Brian Clegg

A Brief History of Infinity

The concept of infinity has challenged thinkers throughout the ages. If you are interested in the infinite, but have found it to be too challenging then you would do well to read Brian Clegg's A Brief History of Infinity: The Quest to Think the Unthinkable

The book starts with the early ideas of the Greeks such as Zeno, and goes on to look at Archimedes use of huge numbers in The Sand Reckoner. Moving on to the middle ages we see the links between religion and the infinite, and how some of Galileo's troubles were due to his ideas of infinity. Clegg goes on to discuss the development of the calculus, including the battles between Leibniz and Newton. In the nineteenth century the problems with infinitesimals were dealt with, but a new can of worms was opened with Cantor's transfinite numbers.

Note that, as the title says, this is a book on the history of infinity and so doesn't deal with the most up to date research. Non-standard analysis is mentioned, but as Clegg tells us, that subject is now over half a century old. Some might also find the book a bit slow going - it is very much aimed at those without previous knowledge. If that's what you think would suit you, or if you want to see how belief gradually moved from potential to actual infinity then you should take a look at this book.

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Paperback 288 pages
ISBN: 1841196509
Salesrank: 328611
Weight:0.53 lbs

Published: 2003 Robinson Publishing

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Paperback 288 pages
ISBN: 1841196509
Salesrank: 14797
Weight:0.53 lbs

Published: 2003 Robinson Publishing

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Paperback 288 pages
ISBN: 1841196509
Salesrank: 54732
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Published: 2003 Constable

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Enjoyable Excursion into the Realm of Infinity ****
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.

Enjoyable Excursion into the Realm of Infinity ****

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.

To infinity and beyond ................. ***
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 infinity and beyond ................. ***

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 [...]

There are better works around **
If I'd not read Rucker's work on the subject (Infinity and The Mind), I might have thought this was pretty cool. But having said that, we may be reaching saturation point on the books about mathematicians (which this seems to be) - we need more about the maths. I think there's a perception that to keep it readable it needs to be dumbed down. There's a lot of that going on. It's possible to explain everything in simple terms if you try hard enough. Maybe Clegg hasn't tried all that hard, or maybe he's scared of alienating the casual reader. Whatever, he doesn't do much for the mathematically literate who want to get something out of this. There's not actually all that much.

There are better works around **

If I'd not read Rucker's work on the subject (Infinity and The Mind), I might have thought this was pretty cool. But having said that, we may be reaching saturation point on the books about mathematicians (which this seems to be) - we need more about the *maths*.

I think there's a perception that to keep it readable it needs to be dumbed down. There's a lot of that going on. It's possible to explain *everything* in simple terms if you try hard enough. Maybe Clegg hasn't tried all that hard, or maybe he's scared of alienating the casual reader. Whatever, he doesn't do much for the mathematically literate who want to get something out of this. There's not actually all that much.

An easy way to infinity ***
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.

An easy way to infinity ***

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.

definitely worth reading ****
I'm no mathematician but still really enjoyed this book. There were bits I didn't understand (some of the set theory ideas, and the proof for aleph-1), but even so it scratched an intellectual itch about Infinity that I've had for some time. Ever wondered how you can fit a closed shape with an infinite border within a finite circle? Yes? then this is the book for you.

definitely worth reading ****

I'm no mathematician but still really enjoyed this book. There were bits I didn't understand (some of the set theory ideas, and the proof for aleph-1), but even so it scratched an intellectual itch about Infinity that I've had for some time. Ever wondered how you can fit a closed shape with an infinite border within a finite circle? Yes? then this is the book for you.

Excellent ****
This is a very interesting and informative description of the history of infinity. Infinity is a fascinating (and complex) subject but Brian Clegg does an extremely good job of presenting it in a highly readable and essentially non-mathematical way. I have a mathematical background but this book should be accessible to all. I tend to agree with the previous reviewer who criticised presentation of some of the more complex mathematics in particular the higher alephs and Godel. However, I do not believe that these issues significantly detracted from my enjoyment of the book (mainly because I had not expectation that they would be covered well). An excellent popular science/mathematics book - highly recommended to all

Excellent ****

This is a very interesting and informative description of the history of infinity.

Infinity is a fascinating (and complex) subject but Brian Clegg does an extremely good job of presenting it in a highly readable and essentially non-mathematical way. I have a mathematical background but this book should be accessible to all.

I tend to agree with the previous reviewer who criticised presentation of some of the more complex mathematics in particular the higher alephs and Godel. However, I do not believe that these issues significantly detracted from my enjoyment of the book (mainly because I had not expectation that they would be covered well).

An excellent popular science/mathematics book - highly recommended to all