Devlin's Angle

March 2005

Mathematics: a natural pursuit

WARNING: This talk makes reference to evolution. Evolution is a theory, not a fact, regarding the origin of living things. The material should be approached with an open mind, studied carefully and critically considered. So there.

I've known people who refuse to eat meat because we kill animals to obtain it, who nevertheless are happy to eat seafood. And high on the list, for some of them, is a delicious Maine lobster. After all, just look at it. Can you imagine anything more primitive, anything less likely to have a conscious sense of its own existence? Well, next time you sit down for a lobster dinner, ponder this: You will be tucking in to one of nature's most accomplished navigators. For the fact is, the common lobster has a geographical location system that humans can match only with the latest, most sophisticated version of GPS, the hugely expensive navigation system that depends upon satellites that orbit the earth, the most accurate timekeeping devices ever devised, masses of computer power, and a pile of advanced mathematics.

What humans accomplish with mathematics and technology, the lobster achieves by being able to "see" the Earth's magnetic field. Not merely in the sense of detecting the magnetic poles. The lobster's system is much more sophisticated than that. The Earth's magnetic field varies from one place to another, in direction, angle to the Earth, and intensity. The lobster appears to be able to use this variation to determine exactly where it is. This was discovered only a few years ago, by ocean scientist Ken Lohmann of the University of North Carolina and his Ph.D. student Larry Boles.

It took Bolas six years of study of the Caribbean spiny lobster in the waters near the Florida Keys before he was convinced that they had this amazing ability. To demonstrate the fact, he tried all kinds of ruses to confuse them. He removed them from the ocean and put them in a lightproof plastic container, drove them around in circles in his boat, took them ashore and drove them around in the back of his pickup truck, placed them next to powerful magnets to distort the Earth's magnetic field, and then dropped them back in the ocean in a new location. As soon as they were released the lobsters headed off directly towards their home. They did so even when Bolas placed rubber caps over their eyes, so they were not navigating by light. But to be doubly sure, Bolas put some lobsters into a marine tank in his lab and subjected them to an artificial magnetic field that mimicked that of the Earth. The lobsters headed off in exactly the direction they would have had to follow to get home if the field had been the Earth's natural one.

The researchers suspect that the lobster's navigational ability may make use of small particles of magnetite, an iron oxide, located in two masses of nerve tissue toward the front of the creature's body. But whatever the mechanism, now you know that lobsters are born with some pretty sophisticated built-in navigational abilities, do you still fancy that lobster dinner?

The built-in GPS system of the lobster is just one of many built-in, natural mathematical capacities you come across in the natural world, that I describe in my latest book The Math Instinct (Thunder's Mouth Press, April 2005), from which this month's column is taken.

Birds provide another example of remarkable navigational ability that I take a close look at in the book. Every year, millions of birds migrate thousands of miles to and from their winter home. How do they know which direction to fly? There are several possibilities, but most of them seem to require mathematical computations that most humans would find challenging. How do the birds do it?

To put the question another way, why is it that a pilot of a Boeing 747 needs a small battery of maps, computers, radar, radio beacons, and navigation signals from GPS satellites - all heavily dependent on masses of sophisticated mathematics - to do what a small bird can do with seeming ease, namely, fly from point A to point B?

To give you some idea of the distances that can be involved, the Arctic Tern flies an annual round trip that can be as long as 22,000 miles, from the Arctic to the Antarctic and back. On the trip south, they make a regular stopover on the Bay of Fundy, fly a grueling, three-day nonstop leg across the featureless north Atlantic, and make their way along the entire west coast of Africa. They return by a different route, coming up the east coast of south and north America. Other sea birds also make amazingly long trips: the Long-tailed Jaeger flies 5,000 to 9,000 miles in each direction, the Sandhill and Whooping Cranes are both capable of migrating up to 2,500 miles per year, and the Barn Swallow logs more than 6,000 miles annually.

How do they find their way? Scientists still have a long way to go before they understand completely how birds navigate. The evidence available today seems to suggest that they use a combination of different methods.

First, the birds may use visual clues. Many animals learn to recognize their surroundings to determine their way. They remember the shape of mountain ridges, coastlines, or other topographic features on their route, where the rivers and streams lie, and any prominent objects that point to their destination. Birds may use this method to locate their nest, but it seems unlikely that it will support flights over long distances. And it clearly cannot be used for navigating over large bodies of water or for flying at night, both of which many species of birds do every year.

Other methods depend on determining the direction of the North Pole. Modern humans usually do this using a compass. How do the birds know which direction is, say, north? One possibility for setting direction is to use the position of the sun in the sky. Many birds have been shown to use the sun to determine where north is. This is not as simple as it might first appear, since the sun changes its position in the sky throughout the day and the pattern of those daily changes itself varies with the seasons of the year. To use the sun to set the direction to north, you have to know where the sun is located in the sky at each time of the day at the precise time of year the journey takes place. For a human navigator, that task alone requires mastery of trigonometry.

Another possibility is that birds discern polarization patterns in sunlight. As the sun's rays pass though our atmosphere, tiny molecules of air allow light waves traveling in certain directions to pass through, but they absorb others, causing the light to be polarized. We humans can see the polarization effect if we look up into the sky at sunset. The polarized light forms an image like a large bow-tie located directly overhead, pointing north and south. It seems that some birds can detect the gradation in polarization from the nearly unpolarized light in the direction of the sun to the almost 100% polarized light 90 degrees away from the sun, and this provides them with a giant compass in the sky. Honeybees also appear to use the polarized light to find their way on cloudy days, when the sun can't be seen. All they need is a small patch of blue sky to see the sun's rays through, and the polarization effect shows them the way.

One obvious problem with birds using the sun to navigate is what do they do at night? Since many birds fly at night, navigating by the sun is clearly not the only method they use.

One possibility, which works at night as well as by day, is to make use of the Earth's magnetic field. This, of course, is exactly what we do when we use a magnetic compass. Some birds use a similar method to navigate. For instance, inside the skull of a homing pigeon is a small pod of magnetic particles, which provides the bird with a tiny magnetic compass in its head. By attaching small magnets to the heads of test birds, researchers have shown that homing pigeons navigate by means of the Earth's magnetic field. The magnets deflect the Earth's magnetic field around the birds, and cause them to fly off course. (Put crudely, with the magnet attached to its head, the bird thinks that any direction it is facing is north.)

Star navigation provides yet another means of navigating that works at night. This method was used by human sailors in times past. At least one species of birds - Indigo Buntings - is known for sure to use the stars to navigate, and it is generally believed that they all do. It appears that they learn to recognize the pattern of stars in the night sky when they are still fledglings in the nest. A few years ago, a study found that nestling Indigo Buntings in the northern hemisphere watch as the stars in the night sky wheel around Polaris - the north star, which lies due north for those in the northern hemisphere. Scientists speculated that being able to identify Polaris in the night sky could help birds identify north. To test this hypothesis, they showed the birds a natural sky pattern inside a planetarium. The birds flew in a direction consistent with being able to detect the motion of the stars. When the experimenters changed the set up so that Betelgeuse was now the star which the stars rotated around, the birds flew in a direction consistent with Betelgeuse being the pole star. They no longer went where they should have relative to Polaris. So, they weren't using the locations of specific star patterns. They were noticing which star the others rotated around. In other words, it wasn't the star patterns, but how the stars moved that counted. For the birds, "north" was where there was a star around which all other stars moved.

Whatever method the birds use to orient themselves, however, orientatation is just part of navigation. For humans, at least, setting the right course from the orientation requires trigonometry. How do the birds do it?

Scientists don't know the answer to that question. What we do know, is that when we humans try to emulate the navigational feats of lobsters or migrating birds, we have to resort to mathematics. In human terms, those creatures have built-in mathematical ability: they have brains that have evolved to carry out the trigonometrical calculations necessary to determine north from the position of the sun or to set a course based on a knowledge of where the North Pole lies. They are, in short, natural born mathematicians.

Or are they? Is it reasonable to describe as mathematics an activity that is surely purely instinctive? Can we really say that lobsters and migrating birds do math? Here's why I think the answer has to be "yes."

We would all agree, I think, that when we use a calculator or a computer to solve a math problem, we are still doing mathematics. In many instances we would even be prepared to say that the calculator or the computer does the math. What then if some non human living creature solves the same problem? The Indigo Bunting, for example? Is there any justification for denying that it too is doing mathematics?

You might argue that no bird is consciously aware of doing any calculations. But then neither is your hand calculator or your computer. You might then counter, "Ah, but the calculator or computer was designed by human engineers to do mathematics." To which I would retort, "But lobsters and birds were designed by Nature to do (that particular) mathematics."

When we approach mathematics as a purely human endeavor, we focus almost exclusively on the conscious performance of computational processes - numerical, algebraic, geometric, etc. - often carried out with the aid of a pencil and paper, or these days some form of electronic computational device such as a calculator or computer. Those kinds of activity are certainly part of mathematics, but if you start from the fact that mathematics - the science of patterns as I like to call it - is about recognizing and manipulating patterns, then viewing the paper-and- pencil stuff we humans do as being all there is to mathematics is like saying that flying is about having wings and flapping them up and down. True, that is how birds fly, but if you take that as being what flying is about you exclude all those jet aircraft that fly around the globe every day. Flying is more fundamental than either birds or airplanes; it is about leaving the ground and moving through the air for extended periods of time. Feathered wings that flap and metal wings engineered by Boeing (that hopefully do not flap) are just two particular ways of performing that activity.

If you are willing to acknowledge that computers can do math - and it is really hard to deny this when there are computer systems that could pass any high school math test, and many university exams come to that - then there really is no justification for denying the same classification to animals that quite plainly solve problems that we humans solve only by mathematics. After all, on the scale of consciousness, computers lie at the very bottom, well below lobsters and birds. This is precisely the point I make in The Math Instinct. Once you get away from the pencil-and-paper view of mathematics we all get from our school days, and you think about the more fundamental activity that those school methods provide just one way of doing, you find that math is all around us. If you want to find the world's greatest mathematician, you don't need to travel to Harvard or Stanford or Princeton. Just visit the ocean or look up at the birds in the sky. For Mother Nature turns out to be the greatest mathematician of all. Through evolution, Nature has endowed many of the animals and plants around us with built-in mathematical abilities that, from a human perspective, are truly remarkable.

That's not only an amazing feature of Nature; it also provides a radical new perspective on mathematics - on what math is and what it means to do math. A perspective that I think could go a long way to helping people overcome the fear they have of what they wrongly perceive to be an unnatural pursuit.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) will be published in April by Thunder's Mouth Press.