Pi in the Sky: Counting, Thinking and Being

by John D. Barrow

John D. Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of 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.

Genres: Science, Mathematics, Nonfiction, Philosophy, Physics, History, Psychology

Paperback

First published January 1, 1992

About the author

John D. Barrow was a professor of mathematical sciences and director of the Millennium Mathematics Project at Cambridge University and a Fellow of the Royal Society.

He was awarded the 2006 Templeton Prize for "Progress Toward Research or Discoveries about Spiritual Realities" for his "writings about the relationship between life and the universe, and the nature of human understanding [which] have created new perspectives on questions of ultimate concern to science and religion".

He was a member of a United Reformed Church, which he described as teaching "a traditional deistic picture of the universe".

Reviews

[Paige McLoughlin · Rating 5 out of 5]

I have read this book many times. I think my first reading was in the mid-1990s. This book touches on the development of mathematics in human prehistory and history. It touches on the foundation crisis of mathematics in the early twentieth century and is an excellent introduction to the philosophy of mathematics. It was written in the 1990s and is surprisingly early in coming up with Max Tegmark's Mathematical Universe Hypothesis and is representative of all the schools but I have a feeling Barrow's heart is with Platonism as is mine. Of course, it is in tune with the current debate but again is as old as the Pythagoreans.


Update 4/25/2021
Read this not only its history from the babylonians through the Greeks and the invention of zero and Arabic numerals to Dedekind Weierstraus Cantor Frege Russell and Godel and the foundational crisis of mathematics. Not only that but heavily covers the philosophy of mathematics the author covers all the major schools of thought but like me he favors Mathematical Platonism. Excellent book I return to often.
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READING PROGRESS
July 10, 1995 – Started Reading
July 13, 1995 – Finished Reading
October 31, 2020 – Started Reading
November 1, 2020 – Finished Reading
January 18, 2021 – Shelved
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January 18, 2021 – Shelved as: philosophy
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January 18, 2021 – Shelved as: psychology

[David · Rating 5 out of 5]

This is a very interesting book, which in spite of its title, is mostly philosophical, not mathematical. The author, a prominent astronomer/cosmologist, notes that mathematics, spite of concerted efforts, remains remarkably elusive -- due to Godel's theorem we can never "prove" that the axioms of mathematics are consistent -- in much the same way that attempts to "prove" the existence of God have remained elusive.

[Rich · Rating 4 out of 5]

It's been awhile since my read. I remember a few things that continue to fascinate:

Do numbers exist? Or are they human concepts applied to the physical world - ie height, width, length.

Math use to be magic. People got burned at the stake for knowing the square root of 36.

As someone who scored higher on the verbal portion of the SAT, I can say that this book doesn't require a calculator or abacus. Gaining insight into how "numbers" play a role in our lives and the universe at large will prompt more interesting questions that you ones you have now. Enjoy.

[Joe · Rating 5 out of 5]

Opening question: Did humans discover mathematics, or did humans invent it?
I read this book many years ago and many of its ideas have not only stayed with me, but have grown into bigger ideas, or awareness of the physical world we live in. This is a history of mathematics that covers many cultures and reveals a lot about the universe we live in, in what we have to look forward to discovering.

James Barrow is nothing short of a genius.

[Nancy Mills · Rating 2 out of 5]

more philosophy than math. Interesting in parts but not my cup of tea.

[Saturnina · Rating 3 out of 5]

The book itself was fascinating, but at least the Finnish translation was so full of errors that it was a bit frustrating at times to read. I'm not sure whether it was due to this, or Barrow's lack of explanation but I got completely puzzled by what he was trying to say about platonism, though in the end I think he got it.

[Cera · Rating 3 out of 5]

An interesting look at the various theories about what mathematics is and where it comes from. It suffers from a lot of repetition at times, and the counting chapter is painful due to the old-fashioned anthropological foundation the author seems to be working from, but it was still a worthwhile reread.

[C · Rating 5 out of 5]

A very entertaining story of pi. Those who love trascendental numbers will enjoy it, too.

[James · Rating 3 out of 5]

The first chapter on the history of counting and numbers is extremely comprehensive and worth reading by anyone. The rest of the book spins off into logic abstraction that is very hard to follow

[Carmel-by-the-Sea · Rating 4 out of 5]

Co o matematyce sadzą sami matematycy? Skąd ona według nich pochodzi? Czy oni ją wymyślają, czy odkrywają? Czy w opisie ilościowym świata, inna nauka mogłaby ją zastąpić? W jakiej jest ona relacji do świata rzeczywistego?

Fizyk matematyczny John Barrow w książce "Pi razy drzwi. Szkice o liczeniu, myśleniu i istnieniu" prześwietlił matematykę pod kątem tych pytań. Zabrał czytelnika w świat liczb, starożytnych problemów geometrycznych i współczesnych fundamentalnych pytań o granice matematyki. Bardzo cenne wydały mi się partie, w których autor pokazał XX-wieczne dylematy matematyków i spory o poprawny sposób jej uprawiania. Matematyka to żywa nauka, która jest czymś więcej niż przypisywaniem elementom świata nas otaczającego etykiet liczbowych, formuł. Przenika świat swoją niedościgłą logika i ścisłością. Nie jest jednak sama siebie pewna.

Kluczowe problemy, które rozważa Barrow, koncentrują się wokół ustaleń Kurta Gödla, który wykazał jej niedoskonałości. Matematyka jest wystarczająco pojemna w metody, by określać swoje granice. Ponadto okazuje się, że tę słabość można przekuć w siłę nowych wyzwań intelektualnych, które absorbują teoretyków od kilkudziesięciu lat.

Układ książki jest dość czysty w formie. Pierwsze 30% tekstu, to świetna panorama historycznego rozwoju sposobów liczenia i dyskusja nad użytkowymi walorami tej nauki. Autor nie szczędzi jednoznacznych przykładów pomysłowości ludzkiej, która pozwoliła przypisać obiektom materialnym reprezentacje liczbowe. W efekcie człowiek nabywał biegłości w abstrakcyjnym myśleniu. Ta część świetnie zdaje relację z podstaw naszego sukcesu w świecie przyrody. Pozostałe 70% tekstu, to dyskusja nad kondycją samej matematyki. Barrow staje się w tej części bardziej filozoficzny i wymaga od czytelnika większej uwagi. Rozważa kilka konkurencyjnych prądów myślowych, które wypracowali badacze przez lata do opisu tego, co robią. Pochyla się nad konstruktywizmem, formalizmem, intuicjonizmem i inwencjonizm by pokazać, jak rożnie można 'czuć' matematykę. Spory fragment końcowy, to dyskusja nad platonizmem, czyli najpopularniejszą wizją matematyki, w której jest ona odkrywana. W tym podejściu, wszystkie jej struktury istnieją gdzieś, a ludzie muszą je tylko wydobyć na światło dzienne.

Nie da się książki w sposób jednorodny zreferować, odnosząc się do każdego elementu, by opinia pozostała w miarę krótka. Chciałbym jedynie podziękować autorowi szczególnie za dwa fragmenty. Pierwszym jest dyskusja o kompresowalności i wynikających z niej różnych zastosowaniach matematyki w naukach ścisłych i społecznych (str. 234-237). Drugi fragment, to cudowna opowieść o dziedzictwie Georga Cantora, który okiełznał nieskończoność, pokazując jak należy ją rozumieć; a także kiedy może prowadzić nas na manowce (str. 293-308). Świetny esej intuicyjnie przyswajalny przez każdego.

Książka napisana jest językiem dość precyzyjnym, choć z licznymi 'udogodnieniami dla niewtajemniczonych'. Barrow bardzo szeroko szuka kulturowych korzeni matematyki. Pokazuje, czemu w tak na pozór spójnej nauce, dochodzi do kontrowersji. Z tego względu książka jest unikatowa - wymagając wyłącznie skupienia czytelnika (ale bez przesady) - pozwala na dotknięcie sedna pytań o sens matematyki, który absorbuje myśli jej twórców. To bardzo ambitny cel, który w większości opisanych problemów udał się jasno zasygnalizować.

Polubić matematykę, to wyzwanie dla większości z nas. Mogę jedynie każdego zachęcać do jej zgłębiania, na własnym poziomie i w dowolnym tempie. Barrow pięknie jej zalety opisał słowami (str. 416):

"Mistyk skłania się ku celebracji; matematyk celebruje pracę umysłową. Istotą matematycznej działalności jest możliwość powtórzenia jej zarówno przez tego samego człowieka, jak i przez innych ludzi. Nie ma żadnych gnostyckich tajemnic."

Polecam "Pi razy drzwi" każdemu, przede wszystkim tym, którzy matematykę odbierają, jako ciąg liczb i formuł kotłujących się na co dzień wyłącznie w umysłach ludzi chodzących z głową w chmurach.

ŚWIETNE - 8/10

[Dean · Rating 4 out of 5]

This was a slow read for me, not necessarily because of the subject matter, but more so because of the type of my particular edition. (Little Brown and Company: Back Bay Books)
After completion of Barrow's book 'Pi in the Sky', it compels me to read other books he has authored. Many interesting items in the book, one of which declared that the number 6 being a 'perfect number' because all of its divisors (1,2,3) add to the number itself (6). St. Augustine represented a viewpoint of the time that "six was a special number because God chose to create the world in six days into a claim that 'because' six is a 'perfect' number God chose to complete his creative work in a six-day period." Figure 6.1 of the book is indicative of three alternative viewpoints concerning the relationship of God and Mathematics: a) a traditional Western position in which mathematics is part of the created order: b) the position that mathematics is a necessary truth that is larger than God in the sense that He cannot suspend or change the rules of mathematics: and c) the view that God and mathematics are the same.

[Paige McLoughlin · Rating 5 out of 5]

I have read this book many times. I think my first reading was in the mid-1990s. This book touches on the development of mathematics in human prehistory and history. It touches on the foundation crisis of mathematics in the early twentieth century and is an excellent introduction to the philosophy of mathematics. It was written in the 1990s and is surprisingly early in coming up with Max Tegmark's Mathematical Universe Hypothesis and is representative of all the schools but I have a feeling Barrow's heart is with Platonism as is mine. Of course, it is in tune with the current debate but again is as old as the Pythagoreans.

[Jeff Keehr · Rating 3 out of 5]

I have no recollection of this book. I read it in 1993, which was before I began to record my impressions of the books I read.

[Guy D · Rating 4 out of 5]

Lots to ponder

[Rhys · Rating 3 out of 5]

A fun and readable romp through mathematics-as-Idea, mathematics-as-Material Nature, as well as through subjectivity/objectivity dualism and the truth requirement of the outside observer (or a set of truth statements larger than the one being employed). Barrow tantalizes with logical paradoxes, the idea of logical friction, and almost Borgesian ontological arguments for computer simulations.

"Alfred Tarski took Gödel’s arguments a little further to show that logical systems are also semantically incomplete as well. He showed that if a mathematical system is consistent then the notion of truth – that is, the collection of all true theorems of the system – is not definable in the system itself. This shows that there are concepts which just cannot be defined within the compass of some formal systems. … One can always define them using a bigger system, but only at the expense of creating further undefinable concepts within the larger system."

[David Lai · Rating 2 out of 5]

Not the easiest of reads.

The long section on early counting systems became very tedious, and after the bit on Cantor's infinities, which was pretty lucid, the prose became rather opaque -- at least to me -- what with Barrow failing to explain ideas such as "complex enough to include arithmetic".

There must be better expositions of the philosophy of mathematics.

[Todd Stockslager · Rating 4 out of 5]

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.

[Manny Tingplants · Rating 3 out of 5]

Highly interesting subject matter, but the first edition I'm reading appears to not have been thoroughly edited. Redundant statements, copious and not always relevant quotes, miss-labelled diagrams, odd double prepositions of the sort I find in my own writing after not splicing sentences together carefully enough. Very awkwardnessful. It could probably be condensed to 150 pages without losing anything while enhancing readability. The upside is that you really have to pay attention and think critically about what you're reading. And it is a fascinating subject.

[David Buhler · Rating 4 out of 5]

It was an excellent introduction to the origin and philosophy of mathematics to this awed reader. I never dreamed there were so many branches, so many differing ideas about what math really is. The great mystery remains: is math a created or discovered by the human mind? !?

[Kevin Brown · Rating 5 out of 5]

Totally profound.