The Man Behind the Curtain

Perhaps our complacency is due to the fact that we have written down a plausible equation. Physicists have long believed that mathematics is the Rosetta Stone for unlocking the secrets of Nature and since a famous 1960 essay by Eugene Wigner entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” the conviction has become an article of faith. It seems to me, though, that the “God is a mathematician” viewpoint is one of selective perception. The great swindle of introductory physics is that every problem has an exact answer. Not only that, students are expected to find it. Such an approach inculcates our charges with an expectation that is, in fact, exactly contrary to the true state of the world. Vanishingly few problems in physics have exact solutions and a physicist’s career is one of finding approximations and hopefully not being too embarrassed by them.

In a freshman course we introduce the simple pendulum—nothing more than a mass on the end of a string that oscillates back and forth. Initially Newton’s laws lead to an equation that is too hard to solve and so we admonish students to simplify it by assuming that the pendulum is executing small oscillations. Then the exercise becomes easy. Well, not only is the original problem too difficult for freshman, it has no exact solution, at least not in terms of “elementary functions” like sines and cosines. Advanced texts tell you that an exact solution does exist, but the use of the term exact for such animals is debatable. In any case, replace the string by a spring and the problem can easily be made impossible to everyone’s satisfaction. One must distinguish the world from the description afforded by mathematics. As Einstein famously put it, contra Wigner, “As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.”

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