Sunday, April 28, 2013

Mathematical Metaphors

Theories in all areas of science tell us something about the world. They are images, or models, or representations of reality. Theories tell stories about the world and are often associated with stories about their discovery. Like the story (probably apocryphal) that Newton invented the theory of gravity after an apple fell on his head. Or the story (probably true) that Kekule discovered the cyclical structure of benzene after day-dreaming of a snake seizing its tail. Theories are metaphors that explain reality.

A theory is scientific if it is precise, quantitative, and amenable to being tested. A scientific theory is mathematical. Scientific theories are mathematical metaphors.

A metaphor uses a word or phrase to define or extend or focus the meaning of another word or phrase. For example, "The river of time" is a metaphor. We all know that rivers flow inevitably from high to low ground. The metaphor focuses the concept of time on its inevitable uni-directionality. Metaphors make sense because we understand what they mean. We all know that rivers are wet, but we understand that the metaphor does not mean to imply that time drips, because we understand the words and their context. But on the other hand, a metaphor - in the hands of a creative and imaginative person - might mean something unexpected, and we need to think carefully about what the metaphor does, or might, mean. Mathematical metaphors - scientific models - also focus attention in one direction rather than another, which gives them explanatory and predictive power. Mathematical metaphors can also be interpreted in different and surprising ways.

Some mathematical models are very accurate metaphors. For instance, when Galileo dropped a heavy object from the leaning tower of Pisa, the distance it fell increased in proportion to the square of the elapsed time. Mathematical equations sometimes represent reality quite accurately, but we understand the representation only when the meanings of the mathematical terms are given in words. The meaning of the equation tells us what aspect of reality the model focuses on. Many things happened when Galileo released the object - it rotated, air swirled, friction developed - while the equation focuses on one particular aspect: distance versus time. Likewise, the quadratic equation that relates distance to time can also be used to relate energy to the speed of light, or to relate population growth rate to population size. In Galileo's case the metaphor relates to freely falling objects.

Other models are only approximations. For example, a particular theory describes the build up of mechanical stress around a crack, causing damage in the material. While cracks often have rough or ragged shapes, this important and useful theory assumes the crack is smooth and elliptical. This mathematical metaphor is useful because it focuses the analysis on the radius of curvature of the crack that is critical in determining the concentration of stress.

Not all scientific models are approximations. Some models measure something. For example, in statistical mechanics, the temperature of a material is proportional to the average kinetic energy of the molecules in the material. The temperature, in degrees centigrade, is a global measure of random molecular motion. In economics, the gross domestic product is a measure of the degree of economic activity in the country.

Other models are not approximations or measures of anything, but rather graphical portrayals of a relationship. Consider, for example, the competition among three restaurants: Joe's Easy Diner, McDonald's, and Maxim's de Paris. All three restaurants compete with each other: if you're hungry, you've got to choose. Joe's and McDonald's are close competitors because they both specialize in hamburgers but also have other dishes. They both compete with Maxim's, a really swank and expensive boutique restaurant, but the competition is more remote. To model the competition we might draw a line representing "competition", with each restaurant as a dot on the line. Joe's and McDonald's are close together and far from Maxim's. This line is a mathematical metaphor, representing the proximity (and hence strength) of competition between the three restaurants. The distances between the dots are precise, but what the metaphor means, in terms of the real-world competition between Joe, McDonald, and Maxim, is not so clear. Why a line rather than a plane to refine the "axes" of competition (price and location for instance)? Or maybe a hill to reflect difficulty of access (Joe's is at one location in South Africa, Maxim's has restaurants in Paris, Peking, Tokyo and Shanghai, and McDonald's is just about everywhere). A metaphor emphasizes some aspects while ignoring others. Different mathematical metaphors of the same phenomenon can support very different interpretations or insights.

The scientist who constructs a mathematical metaphor - a model or theory - chooses to focus on some aspects of the phenomenon rather than others, and chooses to represent those aspects with one image rather than another. Scientific theories are fascinating and extraordinarily useful, but they are, after all, only metaphors.