SEPTEMBER 2009 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS 1

God? A MathematicianChallenges the Proofs

Reviewed by Samuel OttenMichigan State University

It is the job of mathematicians to thinkcarefully about definitions, to beaware of underlying assumptions, to

construct and analyze arguments, andto ultimately uncover truths. For thesereasons it seems appropriate thatmathematicians have stepped forwardto join the discussion regarding theexistence (or nonexistence) of God.Reasoning and rationality—centralcharacteristics of mathematics—areprecisely the things that distinguish usas humans, and it is only natural thatwe should use them when contemplat-ing questions that we consider mostimportant to humanity.

With his new book Irreligion, mathe-matician John Allen Paulos has enteredthe swirling debate regarding the role ofreligion in society and of science inreligion. Paulos is by no means the firstto make an intellectual case for atheism(Paulos himself calls attention to thework of people like Sam Harris andChristopher Hitchens) but he does offersomething unique and personal: amathematical perspective leading tothe conclusion that arguments forGod’s existence “don’t add up.”

Making his position eminently clear,Paulos defines the term irreligion as“topics, arguments, and questions thatspring from an incredulity, not onlyabout religion, but also about others’credulity.” He asserts that “the first stepin untangling religious absurdities” is toexamine the definition of God. Paulosclarifies that the arguments addressedin Irreligion are essentially gearedtoward the familiar monotheistic God—a personal creator with a wise and

powerful hand in daily events. Thisbecomes vitally important later in thebook. For instance, in the chapter onthe Argument from Redefinition, Paulosadmits that a sufficient watering downof the concept of God assures God’sexistence, albeit in a “very strainedPickwickian sense.” Similarly, theArgument from First Cause, even if itdid hold, would only prove theexistence of a God who is the cause ofthe universe; none of the benevolent ormoral characteristics ascribed to sucha God would necessarily follow.

Clarity of definition comes into playelsewhere in the book. While address-ing the Argument from Interventions,Paulos unpacks the term “miracle.” If amiracle is a highly unlikely event, thenmiracles happen every day. Manypeople, however, use the term to referto an act of God. They then claim thatthe existence of miracles (even underthe first definition) proves God’sexistence. Paulos points out thisconflation of concepts and, concerningthe divine interpretation of miracles,asks, “Why do so many in the mediaand elsewhere refer to the rescuing of afew children after an earthquake or atsunami as a miracle when theyattribute the death of perhapshundreds of equally innocent childrenin the same disaster to a geophysicalfault line? It would seem either both arethe result of divine intervention or bothare a consequence of the earth’s platesshifting.”

Another mathematical skill on display inIrreligion is the detection of latentassumptions. For example, take theArgument from First Cause:

1. Everything has a cause.

2. Nothing is its own cause.

3. Causal chains can’t go on forever.

4. So there has to be a first cause, namely, God.

Paulos points out several problemswith this argument, starting with the“gaping hole” in assumption 1. Whenconsidered with assumption 4, we seethat assumption 1 should actually read“Either everything has a cause orthere’s something that doesn’t.” Pauloscontinues, “If everything has a cause,then God does too, and there is no firstcause. And if something doesn’t have acause, it may as well be the physicalworld as God.”

Two other common flaws witharguments for the existence of Godthat Paulos engages are logicalfallacies and errant notions ofprobability. From a logical standpoint,he quickly uncovers circular reasoningwithin the Argument from the AnthropicPrinciple and the Argument fromProphecy. There is also a chapterconcerning the logic of self-referenceand recursion, which contains familiarbut nonetheless interesting thoughtsfrom Bertrand Russell. (See Russell’s1957 book, Why I Am Not a Christian,for another mathematician’s take onreligion.)

Irreligion: A Mathematician ExplainsWhy the Arguments for God Just Don’tAdd Up, by John Allen Paulos, Hill andWang, ISBN: 978-0-8090-5919-5

E

BookReviews

Book Reviews 8/17/09 11:21 PM Page 1

2 SEPTEMBER 2009 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS

appeared in connection with two-person games in which 21 + 27 = 14 and 8 � 8 = 13.

One more particularly appealing geminvolves the so-called multiplicationfrieze patterns (pp. 74-76). On a pieceof graph paper, draw a pattern bound-ed by a zigzag of ones on the left andhorizontal lines of ones above andbelow. Fill in the empty cells within thispattern by the rule that the numbers a,b, c and d in the diamond pattern

ba d

c

satisfy the equality ad = bc + 1. It turnsout that such a pattern repeats itselfeventually, and as a bonus, the entriesare all integers. The authors then ask,“See if you can work out why.”

The joy that the authors take in theirsubject comes through on every page.Some of the material takes more than alittle sophistication to understand, butthe authors’ explanations are carefullywritten and invariably come with helpfuldiagrams. Even with all of its technicalmaterial and nontrivial content, this isstill a wonderful book, and I highlyrecommend it.

10.4169/194762109X468355

The Book of Numbers, by John H.Conway and Richard Guy, Springer-Verlag, ISBN: 0-38797993-X

Probabilistic reasoning is wellrepresented in Irreligion because, sadly,it is so often misused in argumentation.Proponents of the Argument fromProphecy maintain that the couching oftales about God in an enormousamount of detail makes them morelikely to be true. This is an example ofthe conjugation fallacy, which is theerrant belief that the probability ofevents A, B, and C occurring is greaterthan the probability of A alone (becauseB and C seem to support A).

In Irreligion, Paulos not only exhibitsthe clarity of thought expected of amathematician, but also an ease ofcommunication and a healthy dose ofwit as he takes on such prominentproponents of God as C.S. Lewis,Francis Collins, James Redfield, andeven Mel Gibson. Reading theserebuttals and refutations confirms thatit is beneficial to have a mathematicianfront-and-center in the debate aboutthe existence of God because of theinsight and rationality that ourprofession affords.

A New Testament for theBook of Numbers

Reviewed by Ezra BrownVirginia Tech

Although the majority of MathHorizons book reviews aredevoted to new releases, there

are many older titles deserving of ashout-out to today’s newer readers.The Book of Numbers by John Conwayand Richard Guy is certainly on this list.The authors, who are both celebratedmathematicians and excellentexpositors, have assembled animpressive collection of numberpatterns, properties, pictures, andproofs aimed at the inquisitive readerwith no specific mathematicalbackground. Just to give you the flavorof the collection, here are a fewexamples from each of the ten chaptersthat make up the book.

1. The probable origins of “Hickory, Dickory, Dock” and “Eeny, meeny, miney, mo,” and what sheaves (of wheat) and shocks (of corn) have to do with the number sixty.

2. Forty-two lovely full-color diagrams that reveal many number patterns from the familiar triangular numbers n(n+1)/2 and the squares n2 to the stella octangula numbers n(2n2+1) and the truncated octahedral numbers 16n3– 33n2 + 24n–6.

3. Why the next number in the sequence 1, 2, 4, 8, 16, is 31 and not 32.

4. A clear explanation of the reasons why Fibonacci numbers occur in the systems of spirals on pineapples and in sunflower heads.

5. Why the distribution of prime numbers is more orderly than you think.

6. What fractions have to do with card shuffling and the so-called Metonic calendar cycle, in which 235 months is very nearly equal to 19 years.

7. How the interaction between algebra and geometry led to the proof that it is impossible to trisect an arbitrary angle using only a compass and straightedge; also, the “look and say” sequence 1,11,21211,111221,312211,…and why the number of digits in the nth

term of this sequence is roughly proportional to a root of an algebraic equation of degree 71.

8. Why a few simple pictures show how “imaginary numbers” are very real.

9. The celebrated “transcendental” numbers e and π together with a compelling argument, complete with four diagrams, that dispels the mystery about Euler’s famous equation eπi + 1 = 0.

10. A system of adding and multiplying called Nim arithmetic that first

Book Reviews 8/17/09 11:21 PM Page 2