Making the Invisible Visible
Commencement address delivered to the mathematics
graduating class of the University of California at Berkeley,
May 23, 1997.
All of you graduating here today have a good head for
You like adding up long columns of numbers in your
You have always found it easy to balance your
You revel in solving ten simultaneous linear equations in
ten unknowns. In your head.
You are all going to be math teachers or accountants.
You have no sense of humor.
You concentrated on mathematics because it is predictable,
because there is always a right answer you can check in
the back of the book, because you like following very
precise rules, because it allows you to escape from
everyday life into a world that has nothing to do with
everyday life, and because mathematics does not require
the creativity that you completely lack.
That's not me speaking. That's most of society.
Today, those of you graduating become official, card
carrying members of the community of mathematicians. As
a result, you have set yourself up for all the caricatures I just
mentioned. For now, you'd better get used to it. As far as the
future is concerned, you have two choices. One is to
continue to put up with it. The other is to try to do something
about it, to try to change the widespread public
misperceptions about our profession.
I think you have no alternative than to take the second
choice: to do something about it. And I think you have to
start right away. We all have to start right away. My sense of
urgency comes from a change in the public perception of
mathematics that I have seen develop over the past few
When I received my bachelors degree in mathematics from
King's College London in 1968, all of those caricatures I
mentioned a moment ago were common. But back then,
everyone accepted that mathematics was terribly important
-- important for science, important for technology, important
for defense, important for the space race, and important for
These days, mathematics is generally thought to be
irrelevant to most of life. When the popular press clamors for
better math skills among schoolchildren, what they mean is
basic numeracy, not mathematical thinking.
The perception that mathematics is largely irrelevant makes
it doubly important for us to act. First, there is a selfish,
aesthetic reason. None of us who knows what mathematics
is -- what pleasure it can bring -- wants to see it die. We
want to be able to continue to pursue mathematics, and we
want as many others as possible to share our joy in the
Then there is an altruistic, utilitarian reason. As
mathematicians, we know, even if most other people don't,
that mathematics is more important in today's society than
at any other time in history. As members of the mathematics
profession, therefore, it is our responsibility -- and ours
alone -- to ensure that society doesn't blow it. We must
ensure that mathematics continues to receive support, and
that enough people continue to pursue it.
The question is: How do we set about rectifying the result of
hundreds of years of bad press?
Writing articles in scholarly journals won't do it. Nor will
writing books, even popular books. Nor will giving
speeches like this one. Those activities are all largely
preaching to the converted. If enough of us do those things,
and if we do them often enough, they will have some effect.
But not much. And nothing like enough.
Television is a more promising medium. Here, the good
news is that for the past four years a project has been
underway to use prime time television to provide
mathematics with a massive dose of unadulterated and
highly positive P.R. That project is almost completed. Next
spring, PBS will broadcast a series of six one hour
programs on mathematics called Life by the
The series is being produced by WQED television in
Pittsburgh, with major funding from Texas Instruments and
the National Science Foundation. Over the past two years, I
have been involved in the project as a consultant, and I am
currently writing a book that will accompany the series.
This is the first time there has ever been an investment on
this scale to use television to try to change people's
perceptions of mathematics. It may be the only such effort
we will see in our lifetime. Having seen all of the segments
that have been produced so far, I think that it stands a very
good chance of making some headway.
But one television series alone will still not be enough. It will
only reach those people who regularly watch PBS, and
that's still a small portion of the population.
The difficulty we face is that public perceptions are formed
by what the evolutionary biologist Richard Dawkins has
called memes. Memes are the thoughts and ideas that
people produce and make public -- stories, tunes, poems,
myths, beliefs, religions, scientific theories, and the like.
The name "meme" is meant to emphasize their similarity to
genes. Memes are the mental equivalent of genes -- self
replicating entities that multiply and spread through society,
shaping the future development of that society and helping
to determine its culture.
Like our genes, memes need human hosts in order to
propagate. Also like genes, some memes can outlive many
generations of individual biological members of that society.
A meme may also mutate to produce a new variant. This
happens with scientific theories and with religions, to give
just two examples.
For many thousands of years, memes were transmitted from
person to person in a direct fashion, from mouth to ear.
Then, with the invention of writing and postal services,
memes started to be transmitted from person to person over
greater distances, with paper as the carrier. With the
invention of the printing press, a single meme could spread
across an entire country in a matter or days. Today, a new
meme can grow and spread around the globe in a matter of
seconds, travelling at the speed of radio waves through the
air and light waves along fibre optic cables.
Of course, most memes do not live very long. But some do,
and they are the ones that shape and form our culture. Like
the advertising jingle that enters our heads and stubbornly
refuses to go away all day long, a meme, once established,
can be hard to get rid of.
In most western societies, the mathematical memes I listed
at the start of my address are now fully developed and well
established. They will not be easily eradicated. The only
hope, I believe, is to introduce a deadly virus meme to kill
them off. Or even better, a whole range of such virus
What would a virus meme look like? What will it take to get
an idea to spread and grow rapidly enough to stand a
chance of overcoming memes that are already embedded
in our culture?
I think there is only one possible answer. It's the same thing
that advertisers use to persuade us to buy their wares, and
it's the same thing that gets presidents elected.
I mean this in all seriousness. Sound bites.
If you take a look at the way public opinion is shaped these
days, I think you will find the evidence inescapable that the
only chance we have of changing public opinion is by
releasing a small band of sound-bite, memetic viruses,
crafted sufficiently well that they will be able to survive and
Remember, I am not trying to increase public
understanding of mathematics. As it happens, I don't
think there is much of a case to be made in favor of trying to
do that. What I want to change is the public
perception of mathematics. And that's a very different
thing from understanding, and far more important.
So how do we set about creating mathematical virus
memes? The trick is to capture in a single, easily
remembered slogan, the very essence of mathematics.
Right now, if you pick someone at random on the street and
ask them to describe mathematics in a single sentence, the
answer you are likely to get is something along the lines:
"Mathematics is using numbers."
If we want to improve the public perception of mathematics,
we need to come up with one or more equally memorable
virus memes that will kill off that wildly inaccurate
description once and for all.
I'm going to give you two mathematical virus memes that I
think might do the trick. I ask you to help create a memetic
epidemic by passing them on to your friends and
The first meme is the phrase: The science of
The phrase is not mine. I first saw it in print as the title of an
article in Science magazine, written by Lynn Steen
in the late 1980s, but Steen says it did not originate with
him either. But, whoever is the parent or creator of this
particular meme, I did my best to help spread the meme four
years ago when I wrote a Scientific American Library book
with the title Mathematics: The Science of Patterns.
I think it's a good slogan. That's why I took it for the title of
my book. It captures both the nature and the scope of
- Arithmetic and number theory study the patterns of
number and counting.
- Geometry studies the patterns of shape.
- Calculus allows us to handle patterns of motion.
- Logic studies patterns of reasoning.
- Probability theory deals with patterns of chance.
- Topology studies patterns of closeness and position.
- And so forth.
And it's starting to catch on. These days, you often see the
word "patterns" linked to mathematics. It forms a strong
theme throughout the TV series I mentioned a moment ago.
Let's wish it a long life and hope it continues to spread
through the population.
My second memetic virus is this: Mathematics makes the
This is a new meme. If it survives, you will have been
present at its moment of birth. So let me give you some
examples of what I mean.
Without mathematics, there is no way you can understand
what keeps a jumbo jet in the air. As we all know, large
metal objects don't stay above the ground without
something to support them. But when you look at a jet
aircraft flying overhead, you can't see anything holding it
up. It takes mathematics to "see" what keeps an airplane
aloft. In this case, what lets you "see" the invisible is an
equation discovered by the mathematician Daniel Bernoulli
early in the eighteenth century.
While I'm on the subject of flying, what is it that causes
objects other than aircraft to fall to the ground when we
release them? "Gravity," you all answer. But that's just
giving it a name. It doesn't help us to understand it. It's still
invisible. We might as well call it magic. To understand it,
you have to "see" it. That's exactly what Newton did with his
equations of motion and mechanics in the seventeenth
century. Newton's mathematics enabled us to "see" the
invisible forces that keep the earth rotating around the sun
and cause an apple to fall from the tree onto the
Both Bernoulli's equation and Newton's equations use
calculus. Calculus works by making visible the
infinitesimally small. That's another example of making the
Here's another: Two thousand years before we could send
spacecraft into outer space to provide us with pictures of our
planet, the Greek mathematician Eratosthenes used
mathematics to show that the earth was round. Indeed, he
calculated its diameter, and hence its curvature, with 99%
Today, we may be close to repeating Eratosthenes' feat and
discover whether the universe is curved. Using
mathematics and powerful telescopes, we can "see" into the
outer reaches of the universe. According to the astronomer
Robert Kirschner, we will soon see far enough to be able to
detect any curvature in space, and to measure any
curvature that we find.
Knowing the curvature of space has a corollary, as we say
in the mathematics business. As the MSRI (Berkeley)
mathematician Bob Osserman explains in his excellent little
book The Poetry of the Universe, if we can calculate
the curvature of space, then we can use mathematics to see
into the future to the day the universe comes to an end.
Using mathematics, we have already been able to see into
the distant past, making visible the otherwise invisible
moments when the universe was first created in what we
call the Big Bang.
Coming back to earth at the present time, how do you "see"
what makes pictures and sound of a Stanford--Cal football
game miraculously appear on a television screen on the
other side of the Bay? One answer is that the pictures and
sound are transmitted by radio waves -- a special case of
what we call electromagnetic radiation. But as with gravity,
that just gives the phenomenon a name, it doesn't help us
to "see" it. In order to "see" radio waves, you have to use
mathematics. Maxwell's equations, discovered in the last
century, make visible to us the otherwise invisible radio
Here are some human patterns:
- Aristotle used mathematics to try to "see" the invisible
patterns of sound that we recognize as music.
- He also used mathematics to try to describe the
invisible structure of a dramatic performance.
- In the 1950s, the linguist Noam Chomsky used
mathematics to "see" and describe the invisible, abstract
patterns of words that we recognize as a grammatical
sentence. He thereby turned linguistics from a fairly obscure
branch of anthropology into a thriving mathematical
Finally, using mathematics, we are able to look into the
- Probability theory and mathematical statistics let us
predict the outcomes of elections, often with remarkable
- We use calculus to predict tomorrow's weather.
- Market analysts use various mathematical theories to
try to predict the future behavior of the stock market.
- Insurance companies uses statistics and probability
theory to predict the likelihood of an accident during the
coming year, and set their premiums accordingly.
When it comes to looking into the future, mathematics
allows us to make visible another invisible -- the invisible of
the not yet happened. In that case, our mathematical vision
is not perfect. Our predictions are sometimes wrong. But
without mathematics, we cannot even see poorly.
So there you have it -- my new meme: Mathematics
makes the invisible visible.
I'll leave it to you, the University of California at Berkeley
graduating class of '97, to provide more examples of this
meme -- examples of how mathematics makes the invisible
And I urge you help change the popular misperceptions of
our subject by spreading those two memes. Remember,
memes, like other viruses, spread exponentially, and as
mathematicians you know the power of exponential growth,
once it gets going.
Thank you, and congratulations.
MAA Online is edited by Fernando Q. GouvÃªa
Copyright ©1997 by Keith Devlin
Reprinted with permission of the author.