Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Not really what I was expecting, 29 Nov 2008
From the blurb and the title I was expecting a book that would help deepen my understanding of quantum mechanics, and give me new and subtle insights into the implications of the Aspect/Gisin quantum entanglement experiments, possibly including some implications for thought and consciousness, a-la-Penrose. This turns out not to be the book's purpose at all however. It's hard to determine who the intended audience is. While the discussion on Quantum Mechaincs is pitched at layman's level, the discussion around it would seem more aimed at academics in the arts and humanities. It is a wide-ranging book touching on far more than QM. I found the book, informative, provocative, irritating, and in the end, rather moving. I'm glad I persisted with it though I can't say I agree with everything in it.

The introduction announces a post-modernist malaise in the academic humanities, rooted, the authors claim, in the removal of mind from the material world by Cartesian Dualism. This was surprising for me because, as a reader in Cognitive Science and Philosophy of Mind, I know that the modern scientific currency is reductive materialism. I had no idea that there was a community of folks out there who presumed dualism, and deduced pessimism.

The first half of the book then gives a layman's (non-mathematical) description of quantum mechanics. It's a bit sloppy. Terms are introduced without definition. Conclusions are drawn from premises without explanation. Schrodinger's cat is trotted out again, as usual, without qualification, so yet more credible folks will come away thinking that there is something magical about conscious observerhood that collapses superposed quantum states. The dual slit experiment is explained pretty well. Then we come to an exposition of Bell's inequality theorm as an intro to the Aspect/Gisin experiments. We gather that the implications are that Bohr's Copenhagen Interpretation is now incontrovertible, Einstein's Realism is refuted, and the hopes for deeper breakthroughs, such as hidden variable approaches, restoring it are shattered once and for all. The authors then specify a 'logic of complementarity' required to do constructive thinking about quantum phenomena, and point out along the way how Relativity requires the same kind of logic when thinking about space and time. This latter point I did find quite illuminating.

We then get a couple of chapters looking at aspects of Biology and Human Evolution where similar complementarity logic might be applicable. We are essentially looking at emergence, and how wholes can be greater than than sum of parts. We look at how co-operation operates alongside Darwinian competition as a dynamic in evolution, and we look at how culture could have driven the evolution of the physical substrates of human language in a virtuous spiral. Whether complementarity is fruitful of original insights in these areas, or merely provides analogies is hard to say.

The last third of the book gets to its main point which is to use the logic of complemantarity, derived from quantum mechanics, to bring solace to all these languishing postmodernist academics, and show them a way out of their pessimism.

I should say before I go much further that as a scientist who believes in a world out there, that gets on with it regardless of whether we do or can observe it, I don't have a lot of patience with post-modernist thought. The notion that science is a mythical social construction, promulgated by the power elite, is just institutionalised solipsism, and the money spent maintaining serious academic careers and filling our children's heads with this nonsense would be better spent on alleviating poverty or putting a person on Mars.

We get an intellectual history of post-modernism, tracing a line of descent from Descartes, through Nietzche, Husserl, Sartre and Existentialism, the post-structuralists and then the Derridas, Foucaults, etc. We have discovered that language can only ever refer to itself. That nothing meaningful can be said or deduced about the world outside our minds, and that all our thoughts have been hijacked by the power elite so they can get on with oppressing minorities of various pursuasions. Here, I just lose it. The Power elite does it's thing with violence and the exploitation of ignorance, pure and simple. They don't need to control our thoughts and language to do that, and the fact that they don't is what gives us hope for the future. Eventually we learn that the logic of complementarity allows the meanings of words to signify things in the external world and language is saved from the power elite. This is great because I hate to think of these postmodernists suffering needlessly.

We then get a chapter on the implications of the nonlocality implied by the Copenhagen Interpretation and the Aspect/Gisin experiments. Quantum entanglement from the big bang ensures that all particles/quanta in the universe are ultimately bound up in a single whole across all of space and time which is ultimately unknowable, in principle, to science. There can never be ontology, a science or knowledge of what's actually out there. I'm familiar with this understanding and have made my peace with it. The book makes the point that most of the science community simply adopt an ostrich approach to the full implications of nonlocality, so long as the maths works out.

The authors see things in terms of C.P.Snow's culture war between the disaffected postmodernists and the pragmatic mentality of science, a rift that itself follows the complementarity paradigm. In the final chapter they argue that that a dialogue is required between the two cultures if the ecological catastrophe, for which they present a very incisive analysis, facing humanity is to be confronted successfully. They here make a very moving appeal for the rift to be healed and a new complimentarity based unified system of thought to be developed as the basis for a completely new form of religion, shorn of all anthropomorphism and compatible with science but which speaks to all aspects of the human being. Their logic is that only a belief system with the force of a religion will be powerful enough to transform global society into something that can reach a sustainable realtionship with the world. I kind of agree, which is why I found it moving, but I have little optimism of it happening and less so that a shift in rational perspectives will provide the foundation for it.

So a very wide ranging book with some interesting points to make, none of which you'd suspect from the title. As a layman's introduction to Quantum Mechanics, I know there are better ones out there.


important and well written - perhaps flawed, 17 Jul 2007
Sometimes the language of this book, with its long flowing sentences and abstract ideas sounds a little Hegelian, but the vast majority of it is down-to-earth, well thought out and sticks to the task of describing some of the most difficult conceptual areas in science. Quantum Mechanics can never be easy because it is not visualisable as such. There may be some flaws in the argument however (why I marked it down!). The author's explanation of entanglement is solely in terms of non-locality. However they seemed to have ignored the alternative of retro-causality. They actually describe an important retro-causal experiment, but do not seem to incorporate it into their arguments. A further problem seems to occur when they go on to extend the idea of complementarity beyond physics (following Niels Bohr). They describe how `biological reality' might be affected by the same measurement difficulties as physical reality at the micro level. But biology is far too complex, in my opinion, to be able to isolate such an effect. It seems an unwarranted generalisation.

Simple yet technically superb, 21 Sep 2006
Anyone interested in the area of quantum mechanics should read this book. It is easy to understand, yet detailed and technically superb - explaining the various different interpretations that are available. This book is particularly impressive in bridging the knowledge gap that most books on the subject leave - the gap between quantum mechanics and what it implies for the human mind and our everyday lives. For anyone that thinks quantum mechanics has nothing (or very little) to do with reality - think again!

The main strength of this book is its uncompromising tenacity in explaining and staying with the facts. Where little is known, the authors explain the various thories that are around and their likely implications. For me, this book is the best available explanation of quantum mechanics and its unexpected possibilities.

Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Not really what I was expecting, 29 Nov 2008
From the blurb and the title I was expecting a book that would help deepen my understanding of quantum mechanics, and give me new and subtle insights into the implications of the Aspect/Gisin quantum entanglement experiments, possibly including some implications for thought and consciousness, a-la-Penrose. This turns out not to be the book's purpose at all however. It's hard to determine who the intended audience is. While the discussion on Quantum Mechaincs is pitched at layman's level, the discussion around it would seem more aimed at academics in the arts and humanities. It is a wide-ranging book touching on far more than QM. I found the book, informative, provocative, irritating, and in the end, rather moving. I'm glad I persisted with it though I can't say I agree with everything in it.

The introduction announces a post-modernist malaise in the academic humanities, rooted, the authors claim, in the removal of mind from the material world by Cartesian Dualism. This was surprising for me because, as a reader in Cognitive Science and Philosophy of Mind, I know that the modern scientific currency is reductive materialism. I had no idea that there was a community of folks out there who presumed dualism, and deduced pessimism.

The first half of the book then gives a layman's (non-mathematical) description of quantum mechanics. It's a bit sloppy. Terms are introduced without definition. Conclusions are drawn from premises without explanation. Schrodinger's cat is trotted out again, as usual, without qualification, so yet more credible folks will come away thinking that there is something magical about conscious observerhood that collapses superposed quantum states. The dual slit experiment is explained pretty well. Then we come to an exposition of Bell's inequality theorm as an intro to the Aspect/Gisin experiments. We gather that the implications are that Bohr's Copenhagen Interpretation is now incontrovertible, Einstein's Realism is refuted, and the hopes for deeper breakthroughs, such as hidden variable approaches, restoring it are shattered once and for all. The authors then specify a 'logic of complementarity' required to do constructive thinking about quantum phenomena, and point out along the way how Relativity requires the same kind of logic when thinking about space and time. This latter point I did find quite illuminating.

We then get a couple of chapters looking at aspects of Biology and Human Evolution where similar complementarity logic might be applicable. We are essentially looking at emergence, and how wholes can be greater than than sum of parts. We look at how co-operation operates alongside Darwinian competition as a dynamic in evolution, and we look at how culture could have driven the evolution of the physical substrates of human language in a virtuous spiral. Whether complementarity is fruitful of original insights in these areas, or merely provides analogies is hard to say.

The last third of the book gets to its main point which is to use the logic of complemantarity, derived from quantum mechanics, to bring solace to all these languishing postmodernist academics, and show them a way out of their pessimism.

I should say before I go much further that as a scientist who believes in a world out there, that gets on with it regardless of whether we do or can observe it, I don't have a lot of patience with post-modernist thought. The notion that science is a mythical social construction, promulgated by the power elite, is just institutionalised solipsism, and the money spent maintaining serious academic careers and filling our children's heads with this nonsense would be better spent on alleviating poverty or putting a person on Mars.

We get an intellectual history of post-modernism, tracing a line of descent from Descartes, through Nietzche, Husserl, Sartre and Existentialism, the post-structuralists and then the Derridas, Foucaults, etc. We have discovered that language can only ever refer to itself. That nothing meaningful can be said or deduced about the world outside our minds, and that all our thoughts have been hijacked by the power elite so they can get on with oppressing minorities of various pursuasions. Here, I just lose it. The Power elite does it's thing with violence and the exploitation of ignorance, pure and simple. They don't need to control our thoughts and language to do that, and the fact that they don't is what gives us hope for the future. Eventually we learn that the logic of complementarity allows the meanings of words to signify things in the external world and language is saved from the power elite. This is great because I hate to think of these postmodernists suffering needlessly.

We then get a chapter on the implications of the nonlocality implied by the Copenhagen Interpretation and the Aspect/Gisin experiments. Quantum entanglement from the big bang ensures that all particles/quanta in the universe are ultimately bound up in a single whole across all of space and time which is ultimately unknowable, in principle, to science. There can never be ontology, a science or knowledge of what's actually out there. I'm familiar with this understanding and have made my peace with it. The book makes the point that most of the science community simply adopt an ostrich approach to the full implications of nonlocality, so long as the maths works out.

The authors see things in terms of C.P.Snow's culture war between the disaffected postmodernists and the pragmatic mentality of science, a rift that itself follows the complementarity paradigm. In the final chapter they argue that that a dialogue is required between the two cultures if the ecological catastrophe, for which they present a very incisive analysis, facing humanity is to be confronted successfully. They here make a very moving appeal for the rift to be healed and a new complimentarity based unified system of thought to be developed as the basis for a completely new form of religion, shorn of all anthropomorphism and compatible with science but which speaks to all aspects of the human being. Their logic is that only a belief system with the force of a religion will be powerful enough to transform global society into something that can reach a sustainable realtionship with the world. I kind of agree, which is why I found it moving, but I have little optimism of it happening and less so that a shift in rational perspectives will provide the foundation for it.

So a very wide ranging book with some interesting points to make, none of which you'd suspect from the title. As a layman's introduction to Quantum Mechanics, I know there are better ones out there.


important and well written - perhaps flawed, 17 Jul 2007
Sometimes the language of this book, with its long flowing sentences and abstract ideas sounds a little Hegelian, but the vast majority of it is down-to-earth, well thought out and sticks to the task of describing some of the most difficult conceptual areas in science. Quantum Mechanics can never be easy because it is not visualisable as such. There may be some flaws in the argument however (why I marked it down!). The author's explanation of entanglement is solely in terms of non-locality. However they seemed to have ignored the alternative of retro-causality. They actually describe an important retro-causal experiment, but do not seem to incorporate it into their arguments. A further problem seems to occur when they go on to extend the idea of complementarity beyond physics (following Niels Bohr). They describe how `biological reality' might be affected by the same measurement difficulties as physical reality at the micro level. But biology is far too complex, in my opinion, to be able to isolate such an effect. It seems an unwarranted generalisation.

Simple yet technically superb, 21 Sep 2006
Anyone interested in the area of quantum mechanics should read this book. It is easy to understand, yet detailed and technically superb - explaining the various different interpretations that are available. This book is particularly impressive in bridging the knowledge gap that most books on the subject leave - the gap between quantum mechanics and what it implies for the human mind and our everyday lives. For anyone that thinks quantum mechanics has nothing (or very little) to do with reality - think again!

The main strength of this book is its uncompromising tenacity in explaining and staying with the facts. Where little is known, the authors explain the various thories that are around and their likely implications. For me, this book is the best available explanation of quantum mechanics and its unexpected possibilities.


Review of Letters to a Young Mathematician, 18 Jul 2008
'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter `Is God a Mathematician' brings in historical quotes such as `God is a geometer' by Plato, `God is a mathematician' by Paul Dirac and `God is a Pure Mathematician' by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic `A Mathematicians Apology' by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy's classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies `I want the exact number'.
This is an excellent book and definitely worth buying.



A dual-layered work of education and advice but entertaining more than anything, 13 Feb 2007
For the "non-Mathmo", this book will provide a fascinating insight into the career arc of a Mathematician from high school beginnings right up to post-graduate University and research positions. However, it is so much more than that: as is now the norm for Stewart, the work is generously peppered with accessible, entertaining anecdotes which serve as appropriate interludes at any point he is in danger of entering 'yawn booorrring' territory. These anecdotes also elucidate exactly what it is that Mathematicians do and go some way to explaining to the layman just what the appeal of the subject is. Particularly worthy of praise is the fact that Stewart has not relied on the clichés of popular Mathematics which we have all seen before (The Death of Evariste Galois, Sophie Germain etc.) and instead injects some new, lesser known stories into the mix.

Much more narratively interesting than his other works, the apparently ageless Stewart corresponds with a young Mathematician giving advice as she travels, over the course of a decade or so, along the path of a Maths student, eventually to the career ladder. The nature of their correspondence at first has a sinister edge, as if he is grooming her, but any such suspicions are swiftly dropped as it becomes more apparent that the letter format is merely a tool Stewart is employing to spread his message about the world of Mathematics. The personal touch gives Stewart's page-voice a warmth that successfully dampens the often smug and arrogant tone present in his other works.

To the Mathematics student, this work is both an invaluable motivational tool and a useful tome of advice for any career or study choices that the student finds themselves having to make. As a Mathematics student myself I often felt while reading the book that I could slip into the role of Meg (the eponymous "young Mathematican") and feel the wisdom was being addressed to me personally. Chapter 14's anthropological approach to the academic environment (a clever, almost tribal analysis of "gift giving" and "tribute") is essential reading for anyone in the academic field intending to manipulate the machine in their favour. Academia is a highly politicised field and Stewart's insights will minimise frustrations if carefully considered.

While it is a perpetual problem for the popular Mathematics writer to dumb down the subject enough to make it accessible, Stewart has achieved something truly praiseworthy here, a triumph of style that manages to address two audiences on two separate levels with one single voice. If before his work and the work of his popular Mathematics writing peers (especially Keith Devlin) were really only suitable for Mathematicians and students in Mathematical Sciences then Letters to a Young Mathematician marks an important turning point in contemporary popular Mathematical literature for being a work that can be appreciated truly outside of the academic context. Stewart is finally taking Mathematical literature to the places that Russell Stannard did with his Uncle Albert books and Jostein Gaarder did with Sophie's World and for this he can only be praised.

Ian Stewart at his absolute best., 09 Sep 2006
I don't think there is anyone better than Ian Stewart at making mathematics accessible to the masses This latest offering offers a thoughtful insight into the academic environment, in a most entertaining manner - a series of letters to a young, aspiring mathematician, tracing her career from undergraduate studies, through doctorate, research contract staff and member of academic staff. A truly wonderful insight into the academic environment and lifestyle from one of the community's greatest.

Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Not really what I was expecting, 29 Nov 2008
From the blurb and the title I was expecting a book that would help deepen my understanding of quantum mechanics, and give me new and subtle insights into the implications of the Aspect/Gisin quantum entanglement experiments, possibly including some implications for thought and consciousness, a-la-Penrose. This turns out not to be the book's purpose at all however. It's hard to determine who the intended audience is. While the discussion on Quantum Mechaincs is pitched at layman's level, the discussion around it would seem more aimed at academics in the arts and humanities. It is a wide-ranging book touching on far more than QM. I found the book, informative, provocative, irritating, and in the end, rather moving. I'm glad I persisted with it though I can't say I agree with everything in it.

The introduction announces a post-modernist malaise in the academic humanities, rooted, the authors claim, in the removal of mind from the material world by Cartesian Dualism. This was surprising for me because, as a reader in Cognitive Science and Philosophy of Mind, I know that the modern scientific currency is reductive materialism. I had no idea that there was a community of folks out there who presumed dualism, and deduced pessimism.

The first half of the book then gives a layman's (non-mathematical) description of quantum mechanics. It's a bit sloppy. Terms are introduced without definition. Conclusions are drawn from premises without explanation. Schrodinger's cat is trotted out again, as usual, without qualification, so yet more credible folks will come away thinking that there is something magical about conscious observerhood that collapses superposed quantum states. The dual slit experiment is explained pretty well. Then we come to an exposition of Bell's inequality theorm as an intro to the Aspect/Gisin experiments. We gather that the implications are that Bohr's Copenhagen Interpretation is now incontrovertible, Einstein's Realism is refuted, and the hopes for deeper breakthroughs, such as hidden variable approaches, restoring it are shattered once and for all. The authors then specify a 'logic of complementarity' required to do constructive thinking about quantum phenomena, and point out along the way how Relativity requires the same kind of logic when thinking about space and time. This latter point I did find quite illuminating.

We then get a couple of chapters looking at aspects of Biology and Human Evolution where similar complementarity logic might be applicable. We are essentially looking at emergence, and how wholes can be greater than than sum of parts. We look at how co-operation operates alongside Darwinian competition as a dynamic in evolution, and we look at how culture could have driven the evolution of the physical substrates of human language in a virtuous spiral. Whether complementarity is fruitful of original insights in these areas, or merely provides analogies is hard to say.

The last third of the book gets to its main point which is to use the logic of complemantarity, derived from quantum mechanics, to bring solace to all these languishing postmodernist academics, and show them a way out of their pessimism.

I should say before I go much further that as a scientist who believes in a world out there, that gets on with it regardless of whether we do or can observe it, I don't have a lot of patience with post-modernist thought. The notion that science is a mythical social construction, promulgated by the power elite, is just institutionalised solipsism, and the money spent maintaining serious academic careers and filling our children's heads with this nonsense would be better spent on alleviating poverty or putting a person on Mars.

We get an intellectual history of post-modernism, tracing a line of descent from Descartes, through Nietzche, Husserl, Sartre and Existentialism, the post-structuralists and then the Derridas, Foucaults, etc. We have discovered that language can only ever refer to itself. That nothing meaningful can be said or deduced about the world outside our minds, and that all our thoughts have been hijacked by the power elite so they can get on with oppressing minorities of various pursuasions. Here, I just lose it. The Power elite does it's thing with violence and the exploitation of ignorance, pure and simple. They don't need to control our thoughts and language to do that, and the fact that they don't is what gives us hope for the future. Eventually we learn that the logic of complementarity allows the meanings of words to signify things in the external world and language is saved from the power elite. This is great because I hate to think of these postmodernists suffering needlessly.

We then get a chapter on the implications of the nonlocality implied by the Copenhagen Interpretation and the Aspect/Gisin experiments. Quantum entanglement from the big bang ensures that all particles/quanta in the universe are ultimately bound up in a single whole across all of space and time which is ultimately unknowable, in principle, to science. There can never be ontology, a science or knowledge of what's actually out there. I'm familiar with this understanding and have made my peace with it. The book makes the point that most of the science community simply adopt an ostrich approach to the full implications of nonlocality, so long as the maths works out.

The authors see things in terms of C.P.Snow's culture war between the disaffected postmodernists and the pragmatic mentality of science, a rift that itself follows the complementarity paradigm. In the final chapter they argue that that a dialogue is required between the two cultures if the ecological catastrophe, for which they present a very incisive analysis, facing humanity is to be confronted successfully. They here make a very moving appeal for the rift to be healed and a new complimentarity based unified system of thought to be developed as the basis for a completely new form of religion, shorn of all anthropomorphism and compatible with science but which speaks to all aspects of the human being. Their logic is that only a belief system with the force of a religion will be powerful enough to transform global society into something that can reach a sustainable realtionship with the world. I kind of agree, which is why I found it moving, but I have little optimism of it happening and less so that a shift in rational perspectives will provide the foundation for it.

So a very wide ranging book with some interesting points to make, none of which you'd suspect from the title. As a layman's introduction to Quantum Mechanics, I know there are better ones out there.


important and well written - perhaps flawed, 17 Jul 2007
Sometimes the language of this book, with its long flowing sentences and abstract ideas sounds a little Hegelian, but the vast majority of it is down-to-earth, well thought out and sticks to the task of describing some of the most difficult conceptual areas in science. Quantum Mechanics can never be easy because it is not visualisable as such. There may be some flaws in the argument however (why I marked it down!). The author's explanation of entanglement is solely in terms of non-locality. However they seemed to have ignored the alternative of retro-causality. They actually describe an important retro-causal experiment, but do not seem to incorporate it into their arguments. A further problem seems to occur when they go on to extend the idea of complementarity beyond physics (following Niels Bohr). They describe how `biological reality' might be affected by the same measurement difficulties as physical reality at the micro level. But biology is far too complex, in my opinion, to be able to isolate such an effect. It seems an unwarranted generalisation.

Simple yet technically superb, 21 Sep 2006
Anyone interested in the area of quantum mechanics should read this book. It is easy to understand, yet detailed and technically superb - explaining the various different interpretations that are available. This book is particularly impressive in bridging the knowledge gap that most books on the subject leave - the gap between quantum mechanics and what it implies for the human mind and our everyday lives. For anyone that thinks quantum mechanics has nothing (or very little) to do with reality - think again!

The main strength of this book is its uncompromising tenacity in explaining and staying with the facts. Where little is known, the authors explain the various thories that are around and their likely implications. For me, this book is the best available explanation of quantum mechanics and its unexpected possibilities.


Review of Letters to a Young Mathematician, 18 Jul 2008
'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter `Is God a Mathematician' brings in historical quotes such as `God is a geometer' by Plato, `God is a mathematician' by Paul Dirac and `God is a Pure Mathematician' by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic `A Mathematicians Apology' by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy's classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies `I want the exact number'.
This is an excellent book and definitely worth buying.



A dual-layered work of education and advice but entertaining more than anything, 13 Feb 2007
For the "non-Mathmo", this book will provide a fascinating insight into the career arc of a Mathematician from high school beginnings right up to post-graduate University and research positions. However, it is so much more than that: as is now the norm for Stewart, the work is generously peppered with accessible, entertaining anecdotes which serve as appropriate interludes at any point he is in danger of entering 'yawn booorrring' territory. These anecdotes also elucidate exactly what it is that Mathematicians do and go some way to explaining to the layman just what the appeal of the subject is. Particularly worthy of praise is the fact that Stewart has not relied on the clichés of popular Mathematics which we have all seen before (The Death of Evariste Galois, Sophie Germain etc.) and instead injects some new, lesser known stories into the mix.

Much more narratively interesting than his other works, the apparently ageless Stewart corresponds with a young Mathematician giving advice as she travels, over the course of a decade or so, along the path of a Maths student, eventually to the career ladder. The nature of their correspondence at first has a sinister edge, as if he is grooming her, but any such suspicions are swiftly dropped as it becomes more apparent that the letter format is merely a tool Stewart is employing to spread his message about the world of Mathematics. The personal touch gives Stewart's page-voice a warmth that successfully dampens the often smug and arrogant tone present in his other works.

To the Mathematics student, this work is both an invaluable motivational tool and a useful tome of advice for any career or study choices that the student finds themselves having to make. As a Mathematics student myself I often felt while reading the book that I could slip into the role of Meg (the eponymous "young Mathematican") and feel the wisdom was being addressed to me personally. Chapter 14's anthropological approach to the academic environment (a clever, almost tribal analysis of "gift giving" and "tribute") is essential reading for anyone in the academic field intending to manipulate the machine in their favour. Academia is a highly politicised field and Stewart's insights will minimise frustrations if carefully considered.

While it is a perpetual problem for the popular Mathematics writer to dumb down the subject enough to make it accessible, Stewart has achieved something truly praiseworthy here, a triumph of style that manages to address two audiences on two separate levels with one single voice. If before his work and the work of his popular Mathematics writing peers (especially Keith Devlin) were really only suitable for Mathematicians and students in Mathematical Sciences then Letters to a Young Mathematician marks an important turning point in contemporary popular Mathematical literature for being a work that can be appreciated truly outside of the academic context. Stewart is finally taking Mathematical literature to the places that Russell Stannard did with his Uncle Albert books and Jostein Gaarder did with Sophie's World and for this he can only be praised.

Ian Stewart at his absolute best., 09 Sep 2006
I don't think there is anyone better than Ian Stewart at making mathematics accessible to the masses This latest offering offers a thoughtful insight into the academic environment, in a most entertaining manner - a series of letters to a young, aspiring mathematician, tracing her career from undergraduate studies, through doctorate, research contract staff and member of academic staff. A truly wonderful insight into the academic environment and lifestyle from one of the community's greatest.

An Excellent book that I have ever red, 18 Nov 2008
Srinivasa Ramanujan!What a great name and legend in the mathematical history. He is one of my favourite mathematician from my childhood. I tried to find his biography before and I couldnt find proper biography and this book is very very excellent book about the Genius. At one time, I have got tears on my eyes while reading the book and his troubles in his life. He found so many mathematical formulae in his time without proper facilities as we have today.

One thing I can say for sure, he lived only 33 years and If he'd lived more, then where would be the mathematical world?

Disappointing and lacking in mathematical insight., 27 May 2005
I bought this book after reading many others on popular mathematics. I found the book very slow, with heavy repetition of only a few themes. While Ramanujan was clearly an incredible man the book concentrates mostly on the incidental parts of his life and largely ignores the mathematical details the area I believe that defined him as an incredible individual. There was virtually no detail of any of his mathematical achievements making rather a dry biography lacking in interesting detail.

An Inspirational account of the life of a Mathematical geniu, 08 Jan 2003
Srinivasa Ramanujan is rightly a member of the Mathematicians’ Hall of Fame. From humble beginnings in the small town of Kumbhakaon in Tamil Nadu to the hallowed cloisters of Trinity College, Cambridge, this magnificent book narrates the story of Ramanujan’s trails, tribulations and triumphs.

Central to the story are the powerful influences of Ramanujan’s mother and the great English Mathematician, Godfrey Harold Hardy. If his mother, Komala shaped the first part of Ramanujan’s life, then surely Hardy must take full credit for bringing Ramanujan’s prodigious talents to the attention of the world Mathematical community. Other prominent characters also figure in the story – notably Ramanujan’s many friends, Narayana Aiyer, Gopalachari, leading lights in the Indian Mathematical establishment, members of the ruling British classes, Sir Francis Spring, the Governor of Madras Presidency, xxxx, and Cambridge Mathematicians, Neville, Littlewood.

The book presents a touching portrait of Ramanujan the man: an orthodox Vaishav Bhraman, steeped in Hindu culture with all the attendant characteristics of a deeply spiritual outlook, a calm self-assurance about his abilities, and most of all, an obsession with Mathematics. Hardy, his mentor, is also biographed as the passionately atheist, Winchester educated son of a middle class schoolmaster who went up to Cambridge, and at the turn of the 20th century, almost single handed masterminded the rise of English Pure Mathematics.

A book you will read in one go, 12 Jul 2002
I had heard of Ramanujan before, but not being a mathematician myself, I never read anything on him so I bought this biography by Kanigel. I like the style in which it is written, it makes easy reading and keeps you fascinated throughout the book. I read it in 1-2 days. The book not only covers the mathematics and collaboration with Hardy in detail but also the tremendous 'sufferings' Ramanujan had to undergo, and the culture clash between the West and India.

The book is worth the money. The only drawback I can think of is the cheap look and feel of the paperback edition. This book is certainly worth to be published in hardcover edition. I give it 4 stars because way too many books in Amazon get overrated by 5 stars and I don't want to fool people. If you are like me, with no background in Ramanujan, just buy this book, it is very good.

A superb account of the life of the genius Ramanujan., 10 Jul 2001
Kanigel weaves an entriguing biography from his extensive research into the life of Ramanujan. This text follows Ramanujan's journey from intellectual isolation to mathematical enlightenment and the universal acclaim he deserved and desired. Cultural aspects of his Indian background and the ensuing shock of Cambridge are conveyed convincingly. The author makes an unusual effort to explain mathematical concepts and he succeeds in creating a book that will enthrall mathematicians and non-mathematicians alike.

Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Not really what I was expecting, 29 Nov 2008
From the blurb and the title I was expecting a book that would help deepen my understanding of quantum mechanics, and give me new and subtle insights into the implications of the Aspect/Gisin quantum entanglement experiments, possibly including some implications for thought and consciousness, a-la-Penrose. This turns out not to be the book's purpose at all however. It's hard to determine who the intended audience is. While the discussion on Quantum Mechaincs is pitched at layman's level, the discussion around it would seem more aimed at academics in the arts and humanities. It is a wide-ranging book touching on far more than QM. I found the book, informative, provocative, irritating, and in the end, rather moving. I'm glad I persisted with it though I can't say I agree with everything in it.

The introduction announces a post-modernist malaise in the academic humanities, rooted, the authors claim, in the removal of mind from the material world by Cartesian Dualism. This was surprising for me because, as a reader in Cognitive Science and Philosophy of Mind, I know that the modern scientific currency is reductive materialism. I had no idea that there was a community of folks out there who presumed dualism, and deduced pessimism.

The first half of the book then gives a layman's (non-mathematical) description of quantum mechanics. It's a bit sloppy. Terms are introduced without definition. Conclusions are drawn from premises without explanation. Schrodinger's cat is trotted out again, as usual, without qualification, so yet more credible folks will come away thinking that there is something magical about conscious observerhood that collapses superposed quantum states. The dual slit experiment is explained pretty well. Then we come to an exposition of Bell's inequality theorm as an intro to the Aspect/Gisin experiments. We gather that the implications are that Bohr's Copenhagen Interpretation is now incontrovertible, Einstein's Realism is refuted, and the hopes for deeper breakthroughs, such as hidden variable approaches, restoring it are shattered once and for all. The authors then specify a 'logic of complementarity' required to do constructive thinking about quantum phenomena, and point out along the way how Relativity requires the same kind of logic when thinking about space and time. This latter point I did find quite illuminating.

We then get a couple of chapters looking at aspects of Biology and Human Evolution where similar complementarity logic might be applicable. We are essentially looking at emergence, and how wholes can be greater than than sum of parts. We look at how co-operation operates alongside Darwinian competition as a dynamic in evolution, and we look at how culture could have driven the evolution of the physical substrates of human language in a virtuous spiral. Whether complementarity is fruitful of original insights in these areas, or merely provides analogies is hard to say.

The last third of the book gets to its main point which is to use the logic of complemantarity, derived from quantum mechanics, to bring solace to all these languishing postmodernist academics, and show them a way out of their pessimism.

I should say before I go much further that as a scientist who believes in a world out there, that gets on with it regardless of whether we do or can observe it, I don't have a lot of patience with post-modernist thought. The notion that science is a mythical social construction, promulgated by the power elite, is just institutionalised solipsism, and the money spent maintaining serious academic careers and filling our children's heads with this nonsense would be better spent on alleviating poverty or putting a person on Mars.

We get an intellectual history of post-modernism, tracing a line of descent from Descartes, through Nietzche, Husserl, Sartre and Existentialism, the post-structuralists and then the Derridas, Foucaults, etc. We have discovered that language can only ever refer to itself. That nothing meaningful can be said or deduced about the world outside our minds, and that all our thoughts have been hijacked by the power elite so they can get on with oppressing minorities of various pursuasions. Here, I just lose it. The Power elite does it's thing with violence and the exploitation of ignorance, pure and simple. They don't need to control our thoughts and language to do that, and the fact that they don't is what gives us hope for the future. Eventually we learn that the logic of complementarity allows the meanings of words to signify things in the external world and language is saved from the power elite. This is great because I hate to think of these postmodernists suffering needlessly.

We then get a chapter on the implications of the nonlocality implied by the Copenhagen Interpretation and the Aspect/Gisin experiments. Quantum entanglement from the big bang ensures that all particles/quanta in the universe are ultimately bound up in a single whole across all of space and time which is ultimately unknowable, in principle, to science. There can never be ontology, a science or knowledge of what's actually out there. I'm familiar with this understanding and have made my peace with it. The book makes the point that most of the science community simply adopt an ostrich approach to the full implications of nonlocality, so long as the maths works out.

The authors see things in terms of C.P.Snow's culture war between the disaffected postmodernists and the pragmatic mentality of science, a rift that itself follows the complementarity paradigm. In the final chapter they argue that that a dialogue is required between the two cultures if the ecological catastrophe, for which they present a very incisive analysis, facing humanity is to be confronted successfully. They here make a very moving appeal for the rift to be healed and a new complimentarity based unified system of thought to be developed as the basis for a completely new form of religion, shorn of all anthropomorphism and compatible with science but which speaks to all aspects of the human being. Their logic is that only a belief system with the force of a religion will be powerful enough to transform global society into something that can reach a sustainable realtionship with the world. I kind of agree, which is why I found it moving, but I have little optimism of it happening and less so that a shift in rational perspectives will provide the foundation for it.

So a very wide ranging book with some interesting points to make, none of which you'd suspect from the title. As a layman's introduction to Quantum Mechanics, I know there are better ones out there.


important and well written - perhaps flawed, 17 Jul 2007
Sometimes the language of this book, with its long flowing sentences and abstract ideas sounds a little Hegelian, but the vast majority of it is down-to-earth, well thought out and sticks to the task of describing some of the most difficult conceptual areas in science. Quantum Mechanics can never be easy because it is not visualisable as such. There may be some flaws in the argument however (why I marked it down!). The author's explanation of entanglement is solely in terms of non-locality. However they seemed to have ignored the alternative of retro-causality. They actually describe an important retro-causal experiment, but do not seem to incorporate it into their arguments. A further problem seems to occur when they go on to extend the idea of complementarity beyond physics (following Niels Bohr). They describe how `biological reality' might be affected by the same measurement difficulties as physical reality at the micro level. But biology is far too complex, in my opinion, to be able to isolate such an effect. It seems an unwarranted generalisation.

Simple yet technically superb, 21 Sep 2006
Anyone interested in the area of quantum mechanics should read this book. It is easy to understand, yet detailed and technically superb - explaining the various different interpretations that are available. This book is particularly impressive in bridging the knowledge gap that most books on the subject leave - the gap between quantum mechanics and what it implies for the human mind and our everyday lives. For anyone that thinks quantum mechanics has nothing (or very little) to do with reality - think again!

The main strength of this book is its uncompromising tenacity in explaining and staying with the facts. Where little is known, the authors explain the various thories that are around and their likely implications. For me, this book is the best available explanation of quantum mechanics and its unexpected possibilities.


Review of Letters to a Young Mathematician, 18 Jul 2008
'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter `Is God a Mathematician' brings in historical quotes such as `God is a geometer' by Plato, `God is a mathematician' by Paul Dirac and `God is a Pure Mathematician' by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic `A Mathematicians Apology' by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy's classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies `I want the exact number'.
This is an excellent book and definitely worth buying.



A dual-layered work of education and advice but entertaining more than anything, 13 Feb 2007
For the "non-Mathmo", this book will provide a fascinating insight into the career arc of a Mathematician from high school beginnings right up to post-graduate University and research positions. However, it is so much more than that: as is now the norm for Stewart, the work is generously peppered with accessible, entertaining anecdotes which serve as appropriate interludes at any point he is in danger of entering 'yawn booorrring' territory. These anecdotes also elucidate exactly what it is that Mathematicians do and go some way to explaining to the layman just what the appeal of the subject is. Particularly worthy of praise is the fact that Stewart has not relied on the clichés of popular Mathematics which we have all seen before (The Death of Evariste Galois, Sophie Germain etc.) and instead injects some new, lesser known stories into the mix.

Much more narratively interesting than his other works, the apparently ageless Stewart corresponds with a young Mathematician giving advice as she travels, over the course of a decade or so, along the path of a Maths student, eventually to the career ladder. The nature of their correspondence at first has a sinister edge, as if he is grooming her, but any such suspicions are swiftly dropped as it becomes more apparent that the letter format is merely a tool Stewart is employing to spread his message about the world of Mathematics. The personal touch gives Stewart's page-voice a warmth that successfully dampens the often smug and arrogant tone present in his other works.

To the Mathematics student, this work is both an invaluable motivational tool and a useful tome of advice for any career or study choices that the student finds themselves having to make. As a Mathematics student myself I often felt while reading the book that I could slip into the role of Meg (the eponymous "young Mathematican") and feel the wisdom was being addressed to me personally. Chapter 14's anthropological approach to the academic environment (a clever, almost tribal analysis of "gift giving" and "tribute") is essential reading for anyone in the academic field intending to manipulate the machine in their favour. Academia is a highly politicised field and Stewart's insights will minimise frustrations if carefully considered.

While it is a perpetual problem for the popular Mathematics writer to dumb down the subject enough to make it accessible, Stewart has achieved something truly praiseworthy here, a triumph of style that manages to address two audiences on two separate levels with one single voice. If before his work and the work of his popular Mathematics writing peers (especially Keith Devlin) were really only suitable for Mathematicians and students in Mathematical Sciences then Letters to a Young Mathematician marks an important turning point in contemporary popular Mathematical literature for being a work that can be appreciated truly outside of the academic context. Stewart is finally taking Mathematical literature to the places that Russell Stannard did with his Uncle Albert books and Jostein Gaarder did with Sophie's World and for this he can only be praised.

Ian Stewart at his absolute best., 09 Sep 2006
I don't think there is anyone better than Ian Stewart at making mathematics accessible to the masses This latest offering offers a thoughtful insight into the academic environment, in a most entertaining manner - a series of letters to a young, aspiring mathematician, tracing her career from undergraduate studies, through doctorate, research contract staff and member of academic staff. A truly wonderful insight into the academic environment and lifestyle from one of the community's greatest.

An Excellent book that I have ever red, 18 Nov 2008
Srinivasa Ramanujan!What a great name and legend in the mathematical history. He is one of my favourite mathematician from my childhood. I tried to find his biography before and I couldnt find proper biography and this book is very very excellent book about the Genius. At one time, I have got tears on my eyes while reading the book and his troubles in his life. He found so many mathematical formulae in his time without proper facilities as we have today.

One thing I can say for sure, he lived only 33 years and If he'd lived more, then where would be the mathematical world?

Disappointing and lacking in mathematical insight., 27 May 2005
I bought this book after reading many others on popular mathematics. I found the book very slow, with heavy repetition of only a few themes. While Ramanujan was clearly an incredible man the book concentrates mostly on the incidental parts of his life and largely ignores the mathematical details the area I believe that defined him as an incredible individual. There was virtually no detail of any of his mathematical achievements making rather a dry biography lacking in interesting detail.

An Inspirational account of the life of a Mathematical geniu, 08 Jan 2003
Srinivasa Ramanujan is rightly a member of the Mathematicians’ Hall of Fame. From humble beginnings in the small town of Kumbhakaon in Tamil Nadu to the hallowed cloisters of Trinity College, Cambridge, this magnificent book narrates the story of Ramanujan’s trails, tribulations and triumphs.

Central to the story are the powerful influences of Ramanujan’s mother and the great English Mathematician, Godfrey Harold Hardy. If his mother, Komala shaped the first part of Ramanujan’s life, then surely Hardy must take full credit for bringing Ramanujan’s prodigious talents to the attention of the world Mathematical community. Other prominent characters also figure in the story – notably Ramanujan’s many friends, Narayana Aiyer, Gopalachari, leading lights in the Indian Mathematical establishment, members of the ruling British classes, Sir Francis Spring, the Governor of Madras Presidency, xxxx, and Cambridge Mathematicians, Neville, Littlewood.

The book presents a touching portrait of Ramanujan the man: an orthodox Vaishav Bhraman, steeped in Hindu culture with all the attendant characteristics of a deeply spiritual outlook, a calm self-assurance about his abilities, and most of all, an obsession with Mathematics. Hardy, his mentor, is also biographed as the passionately atheist, Winchester educated son of a middle class schoolmaster who went up to Cambridge, and at the turn of the 20th century, almost single handed masterminded the rise of English Pure Mathematics.

A book you will read in one go, 12 Jul 2002
I had heard of Ramanujan before, but not being a mathematician myself, I never read anything on him so I bought this biography by Kanigel. I like the style in which it is written, it makes easy reading and keeps you fascinated throughout the book. I read it in 1-2 days. The book not only covers the mathematics and collaboration with Hardy in detail but also the tremendous 'sufferings' Ramanujan had to undergo, and the culture clash between the West and India.

The book is worth the money. The only drawback I can think of is the cheap look and feel of the paperback edition. This book is certainly worth to be published in hardcover edition. I give it 4 stars because way too many books in Amazon get overrated by 5 stars and I don't want to fool people. If you are like me, with no background in Ramanujan, just buy this book, it is very good.

A superb account of the life of the genius Ramanujan., 10 Jul 2001
Kanigel weaves an entriguing biography from his extensive research into the life of Ramanujan. This text follows Ramanujan's journey from intellectual isolation to mathematical enlightenment and the universal acclaim he deserved and desired. Cultural aspects of his Indian background and the ensuing shock of Cambridge are conveyed convincingly. The author makes an unusual effort to explain mathematical concepts and he succeeds in creating a book that will enthrall mathematicians and non-mathematicians alike.

Magnificently clear., 02 Jan 2008
I recommend this book without reservation to anyone who wants to know about incompleteness, or about computability.

The main subjects of the book are as follows:

(1) It provides an excellent introduction to Godel's Theorems. A beginner in mathematical logic will easily be able to follow the introductory chapters, and thereby gain a good grasp of the incompleteness theorems. In particular, Smith gently introduces the technique of proof by diagonalisation, and shows how to use this to sketch a very easy and elegant incompleteness theorem).

(2) It provides systematic and detailed developments/reconstructions of Godel's proofs. In this sense, do not be fooled by the title's claim that this is an "Introduction"; it covers material far beyond introductory level. (Where it does so, this is clearly flagged.)

(3) It explains the relationship between incompleteness results and computability. Starting from scratch, the reader is introduced gently to Turing's Halting Problem, and then shown the link between this and the incompleteness results. Smith closes with an excellent philosophical commentary on the Church-Turing thesis; the rest of the book is lightly peppered with lucid philosophical commentary.

The single greatest achievement of this book is its clarity. Many books on mathematical logic present incredibly compressed proofs, and any commentary is likewise dry and terse. By contrast, Smith's writing style is cheerful and incredibly clear. He normally proceeds as follows: he first explains the proof-strategy in chatty, conversational English. This is followed by a full and technically rigorous proof, but written (again) in a manner that is easy to follow. He then closes with a simple summary, and a discussion of what the proof shows.

Smith's subject matter is complex and beautiful. The book is beautifully written, but simple to follow. For this reason, it deserves to be the textbook of choice in this area.

Zero and infinity, 07 Aug 2007
Babylonians invented it, Indians worshipped it, Greeks abhorred it. Zero has been a problematic number for a long time. European mathematicians followed Greek footsteps, until they finally realized how important thing zero was for advanced mathematics.

Seife presents us the history of zero and its sister concept infinity, not only in mathematics, but also in physics and quantum mechanics. Zero is an entertaining book, if a bit light. For quick popular science entertainment purposes it's a good choice. (Review based on the Finnish translation.)

one of the best maths books around, 08 Dec 2006
Being an undergraduate philosopher I've had to read a lot of maths books, and this is by far the best. It's true that you don't need much maths background to understand it, but it's also highly enjoyable for those with a lot of maths or physics knowledge - it links up and explains general assumptions in a way which seems never to occur to most teachers of sciences courses. The proof of 0=1 (and, extrapolating, that winston churchill = a carrot) is excellent and well worth committing to memory just to freak out any maths nerds one knows. Also worth a go is the step-by-step guide to making your own wormhole time machine (Step 1: Make a small wormhole, and attach one end to something really heavy). Really excellent, buy everyone you know a copy for christmas.

Mathematics history, 11 Apr 2003
A very readable book. This book covers the life story of the number zero, and it is a facinating story which is being told.

You do not need to have a better than average understanding of maths to be able to appreciate this book.

A good read, highly recommended.

Review for Zero:the Biography of a Dangerous Idea, 30 Jul 2002
This book was absolutely wonderful, it delves into the history of mathematics, as far back as the creation of numbers themselves. It looks at the contribution that the Greeks, Babylonians and Hindus made to mathematics, and how religion had restricted the development of mathematics. The book was written very well, it felt like a story book, rather than a factual book. I recommend this book for everyone with an interest in Maths, you do not need to be a mathematician to enjoy this book.

Highly Recommended, 23 Sep 2001
This is an excellent history of number Zero. Charles Seife takes you from the start, tracing the ideas of zero and inifity through time and how their concepts have been feared and embraced, how they've affected and forced evolution upon religious, philosophical, societal, and scientific ideas. I think this book should be part of any mathematics course. Highly recommend this book!

Not really what I was expecting, 29 Nov 2008
From the blurb and the title I was expecting a book that would help deepen my understanding of quantum mechanics, and give me new and subtle insights into the implications of the Aspect/Gisin quantum entanglement experiments, possibly including some implications for thought and consciousness, a-la-Penrose. This turns out not to be the book's purpose at all however. It's hard to determine who the intended audience is. While the discussion on Quantum Mechaincs is pitched at layman's level, the discussion around it would seem more aimed at academics in the arts and humanities. It is a wide-ranging book touching on far more than QM. I found the book, informative, provocative, irritating, and in the end, rather moving. I'm glad I persisted with it though I can't say I agree with everything in it.

The introduction announces a post-modernist malaise in the academic humanities, rooted, the authors claim, in the removal of mind from the material world by Cartesian Dualism. This was surprising for me because, as a reader in Cognitive Science and Philosophy of Mind, I know that the modern scientific currency is reductive materialism. I had no idea that there was a community of folks out there who presumed dualism, and deduced pessimism.

The first half of the book then gives a layman's (non-mathematical) description of quantum mechanics. It's a bit sloppy. Terms are introduced without definition. Conclusions are drawn from premises without explanation. Schrodinger's cat is trotted out again, as usual, without qualification, so yet more credible folks will come away thinking that there is something magical about conscious observerhood that collapses superposed quantum states. The dual slit experiment is explained pretty well. Then we come to an exposition of Bell's inequality theorm as an intro to the Aspect/Gisin experiments. We gather that the implications are that Bohr's Copenhagen Interpretation is now incontrovertible, Einstein's Realism is refuted, and the hopes for deeper breakthroughs, such as hidden variable approaches, restoring it are shattered once and for all. The authors then specify a 'logic of complementarity' required to do constructive thinking about quantum phenomena, and point out along the way how Relativity requires the same kind of logic when thinking about space and time. This latter point I did find quite illuminating.

We then get a couple of chapters looking at aspects of Biology and Human Evolution where similar complementarity logic might be applicable. We are essentially looking at emergence, and how wholes can be greater than than sum of parts. We look at how co-operation operates alongside Darwinian competition as a dynamic in evolution, and we look at how culture could have driven the evolution of the physical substrates of human language in a virtuous spiral. Whether complementarity is fruitful of original insights in these areas, or merely provides analogies is hard to say.

The last third of the book gets to its main point which is to use the logic of complemantarity, derived from quantum mechanics, to bring solace to all these languishing postmodernist academics, and show them a way out of their pessimism.

I should say before I go much further that as a scientist who believes in a world out there, that gets on with it regardless of whether we do or can observe it, I don't have a lot of patience with post-modernist thought. The notion that science is a mythical social construction, promulgated by the power elite, is just institutionalised solipsism, and the money spent maintaining serious academic careers and filling our children's heads with this nonsense would be better spent on alleviating poverty or putting a person on Mars.

We get an intellectual history of post-modernism, tracing a line of descent from Descartes, through Nietzche, Husserl, Sartre and Existentialism, the post-structuralists and then the Derridas, Foucaults, etc. We have discovered that language can only ever refer to itself. That nothing meaningful can be said or deduced about the world outside our minds, and that all our thoughts have been hijacked by the power elite so they can get on with oppressing minorities of various pursuasions. Here, I just lose it. The Power elite does it's thing with violence and the exploitation of ignorance, pure and simple. They don't need to control our thoughts and language to do that, and the fact that they don't is what gives us hope for the future. Eventually we learn that the logic of complementarity allows the meanings of words to signify things in the external world and language is saved from the power elite. This is great because I hate to think of these postmodernists suffering needlessly.

We then get a chapter on the implications of the nonlocality implied by the Copenhagen Interpretation and the Aspect/Gisin experiments. Quantum entanglement from the big bang ensures that all particles/quanta in the universe are ultimately bound up in a single whole across all of space and time which is ultimately unknowable, in principle, to science. There can never be ontology, a science or knowledge of what's actually out there. I'm familiar with this understanding and have made my peace with it. The book makes the point that most of the science community simply adopt an ostrich approach to the full implications of nonlocality, so long as the maths works out.

The authors see things in terms of C.P.Snow's culture war between the disaffected postmodernists and the pragmatic mentality of science, a rift that itself follows the complementarity paradigm. In the final chapter they argue that that a dialogue is required between the two cultures if the ecological catastrophe, for which they present a very incisive analysis, facing humanity is to be confronted successfully. They here make a very moving appeal for the rift to be healed and a new complimentarity based unified system of thought to be developed as the basis for a completely new form of religion, shorn of all anthropomorphism and compatible with science but which speaks to all aspects of the human being. Their logic is that only a belief system with the force of a religion will be powerful enough to transform global society into something that can reach a sustainable realtionship with the world. I kind of agree, which is why I found it moving, but I have little optimism of it happening and less so that a shift in rational perspectives will provide the foundation for it.

So a very wide ranging book with some interesting points to make, none of which you'd suspect from the title. As a layman's introduction to Quantum Mechanics, I know there are better ones out there.


important and well written - perhaps flawed, 17 Jul 2007
Sometimes the language of this book, with its long flowing sentences and abstract ideas sounds a little Hegelian, but the vast majority of it is down-to-earth, well thought out and sticks to the task of describing some of the most difficult conceptual areas in science. Quantum Mechanics can never be easy because it is not visualisable as such. There may be some flaws in the argument however (why I marked it down!). The author's explanation of entanglement is solely in terms of non-locality. However they seemed to have ignored the alternative of retro-causality. They actually describe an important retro-causal experiment, but do not seem to incorporate it into their arguments. A further problem seems to occur when they go on to extend the idea of complementarity beyond physics (following Niels Bohr). They describe how `biological reality' might be affected by the same measurement difficulties as physical reality at the micro level. But biology is far too complex, in my opinion, to be able to isolate such an effect. It seems an unwarranted generalisation.

Simple yet technically superb, 21 Sep 2006
Anyone interested in the area of quantum mechanics should read this book. It is easy to understand, yet detailed and technically superb - explaining the various different interpretations that are available. This book is particularly impressive in bridging the knowledge gap that most books on the subject leave - the gap between quantum mechanics and what it implies for the human mind and our everyday lives. For anyone that thinks quantum mechanics has nothing (or very little) to do with reality - think again!

The main strength of this book is its uncompromising tenacity in explaining and staying with the facts. Where little is known, the authors explain the various thories that are around and their likely implications. For me, this book is the best available explanation of quantum mechanics and its unexpected possibilities.


Review of Letters to a Young Mathematician, 18 Jul 2008
'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter `Is God a Mathematician' brings in historical quotes such as `God is a geometer' by Plato, `God is a mathematician' by Paul Dirac and `God is a Pure Mathematician' by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic `A Mathematicians Apology' by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy's classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies `I want the exact number'.
This is an excellent book and definitely worth buying.



A dual-layered work of education and advice but entertaining more than anything, 13 Feb 2007
For the "non-Mathmo", this book will provide a fascinating insight into the career arc of a Mathematician from high school beginnings right up to post-graduate University and research positions. However, it is so much more than that: as is now the norm for Stewart, the work is generously peppered with accessible, entertaining anecdotes which serve as appropriate interludes at any point he is in danger of entering 'yawn booorrring' territory. These anecdotes also elucidate exactly what it is that Mathematicians do and go some way to explaining to the layman just what the appeal of the subject is. Particularly worthy of praise is the fact that Stewart has not relied on the clichés of popular Mathematics which we have all seen before (The Death of Evariste Galois, Sophie Germain etc.) and instead injects some new, lesser known stories into the mix.

Much more narratively interesting than his other works, the apparently ageless Stewart corresponds with a young Mathematician giving advice as she travels, over the course of a decade or so, along the path of a Maths student, eventually to the career ladder. The nature of their correspondence at first has a sinister edge, as if he is grooming her, but any such suspicions are swiftly dropped as it becomes more apparent that the letter format is merely a tool Stewart is employing to spread his message about the world of Mathematics. The personal touch gives Stewart's page-voice a warmth that successfully dampens the often smug and arrogant tone present in his other works.

To the Mathematics student, this work is both an invaluable motivational tool and a useful tome of advice for any career or study choices that the student finds themselves having to make. As a Mathematics student myself I often felt while reading the book that I could slip into the role of Meg (the eponymous "young Mathematican") and feel the wisdom was being addressed to me personally. Chapter 14's anthropological approach to the academic environment (a clever, almost tribal analysis of "gift giving" and "tribute") is essential reading for anyone in the academic field intending to manipulate the machine in their favour. Academia is a highly politicised field and Stewart's insights will minimise frustrations if carefully considered.

While it is a perpetual problem for the popular Mathematics writer to dumb down the subject enough to make it accessible, Stewart has achieved something truly praiseworthy here, a triumph of style that manages to address two audiences on two separate levels with one single voice. If before his work and the work of his popular Mathematics writing peers (especially Keith Devlin) were really only suitable for Mathematicians and students in Mathematical Sciences then Letters to a Young Mathematician marks an important turning point in contemporary popular Mathematical literature for being a work that can be appreciated truly outside of the academic context. Stewart is finally taking Mathematical literature to the places that Russell Stannard did with his Uncle Albert books and Jostein Gaarder did with Sophie's World and for this he can only be praised.

Ian Stewart at his absolute best., 09 Sep 2006
I don't think there is anyone better than Ian Stewart at making mathematics accessible to the masses This latest offering offers a thoughtful insight into the academic environment, in a most entertaining manner - a series of letters to a young, aspiring mathematician, tracing her career from undergraduate studies, through doctorate, research contract staff and member of academic staff. A truly wonderful insight into the academic environment and lifestyle from one of the community's greatest.

An Excellent book that I have ever red, 18 Nov 2008
Srinivasa Ramanujan!What a great name and legend in the mathematical history. He is one of my favourite mathematician from my childhood. I tried to find his biography before and I couldnt find proper biography and this book is very very excellent book about the Genius. At one time, I have got tears on my eyes while reading the book and his troubles in his life. He found so many mathematical formulae in his time without proper facilities as we have today.

One thing I can say for sure, he lived only 33 years and If he'd lived more, then where would be the mathematical world?

Disappointing and lacking in mathematical insight., 27 May 2005
I bought this book after reading many others on popular mathematics. I found the book very slow, with heavy repetition of only a few themes. While Ramanujan was clearly an incredible man the book concentrates mostly on the incidental parts of his life and largely ignores the mathematical details the area I believe that defined him as an incredible individual. There was virtually no detail of any of his mathematical achievements making rather a dry biography lacking in interesting detail.

An Inspirational account of the life of a Mathematical geniu, 08 Jan 2003
Srinivasa Ramanujan is rightly a member of the Mathematicians’ Hall of Fame. From humble beginnings in the small town of Kumbhakaon in Tamil Nadu to the hallowed cloisters of Trinity College, Cambridge, this magnificent book narrates the story of Ramanujan’s trails, tribulations and triumphs.

Central to the story are the powerful influences of Ramanujan’s mother and the great English Mathematician, Godfrey Harold Hardy. If his mother, Komala shaped the first part of Ramanujan’s life, then surely Hardy must take full credit for bringing Ramanujan’s prodigious talents to the attention of the world Mathematical community. Other prominent characters also figure in the story – notably Ramanujan’s many friends, Narayana Aiyer, Gopalachari, leading lights in the Indian Mathematical establishment, members of the ruling British classes, Sir Francis Spring, the Governor of Madras Presidency, xxxx, and Cambridge Mathematicians, Neville, Littlewood.

The book presents a touching portrait of Ramanujan the man: an orthodox Vaishav Bhraman, steeped in Hindu culture with all the attendant characteristics of a deeply spiritual outlook, a calm self-assurance about his abilities, and most of all, an obsession with Mathematics. Hardy, his mentor, is also biographed as the passionately atheist, Winchester educated son of a middle class schoolmaster who went up to Cambridge, and at the turn of the 20th century, almost single handed masterminded the rise of English Pure Mathematics.

A book you will read in one go, 12 Jul 2002
I had heard of Ramanujan before, but not being a mathematician myself, I never read anything on him so I bought this biography by Kanigel. I like the style in which it is written, it makes easy reading and keeps you fascinated throughout the book. I read it in 1-2 days. The book not only covers the mathematics and collaboration with Hardy in detail but also the tremendous 'sufferings' Ramanujan had to undergo, and the culture clash between the West and India.

The book is worth the money. The only drawback I can think of is the cheap look and feel of the paperback edition. This book is certainly worth to be published in hardcover edition. I give it 4 stars because way too many books in Amazon get overrated by 5 stars and I don't want to fool people. If you are like me, with no background in Ramanujan, just buy this book, it is very good.

A superb account of the life of the genius Ramanujan., 10 Jul 2001
Kanigel weaves an entriguing biography from his extensive research into the life of Ramanujan. This text follows Ramanujan's journey from intellectual isolation to mathematical enlightenment and the universal acclaim he deserved and desired. Cultural aspects of his Indian background and the ensuing shock of Cambridge are conveyed convincingly. The author makes an unusual effort to explain mathematical concepts and he succeeds in creating a book that will enthrall mathematicians and non-mathematicians alike.

Magnificently clear., 02 Jan 2008
I recommend this book without reservation to anyone who wants to know about incompleteness, or about computability.

The main subjects of the book are as follows:

(1) It provides an excellent introduction to Godel's Theorems. A beginner in mathematical logic will easily be able to follow the introductory chapters, and thereby gain a good grasp of the incompleteness theorems. In particular, Smith gently introduces the technique of proof by diagonalisation, and shows how to use this to sketch a very easy and elegant incompleteness theorem).

(2) It provides systematic and detailed developments/reconstructions of Godel's proofs. In this sense, do not be fooled by the title's claim that this is an "Introduction"; it covers mat