CHAPTER ONE
WHAT'S MATH GOT
TO DO WITH IT?
Understanding is a lot like sex. It's got a practical purpose,
but that's not why people do it normally.
--Frank Oppenheimer
Finding out what's true is a central passion of human activity. It's a
question that dominates the stage and the dinner table, the classroom
and the courtroom, the scientific laboratory and the spiritual retreat.
And yet, with the explosion of information reverberating in our
brains, it becomes harder and harder to hear the clear ring of truth
through the competing facts and philosophies.
As it turns out, mathematics offers a singular set of tools for
seeing truth. Indeed, it 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).
People outside the sciences rarely pick up these tools--in part
because math seems intimidating. Even if people are aware that such
tools exist, they don't know how to apply the tools to things they
care about.
But mathematics already underlies many of society's
most-cherished political and social inventions: Ideas about cause and
effect, fairness and justice, selfishness and cooperation, balancing
risks, spending on welfare or national defense, even the nature of
scientific discovery itself.
True, our ideas about the physical and social world do spring
from sources other than numbers: religion, history, family,
psychology. We accept the "truths" revealed by these sources as
intuitively commonsensical, or obviously right; our Declaration of
Independence describes them as "self-evident."
But math--that most logical of sciences--shows us that the
truth can be highly counterintuitive and that sense is hardly
common.
Mathematics is a way of thinking that can help make muddy
relationships clear. It is a language that allows us to translate the
complexity of the world into manageable patterns. In a sense, it works
like turning off the houselights in a theater the better to see a movie.
Certainly, something is lost when the lights go down you can no
longer see the faces of those around you or the inlaid patterns on the
ceiling. But you gain a far better view of the subject at hand.
William Thurston, the director of the Mathematical Sciences
Research Institute (and by some accounts the world's greatest
living geometer) calls math a kind of "mindware." It allows us to see
and articulate concepts we can't handle in any other way. Ingrid
Daubechies--the MacArthur Award-winning Princeton
mathematician who resurrected wavelet analysis (a tool for doing
everything from storing fingerprints to seeing stars)--says it's akin
to poetry: a way of taking a big idea and condensing and honing it
until it communicates exactly the right information.
Mathematics can function as a telescope, a microscope, a sieve
for sorting out the signal from the noise, a template for pattern
perception, a way of seeking and validating truth. It is a lens that can
clarify the obscure, or obscure and distort what was seemingly clear. It can
take you into the core of a star or to the edge of the universe, give
you the outcome of an election or the result of pumping carbon
dioxide into the atmosphere for a hundred years. You can extrapolate
to the end of time, or back to its beginning. You can get there from
here.
Mathematicians do not see their art as a way of simply calculating
or ordering reality. They understand that math articulates,
manipulates, and discovers reality. In that sense, it's both a language
and a literature; a box of tools and the edifices constructed from
them.
Once I was flying in a plane back from the Boston area, where I
had been talking with a cosmologist at MIT about the universe and
all that. I looked down from my window and saw islands that were
clearly connected under the shallow water by strips of land. On the
ground, those links would have been invisible, the islands completely
unconnected. From the air, the paths between them were laid out as
clearly as road maps. There's a reason, I thought, that a lot of
fundamental physics requires looking in higher dimensions. You can
see more from an elevated point of view.
In the same way, the 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?
In some sense, the unfolding story of the universe is a history of
finding hidden connections. The nature of light was discovered
when a certain number (the speed of light) kept popping out of equations
linking electricity to magnetism. Light was exposed as an
electromagnetic fluctuation--an understanding that allowed
experimenters to go looking for others of its same species. Radio
signals, for example, ride on light that vibrates more slowly than the
eye can see; X rays vibrate faster.
Equations speak volumes, teasing out economic trends, patterns
of disease, growth of populations, and the effects of prejudice and
discrimination. Math produces a quite literal expansion of
consciousness. It allows us to see more. With these tools, we can
extrapolate into the future (but there are hazards) and see invisible
things (curved space).
"What do we really observe?" asked Sir Arthur Eddington in 1959,
summing up the lessons of the century's recent revolution in physics:
"Relativity theory has returned the answer--we only observe
relations. Quantum theory returns another answer--we only observe
probabilities."
What we observe, in other words, are mathematical relationships.
Since mathematics is so good at exposing the truth, it's curious how
often it's used to perpetuate misunderstandings and lies. Math has
power because we give more weight to numbers than we do to
words. "Figures often mislead people," says mathematician Keith
Devlin. "There is no shame in that: words can mislead as well. The
problem with numbers is our tendency to treat them with some
degree of awe, as if they are somehow more reliable than words.... This
belief is wholly misplaced."
People often look to mathematics as an objective line of argument
that will rescue them from the uneasiness of ambiguity. If only we put
things in terms of numbers, we hope, perhaps truth will out. But math
only articulates these ambiguities; it is no lifeboat out of
the sea of confusion--only the buoy that marks the shoals. After all,
it was a mathematical theorem (Godel's theorem) that proved some
truths can't be reached by the road of pure logic at all.
A prime case of intimidation by the numbers is the book The Bell
Curve, a treatise so controversial that a half dozen books were
published in response. Written by Charles Murray of the American
Enterprise Institute and the late Richard Herrnstein of Harvard, the
book wheels out an arsenal of mathematical artillery to bolster the
proposition that intelligence is mostly inherited, that blacks have less
of it, and that little can be done about it. Reviewers--not to mention
readers--admitted to shell shock in the face of such a barrage of
statistics, graphs, and multiple-regression analyses.
Yet the fearless few who plunged into the statistics headlong
found that the numbers which seemed to speak so clearly swept
crucial qualifications under the rug, making much of the mathematics
meaningless.
The question I get asked most frequently is: How can you ever
find out what's true short of becoming a mathematician yourself? The
answer is: You don't have to. You merely need the confidence to ask
the questions that were probably on your mind anyway. Such as:
How do you know? Based on what evidence? Compared to what
else? Like the woman who spent a day exploring exhibits at the
Exploratorium in San Francisco--then went home and wired a lamp.
There was nothing in the world-renowned science museum that
taught her how to wire a lamp. What she found there was simply the
belief in her own abilities to figure things out.
Used correctly, math can expose the glitches in our perceptual
apparatus that lead to common illusions--such as our inability to
perceive the true difference between millions and billions--and give
us relatively simple ways of protecting ourselves from our own
ignorance. As the physicist Richard Feynman once said: "Science is
a long history of learning how not to fool ourselves." A knowledge
of the mathematics behind our ideas can help us to fool ourselves a
little less often, with less drastic consequences.
In short, math matters--a lot more than most people think. We
have to make life-and-death decisions based on what numbers tell
us. 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.
As someone who started out interested in social questions, I am
particularly impressed at the power of math to help 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 more
subtle and even obscure (like the relationship between symmetry,
truth, and things that never change, no matter what).
Many different kinds of truths lie in numbers, and exploring them
is the purpose of this book. What does it mean when one number
can be correlated with another? Say: IQ and intelligence, or math
scores with big feet? If one thing makes another thing more probable,
is it fair to call it a cause? What is the most effective strategy for
winning at games? Is endless economic growth really a good thing
(or even possible)? Was there life on ancient Mars? What's the
fairest way to divide the national budget, or the best way to survive
a game of "chicken"? What is the probability of getting killed by a
terrorist? Getting married after forty? Running into your
brother-in-law in Manhattan? In Nome? What, if anything, do
these numbers we attach to things mean?
No doubt about it, mathematics embodies great power. It's no
wonder that the physicist Sir James Jeans concluded: "The Great
Architect of the Universe now begins to appear as a pure
mathematician."
At the same time, it is far from foolproof. Like all science, it
grows and thrives in cultures and is heavily influenced by their
peculiarities. This book focuses on various mathematical guides to
the truth that can be applied to a wide range of questions, from
issues in the news to matters of purely philosophical or aesthetic
interest.
What I personally like best is the way that truth and beauty come
together in the work of Emmy Noether and Albert Einstein: How
deep truths can be defined as invariants--things that do not change
no matter what; how invariants are defined by symmetries, which in
turn define which properties of nature are conserved, no matter what.
These are the selfsame symmetries that appeal to the senses in art
and music and natural forms like snowflakes and galaxies. The
fundamental truths are based on symmetry, and there's a deep kind of
beauty in that.
The journey begins here.