John Barrow

Pi in the Sky

Pi in the Sky by John Barrow is really a combination of two different books. The first is a history of counting from the earliest times. The second is a look at the ideas of Cantor and Göel and their implication for mathematics. I can see the two parts are connected - inventing infinities is no different to inventing zero, or 1,2,3 for that matter - but Barrow doesn't really bring out this connection. The book is a bit philosophical, but it's easy to read an so is suited to those readers who want to find out more about the philosophy of mathematics without things becoming too technical.

One problem with popular mathematics books is that they all tend to deal with the same few subjects - incompleteness, codes, 'Game of life' etc. and to some extent Pi in the Sky follows this path. However, it does have some interesting material on L.E.J. Brouwer, which would be useful for those who want to find out more about intuitionism.

One criticism I have of the book is that Barrow believes in too strong a form of undecidability, for instance his claim that it is never possible to know if a computer program is the shortest one possible for it's task - this just isn't true.

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Paperback 317 pages
ISBN: 0316082597
Salesrank: 588707
Weight:0.95 lbs

Published: 1992 Back Bay Books

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Paperback 317 pages
ISBN: 0316082597
Salesrank: 484960
Weight:0.95 lbs

Published: 1993 Back Bay Books

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Paperback 317 pages
ISBN: 0316082597
Salesrank: 300390
Weight:0.95 lbs

Published: 1993 Back Bay Books

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Book Description
John D. Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of mathematics: where it comes from, what it is, and where it's going to take us if we follow it to the limit in our search for the ultimate meaning of the universe. Barrow begins by investigating whether math is a purely human invention inspired by our practical needs. Or is it something inherent in nature waiting to be discovered? In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats us to a history of counting all over the world, from Egyptian hieroglyphics to logical friction, from number mysticism to Marxist mathematics. And he introduces us to a host of peculiar individuals who have thought some of the deepest and strangest thoughts that human minds have ever thought, from Lao-Tse to Robert Pirsig, Charles Darwin, and Umberto Eco. Barrow thus provides the historical framework and the intellectual tools necessary to an understanding of some of today's weightiest mathematical concepts.

Book Description

John D. Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of mathematics: where it comes from, what it is, and where it's going to take us if we follow it to the limit in our search for the ultimate meaning of the universe. Barrow begins by investigating whether math is a purely human invention inspired by our practical needs. Or is it something inherent in nature waiting to be discovered?

In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats us to a history of counting all over the world, from Egyptian hieroglyphics to logical friction, from number mysticism to Marxist mathematics. And he introduces us to a host of peculiar individuals who have thought some of the deepest and strangest thoughts that human minds have ever thought, from Lao-Tse to Robert Pirsig, Charles Darwin, and Umberto Eco. Barrow thus provides the historical framework and the intellectual tools necessary to an understanding of some of today's weightiest mathematical concepts.

Might have been a classic if .. . ****
Might have been a classic if I had understood more of it. This is an extremely deep subject, searching for the source of mathematics. Is it a closed system of symbols that can solve any problem since it is a self-constructed system, as the formalists claim? Apparently not exactly, as Godel has proven that any system of math is contains unsolvable problems. Is it merely the presence of numbers and definite operations in the mind, learned from human activity? But this essentially limits math, as Barrow points out, to a branch of psychology; it is "finite, shorn of many truths that we had liked, divested of so many devices that were as much a part of human intuition as counting, and divorced from the study of the physical world." Then there's the Platonist view, that mathematics is an ideal, discovered and not invented. "Mathematics exists apart from mathematicians," says Barrow. This view, teetering on the brink of mysticism, is closest to where the philosophy of mathematics is today, according to Barrow, especially among consumers of mathematics such as physicists working at the extreme edge of science. Compare this to my review of Frank Tipler's The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead

Might have been a classic if .. . ****

Might have been a classic if I had understood more of it. This is an extremely deep subject, searching for the source of mathematics. Is it a closed system of symbols that can solve any problem since it is a self-constructed system, as the formalists claim? Apparently not exactly, as Godel has proven that any system of math is contains unsolvable problems.

Is it merely the presence of numbers and definite operations in the mind, learned from human activity? But this essentially limits math, as Barrow points out, to a branch of psychology; it is "finite, shorn of many truths that we had liked, divested of so many devices that were as much a part of human intuition as counting, and divorced from the study of the physical world."

Then there's the Platonist view, that mathematics is an ideal, discovered and not invented. "Mathematics exists apart from mathematicians," says Barrow. This view, teetering on the brink of mysticism, is closest to where the philosophy of mathematics is today, according to Barrow, especially among consumers of mathematics such as physicists working at the extreme edge of science.

Compare this to my review of Frank Tipler's The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead

Is mathematics real? ****
That may be a silly question. After all, most of us use counting and numerical calculations many times a day. However, the reading matter here digs below the surface, and asks such awkward questions. What is the nature of maths? Would there be any maths if there were no mathematicians? Starting with theories of counting, and the origins of methods of enumeration, John Barrow plunges headlong into the philosophy of mathematics. Perhaps the book ought to carry a health warning, for it should not be read accidentally. Readers need to have a grounding in some of the great mathematical movements, and discoveries. (Perhaps it is a bit judgmental to even use the word "discoveries"; are mathematical ideas invented or discovered? That topic is part of the subject matter). I liked the debate, but found the volume hard going. It is not the kind of book to read solidly from cover to cover. A great deal of re-reading is necessary, and picking it up on the train requires a conscious effort to remember what the current debate is about. Some of the arguments are very intricate for those of us who are not mathematicians. The work of some of the pillars of mathematics are described in varying detail, together with the triple crises that hit maths in the early years of the 20th Century. The optimism of Hilbert on the one hand, or Russell and Whitehead on the other was washed away by the work of Kurt Godel. The Austrian Godel, by the way, has been described as one of the most innovative minds of that century. There are some interesting insights into some of the characters from the history of maths. Leopold Kronecker did not believe in negative numbers. However, he had been a BANKER. How did he convince his customers that the problems caused by negative numbers (i.e. too little in their accounts) needed to be solved? There were also some disturbing questions raised by the work of Cantor on set theory. This gives rise to a wonderful paradox called "Hilbert's Hotel". As with many works on philosophy, it is not the answers that are important, it is the questions. Does the entity pi exist, even if there are no mathematicians. Is there really a universal 'pi in the sky', external to any human thought? You decide. Peter Morgan, Bath, UK (morganp@supanet.com)

Is mathematics real? ****

That may be a silly question. After all, most of us use counting and numerical calculations many times a day. However, the reading matter here digs below the surface, and asks such awkward questions. What is the nature of maths? Would there be any maths if there were no mathematicians?

Starting with theories of counting, and the origins of methods of enumeration, John Barrow plunges headlong into the philosophy of mathematics. Perhaps the book ought to carry a health warning, for it should not be read accidentally. Readers need to have a grounding in some of the great mathematical movements, and discoveries. (Perhaps it is a bit judgmental to even use the word "discoveries"; are mathematical ideas invented or discovered? That topic is part of the subject matter).

I liked the debate, but found the volume hard going. It is not the kind of book to read solidly from cover to cover. A great deal of re-reading is necessary, and picking it up on the train requires a conscious effort to remember what the current debate is about. Some of the arguments are very intricate for those of us who are not mathematicians.

The work of some of the pillars of mathematics are described in varying detail, together with the triple crises that hit maths in the early years of the 20th Century. The optimism of Hilbert on the one hand, or Russell and Whitehead on the other was washed away by the work of Kurt Godel. The Austrian Godel, by the way, has been described as one of the most innovative minds of that century.

There are some interesting insights into some of the characters from the history of maths. Leopold Kronecker did not believe in negative numbers. However, he had been a BANKER. How did he convince his customers that the problems caused by negative numbers (i.e. too little in their accounts) needed to be solved? There were also some disturbing questions raised by the work of Cantor on set theory. This gives rise to a wonderful paradox called "Hilbert's Hotel".

As with many works on philosophy, it is not the answers that are important, it is the questions. Does the entity pi exist, even if there are no mathematicians. Is there really a universal 'pi in the sky', external to any human thought? You decide.

Peter Morgan, Bath, UK (morganp@supanet.com)

hey mr. wanton arborcide from iceland ****
doesn't the mathematical concept of greater than come from a human mind? sure, some birds can count and distinguish between object sizes but can they creatively abstract and apply the concept to solve other physical problems? nope. and if you think the whole book is based on a false premise, it still has some interesting views, facts and features. does it warrant a 1 star? i mean you can learn from everything. even mistakes. i mean i learned from you just now.

hey mr. wanton arborcide from iceland ****

doesn't the mathematical concept of greater than
come from a human mind? sure, some birds can count and distinguish between object sizes but can they creatively abstract and apply the concept to solve other physical problems?
nope. and if you think the whole book is based
on a false premise, it still has some interesting views, facts and features.
does it warrant a 1 star?
i mean you can learn from everything.
even mistakes.
i mean i learned from you just now.

wanton arboricide *
The author's leading claim in this book is that "the only mathematics we know is human mind and brain based mathematics." This claim can be understood in either of two ways, which the author does not distinguish from one another. On one hand, by "mathematics" he might mean the practice or family of practices that go by that name (the sort of thing that a math teacher gets paid to teach). In this sense, it is just trivial that the only mathematics we know depends on humans, just as the only civil engineering or basketball we know depends on humans. But let's be charitable, and try to construe the author's claim in a way that does not reduce it to a mere triviality. Let's suppose that by "mathematics" he means not the practice of investigating mathematical fact, but the body of fact thus investigated. But if that is what he means, then his claim is clearly false. It is clearly false, for example, that the fact that 1 is less than a million depends on humans, their minds, or their brains. Thus, the whole book is premised on a fallacy that can be spotted by a second year philosophy major.

wanton arboricide *

The author's leading claim in this book is that "the only mathematics we know is human mind and brain based mathematics." This claim can be understood in either of two ways, which the author does not distinguish from one another. On one hand, by "mathematics" he might mean the practice or family of practices that go by that name (the sort of thing that a math teacher gets paid to teach). In this sense, it is just trivial that the only mathematics we know depends on humans, just as the only civil engineering or basketball we know depends on humans. But let's be charitable, and try to construe the author's claim in a way that does not reduce it to a mere triviality. Let's suppose that by "mathematics" he means not the practice of investigating mathematical fact, but the body of fact thus investigated. But if that is what he means, then his claim is clearly false. It is clearly false, for example, that the fact that 1 is less than a million depends on humans, their minds, or their brains.

Thus, the whole book is premised on a fallacy that can be spotted by a second year philosophy major.

Will There Be Pi in The Sky By and By When You Die? ****
Barrow, an astronomer at the University of Sussex when this book was published, provides an entertaining and informative account of the foundations and philosophy of mathematics. Do mathematicians invent or discover mathematics? What 'reality' do mathematical entities like pi have? What accounts for what physicist Eugene Wigner has called, in a now-famous paper, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (299)? After an interesting account of the history of counting and numbers, Barrow discusses in succeeding chapters the philosophies of formalism, inventionism, intuitionism, and platonism, a sophisticated version of which he seems to favor. Perhaps most mathematical workers follow what Alfred Korzybski called "the 'christian science' school of mathematics, which proceeds by faith and disregards entirely any problems of the epistemological foundations of its supposed `scientific' activities" (Science and Sanity 748). I commend Barrow because he considers these epistemological questions important and writes about them so engagingly. Barrow's discussions of theories and personalities provide useful background for understanding mathematical foundations. As for Barrow's conclusions, from a non-aristotelian view, the appeal of platonism seems understandable as an example of identification, the confusion of orders of abstracting. Barrow doesn't seem to consider that mathematicians may both invent and discover mathematics. He seems so taken with the effectiveness of mathematics in the natural sciences that the notion of mathematical entities existing solely as high-order abstractions in human nervous systems seems insufficient to him. As Korzybski pointed out, we live in a world of multi-dimensional, ordered structures or relations. It does not seem unreasonable, then, that we can map this world with an exact language of relations, i.e., mathematics. But as Korzybski also pointed out many times, "the map is not the territory."

Will There Be Pi in The Sky By and By When You Die? ****

Barrow, an astronomer at the University of Sussex when this book was published, provides an entertaining and informative account of the foundations and philosophy of mathematics. Do mathematicians invent or discover mathematics? What 'reality' do mathematical entities like pi have? What accounts for what physicist Eugene Wigner has called, in a now-famous paper, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (299)? After an interesting account of the history of counting and numbers, Barrow discusses in succeeding chapters the philosophies of formalism, inventionism, intuitionism, and platonism, a sophisticated version of which he seems to favor. Perhaps most mathematical workers follow what Alfred Korzybski called "the 'christian science' school of mathematics, which proceeds by faith and disregards entirely any problems of the epistemological foundations of its supposed `scientific' activities" (Science and Sanity 748). I commend Barrow because he considers these epistemological questions important and writes about them so engagingly. Barrow's discussions of theories and personalities provide useful background for understanding mathematical foundations. As for Barrow's conclusions, from a non-aristotelian view, the appeal of platonism seems understandable as an example of identification, the confusion of orders of abstracting. Barrow doesn't seem to consider that mathematicians may both invent and discover mathematics. He seems so taken with the effectiveness of mathematics in the natural sciences that the notion of mathematical entities existing solely as high-order abstractions in human nervous systems seems insufficient to him. As Korzybski pointed out, we live in a world of multi-dimensional, ordered structures or relations. It does not seem unreasonable, then, that we can map this world with an exact language of relations, i.e., mathematics. But as Korzybski also pointed out many times, "the map is not the territory."

Worth a look *****
As someone who barely got through algebra in high school, I can attest that Barrow's book is lucid and engrossing even for the equation-challanged. The book is entertaining and well-written---he manages to hold the reader's interest because he sticks to the interesting theory that underlies mathematics, rather than the nitty-gritty of blah-equals-blah-blah-blah. Why DOES mathematics work so well to describe the real world? We may never know, but it's good to ask the question.

Worth a look *****

As someone who barely got through algebra in high school, I can attest that Barrow's book is lucid and engrossing even for the equation-challanged. The book is entertaining and well-written---he manages to hold the reader's interest because he sticks to the interesting theory that underlies mathematics, rather than the nitty-gritty of blah-equals-blah-blah-blah. Why DOES mathematics work so well to describe the real world? We may never know, but it's good to ask the question.