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Making the Invisible Visible

Making the Invisible Visible

Keith Devlin


Commencement address delivered to the mathematics graduating class of the University of California at Berkeley, May 23, 1997.


Good afternoon.

All of you graduating here today have a good head for figures.

You like adding up long columns of numbers in your head.

You have always found it easy to balance your checkbook.

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're dull.

You're boring.

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 decades.

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 economic growth.

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 subject.

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 Numbers.

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 memes.

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.

Sound bites.

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 prosper.

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 relatives.

The first meme is the phrase: The science of patterns. 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 mathematics:

  • 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 invisible visible.

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 ground.

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 invisible visible.

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% accuracy.

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 waves.

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 science.
Finally, using mathematics, we are able to look into the future:
  • Probability theory and mathematical statistics let us predict the outcomes of elections, often with remarkable accuracy.
  • 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 visible.

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. Gouvêa

Copyright ©1997 by Keith Devlin
Reprinted with permission of the author.

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