Pi in the Sky: Counting, Thinking, and Being

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 ([email protected])

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."

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.

When I took algebra in high school I didn't like it. My teachers seemed to say "All the interesting problems have been solved so just memorize your textbook for the quiz on Friday, please." Not a presentation that would inspire most teenagers.The books starts with an introduction that really grabbed me. It talks about how most scientific theories are expressed in the language of mathematics and then asks a simple question: Why? What is it about the world that makes it so mathematical? The introduction clearly lays out mathematics deepest secret: Beneath all the formulas and proofs there is something about math that is mysterious and profound. This was not something my high school teacher pointed out.The following chapters present the history of mathematics in an style that manages to inform about important concepts without getting bogged down in formulas. The author strikes a delicate balance between writing about mathematicians as people and writing about their work and its importance. In the end, he doesn't have any answers about the deep questions posed in the introduction. But after reading his book it didn't matter because at least I now understood better what the questions meant and could appreciate their profound, abstract beauty. Sort of like the difference between looking up in the night sky and seeing little points of light versus seeing the vast universe. Excuse my hyperbole, but that's the best I can explain it.Anyway, I highly recommend the book to anyone who might read this review. I especially recommend it to students just beginning their math studies. This book will help them appreciate the subject in a way that no textbook ever could.

This book did more to aid my understanding of modern mathematics than any graduate math course I took. I recommend it for those that want to understand how to understand math. This book, plus "Godel, Escher, Bach" and "Fuzzy Thinking" make will remove the confusion instilled by small minded math teachers.