a broad topic and other readers may well find Clegg's far ranging approach to be stimulating and enjoyable.

Clegg addresses in a credible fashion Galileo's investigations of infinity, the battle between Newton and Leibniz (and Bishop Berkeley too) over infinitesimals, Cantor's transfinite numbers, and Robinson's non-standard analysis. These non-technical discussions are neither too superficial, nor too vague.

For readers willing to delve deeper into the mathematics of infinity, I highly recommend The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise by Mary Tiles. This is not an easy book as it primarily targets advanced students in philosophy and mathematics. Nonetheless, it is within reach of a persistent reader. Some sections can be read stand-alone; three mid-chapters (Numbering the Continuum, Cantor's Transfinite Paradise, and Axiomatic Set Theory) are outstanding and in themselves are worth the price of this book.

Going a step farther, a math major might wish to tackle Georg Cantor's Contributions to the Founding of the Theory of Transfinite Numbers. I particularly recommend the Dover reprint of the same title as it contains a lengthy introduction by the mathematician Philip E. B. Jourdain.
There is yet a third book to consider, one that has the same title as Brian Clegg's book. A Brief History of Infinity by Paolo Zellini (translated by David Marsh) "explores every aspect of infinity, distilling the wisdom of philosophers, artists, mathematicians, and theologians over the millennia". Less mathematical than my other two recommendations, Zellini's work is nonetheless quite challenging. It provides a detailed examination of philosophical issues underlying the concept of infinity.

There is almost a pun in the title: how can you have a brief history of infinity? However, this book by Brian Clegg is part of a series, so it is stretching bounds of probability to suggest that the whole series is named just to get a pun in a title! I have not read any of the others in the series, but from their titles, they seem to be of a historical nature, rather than concepts and ideas. The sub-title of this volume is also interesting: "the quest to think the unthinkable". That statement gives a key to the book. What does the notion of infinity give us?

For me the first 100 pages of Clegg's book do not really work. It is both slow, and nebulous. However, after that, it opens up, and gives a good narrative of the progress towards the unreachable. Strangely, that is not just the infinitely large, but also the infinity small. Somehow "proper" numbers are sandwiched between the immeasurable at either ends of the spectrum. Clegg falls on the side of the usefulness of ideas, rather than the sheer understandability. Calculus, in both the Newtonian and Leibnizian forms, is introduced as "something that works". It was only later that the need for infinitely small is eliminated from calculus - by then this mathematical tool had been very useful in solving many real problems.

After the slow labouring start, the volume CAN be a beginner's guide to the subject, but is probably not be for the faint-hearted. It will help your understanding if you are familiar with ideas from modern mathematical thinking. If not, the notion of there being different `levels' of infinity can come as rather a jolt. How can two numbers, both of which are not countable, be different? How can there be the same number of odd numbers as there are of odd-and-even numbers? It is not surprising that the man behind set theory, Georg Cantor, lost his sanity when investigating the infinitely large.

In essence, the book covers so much more than just infinity. It really does go "to infinity and beyond". Infinity is a strange idea, but one that underlies many of the advances that have taken place since the start of the 20th century. Advances in number theory, cosmology, physics and atomic physics can be better understood if a basic understanding of infinity is grasped. Clegg covers these parts well.

The title was not a pun. However, Clegg does not disappoint in his last sentence. "When it comes to infinity, the possibilities are, perhaps inevitably, endless". I suppose he had to say that, didn't he?

Peter Morgan, Bath, UK [...]

To somebody with no mathematics background, what is available in this book may impress. But there is no reason for that. I could have as easily written down a few formulas and make them "scaring" enough to the layman. Or just mention something like "a pure quantum state is typically represented by a vector in a Hilbert space". Vector? Hilbert space? One could say, "this guy must know more than I do". But I could just be rephrasing something I read elsewhere and that I little understand beyond a basic grasp.

This is what comes to my mind when I read this book. The author has probably no university degree in either physics or mathematics. The emphasis on history and unimportant details seems to account for a lack of a more substantial as well conceptual understanding of the subject matter.

The mathematics in the book has no significance in its context and lacks a proper framework. It seems to be there to impress the reader somehow. It is out of place with the emphasis on history, the lack of focus in so many parts and the often glaring absence of a deeper understanding of the area. The fact that author has written many science books to the layman is not necessarily a proof to the contrary.

One must credit his seemingly genuine interest in science and its fundamental questions and concepts. But good intentions are not enough. Unfortunately, very often this book reads like a cheap science article.

I was a little disappointed that works and discoveries about infinity was not treated in more detail. Instead, many of the pages are used for biographies; The life and doings of a lot of mathematicians are covered from childhood to death. This of course can be (or is) very interesting, but was not what I excpected. I also got a feeling that this was done in part to avoid writing more about infinity, which of course is a much more difficult topic. The book is intended for a reader with little mathematical background, and this may be the reason why the author avoids difficult questions. There are good and readable presentations of some of the wellknown paradoxes, which should make everyone wonder about the strange behavior when we move away from the finite experience.

This book is a very boring read. The main problem is that some verses from other books are simply copy-pasted at various junctures. This makes the book very hard to read. One can read one, two, three ... infinity by George Gamow or some of the Simon Singh's or Carl Sagan's books instead of wasting time on this book.

I have enjoyed this book very much. The subtitle directs the reader to the subject, which is a history of the study of infinity. The history is accurate, to my knowledge, and entertainingly presented. The mathematics is basic but enlightening to its intended readership and serves to further the historical discourse. The quotations, such as those of Galileo, are well chosen and convey the intellectual environment of the time. There are many tomes available which plummet the mathematical intricacies of the subject or which go into minute historical details, but this overview should be a delight to those who would be engagingly introduced to a frequently formidable concept. (One detail: on p.69 we read: "although [transcendence of calculation by equation] is the case with pi, we now know the value of this quirky number to many thousands of decimal places." Pi was, in fact, calculated to a million places in the 1970s and is astronomically farther along now.)