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Excerpts
from The Universe and the Teacup
by K. C. Cole
What’s
Math Got To Do With It?
Chapter 1
We cannot afford to remain dumb about mathematical ideas simply because we hated them in high school—any more than we can remain dumb about computers or AIDS. Mathematics is essential, not peripheral, knowledge.
Math
has the power to sift through evidence and decide what is true in a wide
variety of situations. Some of the tools may be obvious (like probability) while
others are subtler and even obscure (like the relationship between symmetry,
truth, and things that never change, no matter what).
Mathematics
brings surprising clarity to an astonishing range of issues, from cosmic
questions (the fate of the universe) to social controversy (O.J.’s guilt) to
specific matters of public policy (race and IQ scores).
.
. . tools of mathematics allow one to see otherwise invisible patterns and
connections. Mathematics has revealed hidden trends (HIV infection), new
kinds of matter (quarks, dark matter, antimatter), and crucial correlations
(between smoking and lung cancer). It
does this by exposing the bare bones of a situation, overcoming the commonsense
notions that so often lead us astray. Math
allows you to strip off the coverings and get right down to the skeleton. What
is going on underneath that accounts for what you see on the surface?
What’s holding it up? If
you dig deep enough, what do you find?
Mathematics
did not appear our of nothing and nowhere . . .. It was created (or discovered, if you prefer) by human
beings. The ways our brains and bodies work have
molded not only the study of mathematics but also our everyday perceptions of quantitative things.
Most
number systems used in societies around the world are built on multiples of
ten because virtually all human beings are born with ten fingers and ten
toes. (Some cultures expand upon
this idea by making use of wrists, elbows, shoulders, and chest.)
The
world beyond our physical bodies is sculpted by forces that produce omnipresent
mathematical objects. Geometry really does grow on trees. Because of the
way gravitational, electrical, and nuclear forces operate, everything Moon size
or larger is round or roundish. Water
falls from a fountain in parabolas. Soap
bubbles meet at 120-degree angles, and the two hydrogens and oxygen in water
molecules meet at 105 degrees –giving shape to bubbles and snowflakes. Trees
and blood vessels and rivers all branch in strikingly similar ways.
Measures
and notions of time are based on the revolutions of our planet and the
time it takes to wind its way around the Sun. We orient ourselves in space along
Earth’s spin axis (north-south) and magnetic poles. We think of down as a
fixed direction, although down is up for someone living on the other side of the
globe. In truth, down is the direction of the greatest pull of gravity.
If you are alone in space, there is no such thing as down, up, east, or
west.
But
mathematical concepts also go beyond experience.
The perfect circles and right angles favored by geometers do not exist in
the natural world. Numbers can do things that things cannot.
Mathematicians
today rely on all kinds of strange objects that were once thought completely out
of the bounds of common sense: various kinds of infinities; imaginary and
transcendental numbers; higher dimensional geometries; and so forth.
Consider
the extreme difficulty we have with very large or very small numbers.
Anyone who has ever mixed up a billion and a trillion knows that after a
while, all big numbers begin to look alike.
Daily we are bombarded with incomprehensible sums:
The
national debt has grown to trillions of dollars. The Milky Way galaxy contains 200 billion stars, and there
are 200 billion other galaxies in the universe.
The chemical reactions that power everything from fire to human thought
take place in femtoseconds (quadrillionths of a second). Life has evolved over a
period of roughly 4 billion years.
What
are we to make of such numbers?
Our
brains, it appears, may not be engineered to cope with extremely large or small
numbers.
All
of us have trouble grasping how inflation at 5 percent can cut income in half in
a decade or so, or how a population that’s growing at even 3 percent can
rapidly overtake every inch of space on Earth.
If
we can’t readily grasp the real differences between a thousand, a million, a
billion, a trillion, how can we rationally discuss budget priorities?
We can’t understand how tiny changes in survival rates can lead to
extinction of species, how AIDS spread so quickly, or how small changes in
interest rates can make prices soar. We can’t understand the smallness of
subatomic particles or the vastness of interstellar space. We haven’t a clue
how to judge increases in population, firepower of weapons, energy consumption.
If
you draw a line on the chalkboard designating zero on one end and marking a
trillion on the far side, where would a billion fall? Most people put it about a third of the way between zero and
a trillion. Actually, it falls very
near the chalk line that marks the zero.
Compared
to a trillion, a billion is peanuts. The same goes for the difference between a
millionth and a billionth. If the
width of this page represents a millionth of something, then a billionth would
be much less than a pencil line.
S.
George Djogvski, writing in Caltech’s Engineering and Science, offers
the following analogy to help us imagine the
vast distances of space and the
immensity of the national debt.
If
the Sun were an inch across and five feet away from our vantage point on Earth,
“the solar system would be about a fifth of a mile across.
The nearest star would be 260 miles away, almost all the way to San
Francisco (from Los Angeles), and our galaxy would be 6 million miles across.
The next nearest galaxy would be 40 million miles away.
At this point you begin to lose scale, even with this model—the nearest
cluster would be 4 billion miles away, and the size of the observable universe
would be a trillion miles. If you were to ride across it at five dollars per
mile, you could pay off the national debt.”
The
late physicist Sir James Jeans offers the following analogies concerning the
miniscule
scale of molecules, the vastness of space, and the stupendous heat involved in
nuclear fusion:
“The
number of molecules in a pint of water placed end to end . . . would form a
chain capable of encircling the Earth over 200 million times.”
“Empty
Waterloo Station of everything except six specks of dust, and it is still far
more crowded with dust than space is with stars.”
A
pinhead heated to the temperature of the center of the Sun, “would emit enough
heat to kill anyone who ventured within a thousand miles of it.”
Certainly,
exponential scales play a part in human perception of time. We remember last
week more clearly than the week before. At
the same time, each year makes up a smaller fraction of your life—so that one
year at the age of two is half your total life span; at fifty, a year is merely
a fiftieth—and seems to fly by.
All
senses play some part in presenting the world to us in this same “distorted”
way.
We can’t perceive changes that are too gradual, too finely tuned, for
our perceptual system to pick up. We
don’t see mountains move or flowers grow, even though we know that some
mountain peaks were once at the bottom of the sea and that flowers begin as tiny
seeds.
Why
does ignorance concerning exponential amplification harm us?
Think: inflation, epidemics, energy, consumption, nuclear explosions, and
population explosions—you get the idea. Take population growth. People have known since Malthus that population increases
exponentially. But no one worries much because populations are self-limiting.
When people run out of food and places to live, they die or go to war or shop
having children or some combination of all of these. All this to true—to some extent.
“We’re adding a quarter million people every twenty-four hours . . . And no one is really taking action about it” (David Pimentel, Cornell ecologist).
A closely connected problem is consumption. It’s a given in virtually all economic discussions that growth is a desirable goal. To bring Third World countries into the affluence of modernity, we urge them to adopt U.S. style consumption. It doesn’t take much to see where this leads.
“In
a finite environment, inflation cannot continue, however cleverly we may
postpone or disguise the inevitable”(Pimentel). By rethinking our creation
stories in light of current mathematical and scientific research, we may find a
solution of some of humanity’s most pressing problems.
Scientist,
statisticians, and policy makers attach numbers to the risk of getting breast
cancer or AIDS, to flying and food additives, to getting hit by lightning or
falling in the bathtub.
Yet
despite all the numbers floating around, most people are quite properly
confused about risk. I know people who live happily on the San Andres Fault and
yet are afraid to ride the New York subways (and vice versa).
I’ve know smokers who can’t stand to be in the same room with a fatty
steak, and women afraid of the side effects of birth control pills who have sex
with strangers. Risk assessment is rarely based on purely rational
considerations . . .. We worry about flying but not driving. . .
The average male faces three times the threat of early death associated
with not being married as he does from cancer.
(An
airplane crash occurs and the nation is shocked and questions how to prevent it)
Meanwhile, tens of thousands of children die every day around the world from
common causes such as malnutrition and disease. That’s roughly the same as a
hundred exploding jumbo jets full of children every single day. People who care
more about the victims of Flight 800 aren’t callous or ignorant.
It’s just the way our mind works.
As a general principle, people tend to grossly exaggerate the risk of
any danger perceived to be beyond their control, while shrugging off risks they
think they can manage. Thus, we go skiing and skydiving, bur fear asbestos.
We resent and fear the idea that anonymous chemical companies are putting
additives into our food; yet the additives we load onto our own food—salt,
sugar, butter—are millions of times more dangerous.
This
is one reason that airline accidents seem so unacceptable—because strapped
into our seats in a cabin, what happens is completely beyond our control.
In a poll taken after the TWA Flight 800 crash, an overwhelming majority
of people said they’d be willing to pay up to fifty dollars more for a
round-trip ticket if it increased airline safety.
Yet the same people resist moves to improve automobile safety, for
example, especially if it costs money.
The
idea that we can control what happens also influences who we blame when things
go wrong.
Most people don’t like to pay the costs for treating people injured by
cigarettes or riding motorcycles because we think they brought these things on
themselves. Some people also hold
these attitudes toward victims of AIDS, or mental illness, because they think
the illness results from lack of character or personal morals.
For obvious reasons, dramatic or exotic risks seem far more dangerous
than more familiar ones. “A
woman drives down the street with her child romping around in the front seat,”
says John Allen Paulos. “Then
they arrive at the shopping mall, and she grabs the child’s hand so hard it
hurts, because she ‘s afraid he’ll be kidnapped.”
Children
who are kidnapped are far more likely to be whisked away by relatives than
strangers, just as most people are murdered by people they know.
Familiar
risks creep up on us like age and are often difficult to see until it’s too
late to take action. . . a frog
placed in hot water will struggle to escape, but the same frog placed in cool
water that’s slowly warmed up will sit peacefully until it’s cooked. “One
cannot anticipate what one does not perceive.”
It won’t come as a great surprise to anyone that ego also plays a
role in the way we access risks.
Whatever the evidence to the contrary, we think:
“It won’t happen to me.” We
don’t like to see ourselves as vulnerable.
Average people, studies have shown, believe that they will enjoy longer
lives, healthier lives, and longer marriages than the average person. Despite
they themselves are, well, average people too.
According to a recent poll, 3 out of 4 baby boomers think they look
younger than their peers, and 4 out of 5 think they have fewer wrinkles than
other people their age—a statistical impossibility.
We also seem to believe it won’t happen to us if it hasn’t
happened yet. That is, we
extrapolate from the past to the future . . . This is rather like reasoning that
flipping a coin ten times that comes up heads guarantees that heads will come up
indefinitely.
The
Vatican measures saintliness, the military courage, the judicial system
penitence, dog breeders temperament. Both
physicians and policy makers measure human worth.
Juries put dollars and cents on pain and suffering.
Measurement,
it is probably fair to say, is the cornerstone of knowledge.
It allows us to compare things with other things and to quantify
relationships. Mine is bigger than
yours amounts to a mathematical statement (M>Y).
My country/child/car is better/smarter/faster than anyone else’s.
But
measurements have meaning only to the extent that we respect their various
limits. No measurements, when you
get right down to it, are straightforward.
All involve disentangling things that can’t be separated, or
qualifying things that can’t be counted, or defining things you can’t quite
put your finger on.
Indeed,
neither space nor time can be measured in isolation from the other. Other
quantities we try to measure are equally entangled: matter and energy, for example, or mind and brain, or the
influence of genetic and environmental factors on intelligence.
Knowledge
is also lost with the most mundane measurements of everyday life. You cannot chemically analyze your dinner and eat it.
You cannot dissect the mathematics underlying Mozart and at the same time
feel the emotional impact. A Picasso, viewed through a powerful microscope,
dissolves into a grainy pattern of dots. Earth,
viewed from space, reveals itself as a sphere but tells you nothing about
what’s going on in your own backyard. Something
is lost for every measurement that’s gained.
According
to quantum mechanics, your very choice to measure something affects—and even
determines –the measurement you make.
Still, quantum numbers describe atoms a whole lot better than IQ scores describe intelligence. That’s partly because physicists have refined their definitions of atomic properties better than people have defined what it means to be smart (not to mention wise).
Other immeasurables aren’t so obvious. For example, how do you measure success?
To make it easy, let’s just say economic success, and limit comparisons to countries.
On the face of it, this is a snap. You simply measure GNP (gross national product), the sum of all the goods and services the economy produces. But as Harvard economist Amartya Sen has pointed out, traditional measures such as GNP neglect critical factors such as well-being. Famine often exists side by side with plentiful food, and high starvation rates are common in wealthy countries. Sen suggests mortality rates are a much better measure, because they more accurately reflect national well-being. By those measures, American blacks are worse off than the poor of many Third World countries.
One of the most egregious examples of drawing boundaries where none exist pops up in matters of race. Despite the fact that application forms and census forms ask people to choose whether they are black, white, Asian, or other, race is not a biological concept. In a recent survey, 41 percent of anthropologists said that there is no such biological thing as race.
Congregating things into groups can make measurements even more deceptive. For example, since is often isn’t possible to measure every individual within a group, we might substitute the group’s average value. But averages meant to represent every real individual—as in the oft-quoted statistic: The average U.S. family7 has 2.5 children. When someone offers statistics stating, say, that boys on average do better in math than girls on average, that doesn’t say anything at all about the abilities of any particular boy or girl.
Darrell Huff, in his classic book How to Lie with Statistics, gives the unfortunately common example of intelligence tests. An IQ test that gives one person a score of 98 (error of plus or minus 3) and another person a 101 (plus or minus 3) tells you absolutely nothing. And there’s a fifty-fifty chance that the person scoring 98 has an IQ that lies somewhere between 95 and 101. And there’s a fifty-fifty chance that the person scoring 101 has an IQ between 98 and 104. That means the person scoring 98 could be superior to the person scoring 101 by three points (52).
There are good reasons to think a world that’s different in scale will also be different in kind. More or less of something very often adds up to more than simply more or less; quantitative changes can make huge qualitative differences.
When the size of things changes radically, different laws of nature rule, time ticks according to different clocks, new worlds appear out of nowhere while old ones dissolve into invisibility. Consider the strange situation of a giant, for example. Big and strong to be sure, but size comes with distinct disadvantages. According to J.B.S. Haldane in his classic essay, “On Being the Right Size” a sixty-foot giant would break his thighbones at every step. The reason is simple geometry. Height increases only in one dimension, are in two, volume in three. If you doubled the height of a man, the cross section, or thickness, of muscle that supports him against gravity would quadruple (two times two) and his volume—and therefore weight—would increase by a factor of eight. If you made him ten times taller, his weight would be a thousand times greater, but the cross section of bones and muscles to support him would only increase be a factor of one hundred. Result: shattered bones.
Fleas can perform superhuman feats routinely. These puny critters can pull 160,000 times their own weight and jump a hundred times their own height. Small creatures have so little mass compared to the area of their muscles that they seem enormously strong.
A mouse, writes Haldane, could be dropped from a thousand-yard-high cliff and walk away unharmed. A rat would probably suffer enough damage to be killed. A person would certainly be killed. And a horse, he tells us, “splashes”(56).
Water is one of the stickiest substances around. A person coming out of the shower carries about a pound of extra weight, scarcely a burden. But a mouse coming out of the shower would have to lift its weight in water, according to Haldane. For a fly, water is as powerful as flypaper; once it gets wet, it’s stuck for life. That’s one reason, writes Haldane that most insects have a long proboscis (56).
Scale up to molecule-size matters, and electrical forces take over; scale up further and gravity rules. As Philip and Phylis Morrison point out tin the classic Powers of Ten, if you stick your hand in a sugar bowl, your fingers will emerge covered with tiny grains that stick to them due to electrical forces. However, if you stick your hand into a bowl of sugar cubes, you would be very surprised if a cube stuck to your fingers.
We know that gravity takes over in large-scale matters because everything in the universe larger than an asteroid is round or roundish—the result of gravity pulling matter in toward a common center. Everyday objects like houses and mountains come in every old shape, but mountains can only get so high before gravity pulls them down. They can get larger on Mars because gravity is less. Large things lose their rough edges in the fight against gravity. “No such thing as a teacup the diameter of Jupiter is possible in our world,” say the Morrisons. As a teacup grew to Jupiter size, its handle and sides would be pulled into the center by the planet’s huge gravity until it resembled a sphere (57).
Crowd behavior is much more predictable than the behavior of any individual. This fact applies equally to inanimate objects. Toss a coin, and you cannot predict whether it will come up heads or tails; no matter how many times you toss it, the probability of it coming up heads or tails remains fifty-fifty. However, if you toss a coin a million times, you can be certain it will come up tails roughly 500,000 times. While no gambling establishment can predict which number will come up on a single roll of dice, they can predict with some confidence the outcome of a great many rolls—that’s how they make their profit (64).
In a sense, all the patterns of nature, from flowering trees to ocean swells, from mountains to koala bears, are the emergent properties of simple interactions between subatomic particles that over time add up to far more than the sum of their parts(65).
Time may well be the ultimate emergent property. A single particle can go backward or forward in space or time—it makes no difference—and there’s no clear way to tell which is which. The only time that exists is the atom’s own internal clock—the frequency at which it vibrates. But put a bunch of atoms together, and no one has any problem telling which way time flows: It’s always in the direction of disorder. Left to their own devices, food rots, skin wrinkles, paint peels, mountains erode, stockings run. And yet, there is not hint of this large-scale headlong rush toward disorder within any single atom alone.
In one sense, “more is different” is the mathematical version of the old saying about the straw that breaks the camel’s back. At some point, more changes everything(64).
{This change can be negative or positive following the “tipping point.”]
What this means for public policy, Gladwell says, is that we shouldn’t jump to conclusions about the effectiveness or failure of social policies without taking the tipping point into account. We shouldn’t conclude, say, that the welfare system doesn’t help people get out of poverty because it hasn’t accomplished that goal yet. We shouldn’t conclude that money spent on inner city schools is wasted because it hasn’t shown results compared to money put in. It could be that we simply haven’t yet reached the tipping point (66).
The Galileo spacecraft cruised the solar system for six long years before arriving at the giant planet Jupiter in December 1995—the final destination of a 2.3-billion – mile journey that looped twice around the earth and once around Venus.
It was a very tricky maneuver. The probe’s entry had to be as precise as a hypodermic needle slipping in beneath the skin. If the entry angle was too shallow, the probe would bounce off the planet’s atmosphere like a stone skipping from the surface of a pond; too deep, and it would be destroyed before it could phone home any information.
As the world now knows, on December 7, at exactly 5:06 P.M. Pacific time, precisely as planned, the tiny probe dove through Jupiter’s pastel-colored clouds cleanly enough to earn it an Olympic gold. Despite the duration [6 years] and distance [2.3 billion miles], its aim on arrival was picture-perfect (68).
This is the kind of spectacular success that leads people to believe science is good at predicting just about anything: the next earthquake, the next cancer victim, the next stock market crash, the global climate twenty or two hundred years from now.
But prediction is neither the goal nor the forte of science. If truth be told, the physicists can’t even perfectly predict where a Ping-Pong ball will bounce on the other side of the table (69).
The kind of prediction science does so well might better be described as pattern perception. Galileo got to its target not by predicting the future, but by following well-known patterns to their logical conclusions. Objects in motion follow well-understood paths as they coast and fall and loop around bodies in space. If you know the patterns, it becomes a matter of mere calculation to get Galileo to Jupiter on target (72).
Prediction, said physicist Frank Oppenheimer, “is dependent only on the assumption that observed patterns will be repeated. The merchant, the politician, the parent, the artist, and the doctor all depend for their success on the subtleties of pattern recognition. . . . The predictions of the physicists, the psychologist, or the economist in no way set them apart from the rest of humanity” (74).
In these cases, researchers trying to predict the future trends often rely on a technique called curve fitting I which a mathematical function is found that describes an existing pattern.
The problem is, the same curve can be described by very different equations. Author and physiologist Robert Root-Berstein has described the dangers of over reliance on this technique of curve fitting as a way to extrapolate into the future, and he points to some notable failures—for example, in predictions about the spread of AIDS or the threat of global warming. Just twenty years ago, he points out, articles in science journals warned of a coming ice age and the potential for galloping glaciers (75).
He argues instead that we first need a deeper understanding of the basic phenomena involved; that we need to learn much more abut climate and disease and population before we can sensibly begin to predict the future. Predictions based on extrapolation can’t be any more accurate than the models we begin with. The Galileo spacecraft fount its way to Jupiter right on target because celestial dynamics are well understood. The same cannot be said of many other sciences (75).
Encoded into physics is the Heisenberg uncertainty principle, this weird fuzziness of the subatomic realm seems to put a natural limit on what we can know. . . . A simple prediction such as where a particular drop of water will fall as it crashes over Niagara Falls is beyond our capabilities. It simply requires too much information (76).
Consider the information required to predict the trajectory of a ball batted into right field by a baseball player. For openers, you would need precise information about the velocity and spin of the ball, the elasticity of the materials, the interaction of surfaces, the weight and structure of the batter’s hands, not to mention winds, temperature, humidity, and so forth. Newton’s laws are simple enough, but the ingredients added by reality make the problem overwhelmingly complex. Plus, someone could throw a popcorn box at the ball, or a bird could fly into its path (76).
The same tangle obscures the exact origin of many health problems, for example, why one person gets cancer and another doesn’t. Genetics, environment, and behavior all interact in ways too complicated—at least for now—to sort out (78).
According to Ivar Ekeland in the Broken Dice, “If we want to know what the weather will be like one year from today, we have to take everything into account, from the butterflies flying in the Amazon jungles to the candles burning in churches” (79).
Filtering out the static is a process central to both science and human perception. You can’t perceive anything if you can’t block most of the information that comes your way. You can’t hear the voice on the phone if other people are shouting in the background; you couldn’t see anything at all if your iris didn’t shut out most of the light—allowing only a tiny bit to sneak in through the pupil (84).
Where do the stars spend their days? They’re up there, of course, decorating the sky as always, but you can’t see them in the glare of the light from the Sun. You can’t see the stars in the day for the same reason you can’t hear a whisper in a noisy restaurant – the insistent Sun shouts them out (84).
Even at night, there’s a lot of noise in the sky that can make it hard for astronomers to see the stars; there’s light from the Moon and city; there’s heat fro the telescope itself (many are refrigerated); there’s wind that stirs up the images and makes them as blurry as a penny on the bottom of a pool. A lot of the art in astronomy (and other sciences) is figuring out how to get around the noise without losing the signal (84).
To some extent, the noise problem is merely an artifact of choice and circumstance. Consider the pile of dust and dirt accumulating under the refrigerator. The crumbs were recently part of the cake you had for breakfast; the cat hair was part of the animal’s fur; the leaf belonged to a tree; and the paper clip found its way to the floor from something you opened in the mail. You don’t consider any of these things as candidates for the trash heap until they wind up in the “wrong” place (85).
These things we dismiss as noise often have a great deal to tell us, however, Both scientists and artists learn to pay attention to the crumbs that other people are about to sweep under the rug. They learn to be good noticers. The same could be said about good teachers, good parents, effective politicians. (85).
Post-it Notes were invented in much the same way. A chemist investigating ways to make a better glue found instead glue so ineffective it wouldn’t permanently attach to anything. Instead of throwing the invention in the garbage, the chemist saw that it would work perfectly in a different context—sticky notepaper that could be peeled off without a trace (86).
In The Collapse of Chaos, Stewart and Cohen point out that no signal has inherent meaning—outside the context of whatever hears it or sees it or decodes it. A compact disc, for example may contain all the information necessary to play a piece of music. But without a CD player, it is only a pretty silvery disk—suitable for playing Frisbee. In the same way, a strand of DNA contains no message in the absence of molecules that can read the genetic code. “The number 911 has no inherent meaning,” they write. “ In the context of the U.S. telephone system, it means “emergency”; in the context of a lottery it may mean “you lose”; and in the context of housing it means that you live on a fairly long street” (88).
Consider the plight of a team of researchers who finally saw the first wrinkles I space-time coming at us from 10 or so billion years ago, the fossil imprints of the conditions at the origins of the universe (91) . . . various astronomers had been puzzled by an unexpected hiss in the sky. Thinking it was noise, the researchers figured it was the result of a creaky instrument or some interfering sources –perhaps bird droppings in the antenna.
The theorists, meanwhile, had figured out that if the universe really did start with the big bang, it should be possible to detect traces of left over radiation from the explosion; it should still pervade the sky, common at us from the origin, now vastly stretched and cooled over the course of its 10-billion-year history (91).
The punch line, of course, is that the “noise” bothering the astronomers turned out to be that very leftover radiation from the big bang. But that was only the beginning. As soon as people realized that they were reading messages from the origins of the universe, they set about trying to decode them—in particular, to find the seeds of the structure in the universe, the tiny lumps in space-time that eventually grew into the greats strings and clusters of galaxies that drape the darkness like garlands(91).
CHAPTER 9:
VOTING: LANI GUINIER WAS RIGHT
In 1993, Bill Clinton nominated a little-known law professor named Lani Guinier as his candidate to head the Justice Department’s civil rights division. . . . She was called a “radical,” a “mad-woman” a “quota queen,” a “racist” and even “loony Lani.”
In a nutshell, Guinier was accused of questioning the very core of American democracy—the sacred ideal of majority rule. She was telling people what they didn’t want to hear—that our election systems were neither fair nor democratic. What they produced was not democracy, but a tyranny of the majority. That phrase –“tyranny of the majority” was lifted from the lips of none other than founding father James Madison. It was he who argued that the tyranny imposed by 51 percent of the people was every bit as threatening to democracy as the royal tyranny the colonists fought to leave behind (100).
It is easy to show that election results depend directly on the choice of the voting system. Even when the preferences of the voters don’t change, they can choose different winners if they change the details of the way they vote (101).
Plurality means that a candidate needn’t be all that popular to win. In a wide field of primary candidates, it’s possible to get a winner with less than 20 percent of the vote. And if that scenario develops, says New York University political scientist Steven Brams, the winner is likely to be an extremist—“the one with the most vociferous support,” not the one with the broadest appeal to the general electorate. As a result, the system can encourage extremism, reward name-calling, alienate voters, and fail to reflect the wishes of most of the people most of the time (102).
Mathematicians have been studying the flaws of voting systems for two hundred years. They don’t agree on which system is best, but they do agree on which is the worst: It’s our own hallowed tradition that says those with the most votes gets to decide for everyone.. . The subject is well know in academic circles. As far back as 1951, Stanford economist Kenneth Arrow proved mathematically that no democratic voting system can ever be completely fair (and later won the Nobel Prize for his efforts). This notion is known as Arrow’s impossibility theorem because it proves that perfect democracy is impossible (102).
Brams and others are drawn to a voting system that allows a broader spectrum of choices than simply “yes” and “no.” For example, “approval voting,” allows each voter to cast one vote per candidate–changing the “one person, one vote” dictum to “one candidate, one vote.” That way a voter can “approve of” as many candidates as he or she likes (103).
While plurality systems encourage candidates to take extreme positions that develop a hard core of support, approval voting requires candidates to appeal for broad support. As a plus, the system might do a lot to eliminate negative campaigning . . ..
Approval voting is not without its detractors, however. Paulos argues that it tends to produce bland, mediocre winners. . . The system Paulos likes much better is cumulative voting, in which voters can pile up several votes for a single candidate (or issue) they feel strongly about. As in approval voting, each voter has as many votes as candidates. But under cumulative voting, a voter could focus on the most important issue, instead of simply approving or disapproving off each one (103).
Cumulative voting is already widely used in this country to elect corporate boards of directors. It’s the system championed by Lani Guinier, who writes: “It is not a particularly radical idea; thirty states either require or permit corporations to use this election. . cumulative voting is also used in city council elections (103).
[Another system offered is preference voting]. Preference voting is a method already used to elect both houses of the Australian congress, the Irish president, and the Cambridge, Massachusetts, city council. Under preference voting, each voter ranks each candidate first, second, third, and so forth. But if someone’s first place choice seems doomed to defeat then the voter’s second place vote is counted instead (105).
Each of these voting systems skews the results of an election in its own particular way. The founders of our country were aware of the need to balance the flaws in any one system through checks and balances. Indeed, the Constitution provides for a complex of voting schemes specifically designed to protect minority interest: small states are protected by the Senate’s two-member-per-state rules, while populous states wield more power under proportional representation in the house; the president can rein in an out-of-control majority with a veto; and amendments to the Constitution require supermajority victories (105).
Reform is not likely to come easy. Many people feel that there’s something vaguely un-American about choosing voting methods based on predicted outcomes—even though the experts say this is inevitable(106).
Alas, these commonsense conclusions are precisely those that mathematics tells us we can’t avoid. All voting systems play favorites. And even simple mathematical solutions can open up a Pandora’s box of problems (106).