Innumeracy: Mathematical Illiteracy and Its Consequences (Vintage)

88 Pages·1990·1.29 MB·English

by  John Allen Paulos

Praise for INNUMERACY
"Paulos involves the reader in the fun of numeracy in his entertaining,
thought-provokingbook."
—ChristianScienceMonitor
"[John Allen Paulos] takes us a couple of steps closer to numeracy, and it is
allin allan enlighteningplace to be."
—The NewYork Times
"Paulos gives us a readable romp across a varied mathematical landscape . .
. [He] makes numbers, probability and statistics perform like so many
trained seals for the reader's entertainment and enlightenment." —The
PhiladelphiaInquirer
"Paulos's talent lies in his ability not only to take the reader step by step
through the computations necessary to reach a probability but also to
impart—with a droll humor, a lively irreverence, and a literate as well as a
numerate mind—the wonder and groundedness you can derive from a
quantitative wayofseeingtheworld."
—Boston Phoenix Literary Supplement
INNUMERACY
Mathematical Illiteracy and Its Consequences
John Allen Paulos
VintageBooks
ADivision ofRandomHouse,Inc.
New York
FIRSTVINTAGEBOOKSEDITION,JANUARY 1990
Copyright © 1988 byJohn Allen Paulos
All rights reserved under International and Pan-American Copyright
Conventions. Published in the United States by Vintage Books, a division of
Random House, Inc., New York. Originally published in hardcover by Hill and
Wang, a division of Farrar, Straus and Giroux, New York, in 1988. Published
byarrangement with Hill and Wang, a division of Farrar, Straus &Giroux,Inc.
LibraryofCongressCataloging-in-Publication DataPaulos, John Allen.
Innumeracy : mathematical illiteracy and its consequences / John Allen
Paulos.—1st Vintage Booksed.
p. cm.
Reprint.Originallypublished:NewYork :Hill and Wang,1988.
ISBN0-679-72601-2
1.Mathematics—Popularworks. I.Title. QA93.P38 1990
510—dc20 89-40162
CIP
Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1
To Sheila, Leah, and Daniel for numberless reasons
Contents
Introduction 3
1 Examples and Principles 7
2 Probability and Coincidence 33
3 Pseudoscience 66
4 Whence Innumeracy? 97
5 Statistics, Trade-Offs, and Society 133
Close 177
INNUMERACY
Introduction
"Math was always my worstsubject."
"A million dollars, a billion, a trillion, whatever. It doesn't matter as long as we do something
about the problem."
"Jerryand I aren't goingto Europe, what with all the terrorists."
Innumeracy, an inability to deal comfortably with the fundamental notions of number and
chance, plagues far too many otherwise knowledgeable citizens. The same people who cringe
when words such as "imply" and "infer" are confused react without a trace of embarrassment to
even the most egregious of numerical solecisms. I remember once listening to someone at a
party drone on about the difference between "continually" and "continuously." Later that
evening we were watching the news, and the TV weathercaster announced that there was a
50 percent chance of rain for Saturday and a 50 percent chance for Sunday, and concluded that
there was therefore a 100 percent chance of rain that weekend. The remark went right by the
self-styled grammarian, and even after I explained the mistake to him, he wasn't nearly as
indignant as he would have been had the weathercaster left a dangling participle. In fact, unlike
other failings which are hidden, mathematical illiteracy is often flaunted: "I can't even balance
my checkbook." "I'm a people person, not a numbers person." Or "I always hated math."
Part of the reason for this perverse pride in mathematical ignorance is that its consequences
are not usually as obvious as are those of other weaknesses. Because of this, and because I
firmly believe that people respond better to illustrative particulars than they do to general
exposition, this book will examine many real-world examples of innumeracy— stock scams,
choice of a spouse,newspaper psychics, diet and medical claims, the risk of terrorism, astrology,
sports records, elections, sex discrimination, UFOs, insurance and law, psychoanalysis,
parapsychology, lotteries, and drug testing among them.
I've tried not to pontificate excessively or to make too many sweeping generalizations about
popular culture or our educational system (a la Allan Bloom), but I have made a number of
general remarks and observations that I hope are supported by the examples. In my opinion,
some of the blocks to dealing comfortably with numbers and probabilities are due to quite natural
psychological responses to uncertainty, to coincidence, or to how a problem is framed. Others can
be attributed to anxiety, or to romantic misconceptions about the nature and importance of
mathematics.
One rarely discussed consequence of innumeracy is its link with belief in pseudoscience, and
the interrelationship between the two is here explored. In a society where genetic engineering,
laser technology, and microchip circuits are daily adding to our understanding of the world, it's
especially sad that a significant portion of our adult population still believes in Tarot cards,
channeling mediums, and crystalpower.
Even more ominous is the gap between scientists' assessments of various risks and the
popular perceptions of those risks, a gap that threatens eventually to lead either to unfounded and
crippling anxieties or to impossible and economically paralyzing demands for risk-free guarantees.
Politicians are seldom a help in this regard since they deal with public opinion and are therefore
loath to clarify the likely hazards and trade-offsassociated with almost any policy.
Because the book is largely concerned with various inadequacies—a lack of numerical
perspective, an exaggerated appreciation for meaningless coincidence, a credulous acceptance of
pseudosciences, an inability to recognize social trade-offs, and so on— much of the writing has a
debunking flavor to it. Nevertheless, I hope I've avoided the overly earnest and scolding tone
common to many such endeavors.
The approach throughout is gently mathematical, using some elementary ideas from
probability and statistics which, though deep in a sense, will require nothing more than
common sense and arithmetic. Some of the notions presented are rarely discussed in terms
accessible to a wide audience and are the kind of thing that my students, for example, often
enjoy but usually respond to with: "Will we need to know that for the quiz?" There won't be a
quiz, so they can be enjoyed freely, and the occasional difficult passage can be ignored with
impunity.
One contention of the book is that innumerate people characteristically have a strong
tendency to personalize—to be misled by their own experiences, or by the media's focus on
individuals and drama. From this it doesn't necessarily follow that mathematicians are
impersonal or formal. I'm not, and the book isn't either. My goal in writing it has been to
appeal to the educated but innumerate—at least to those whose fear of mathematics is not so
great that (num)(ber) is automatically read as (numb)(er). The book will have been well worth
the effort if it can begin to clarify just how much innumeracy pervades both our private and our
public lives.
1
Examples and Principles
Two aristocrats are out horseback riding and one challenges the other to see which can come
up with the larger number. The second agrees to the contest, concentrates for a few minutes, and
proudly announces, "Three." The proposer of the game is quiet for half an hour, then finally
shrugs and concedesdefeat.
A summer visitor enters a hardware store in Maine and buys a large number of expensive
items. The skeptical, reticent owner doesn't say a word as he adds the bill on the cash register.
When he's finished, he points to the total and watches as the man counts out $1,528.47. He then
methodically recounts the money once, twice, three times. The visitor finally asks if he's given
him the right amount of money, to which the Mainer grudginglyresponds,"Just barely."
The mathematician G. H. Hardy was visiting his protege, the Indian mathematician
Ramanujan, in the hospital. To make small talk, he remarked that 1729, the number of the taxi
which had brought him, was a rather dull number, to which Ramanujan replied immediately, "No,
Hardy!No, Hardy!It is a very interesting number. It is the smallest number expressibleas the sum
oftwo cubes in two differentways."
BIGNUMBERS,SMALLPROBABILITIES
People's facility with numbers ranges from the aristocratic to the Ramanujanian, but it's an
unfortunate fact that most are on the aristocrats' side of our old Mainer. I'm always amazed and
depressed when I encounter students who have no idea what the population of the United States
is, or the approximate distance from coast to coast, or roughly what percentage of the world is
Chinese. I sometimes ask them as an exercise to estimate how fast human hair grows in miles per
hour, or approximately how many people die on earth each day, or how many cigarettes are
smoked annually in this country. Despite some initial reluctance (one student maintained that
hair just doesn't grow in miles per hour), they often improve their feeling for numbers dra-
matically.
Without some appreciation of common large numbers, it's impossible to react with the proper
skepticism to terrifying reports that more than a million American kids are kidnapped each year,
or with the proper sobriety to a warhead carryinga megatonofexplosivepower—theequivalentofa
milliontons(or two billion pounds) ofTNT.