In a famous essay called "On the Unreasonable Effectiveness of Mathematics in the Physical Sciences," Eugene P. Wigner, the Nobel laureate in physics, asked why mathematics does so well in describing and predicting physical phenomena that no one had ever seen before. Newton's laws could predict artificial satellites, Einstein's theory predicted atomic bombs, and 20th-Century cosmology predicted black holes in space.
But why should it be, Wigner asked, that mathematics--a purely logical, self-contained, deductive system--corresponds so closely to nature?
This question, which has no altogether satisfactory answer, came to mind several times recently in our meanderings around town. It started the other morning when we dropped by the workshop on modern mathematics that William F. Lucas of the Claremont Graduate School was giving for high-school math teachers.
Lucas is an expert in game theory--the branch of mathematics devoted to analyzing human decision-making. Noting that "math has always been important in the physical sciences," Lucas said that "in the last few decades it has become important in the social sciences" as well. And he said that this application would be as powerful as math was to physics in the time of Newton and Leibniz.
That evening, as luck would have it, we found ourselves at a small dinner party where the guests included Gian-Carlo Rota, the eminent mathematician from MIT who is spending January at USC. ("Anyone who can get out of Boston in January does," he told us.)
Rota said that one of the central questions in mathematics today is whether math is applicable to anything other than physics. "It certainly isn't applicable to economics," he declared, and right away we knew that we liked him and shared his views.
Several of the people at the table, including the host, were mathematical biologists, and Rota said he doubted that math would prove as useful in biology as it has been in mechanics. This sparked a lively discussion.
A few nights later we were at the Beckman Auditorium of Caltech, where John J. Hopfield, a professor of biology and chemistry at the institute, delivered a public lecture on brains and computers and the inherent differences between them. A reasonable conclusion from what he said is that mathematics, which is very applicable to computers, may not be all that useful in describing what the brain does.
Computers have rigid logic and rigid circuits, and can deal only with absolutes such as on or off, yes or no, true or false. Brains, however, function more as a whole and are very good at recognizing shades of gray. They are not mechanistically determined (at least it seems that way) like the orbits of the planets around the sun, but admit of creativity and new ideas that were not predictable from the laws of physics.
This conclusion supported Rota's view that the correlation between mathematics and the world drops off once you leave physics. Either that or there's something that researchers in math and biology need to learn to strengthen the correspondence between their fields.
In the meantime, it remains an open question just how powerful mathematics really is. But we like the idea that the mysterious brain may respond to a different set of laws than the ones that we know.