I submit for your favorable consideration Frank Wilczek's reflections on the culture of force in physics. (Yes, everybody blogged about the Wilczeks months ago, for one reason or another. I do not so much run with the herd as shamble in its wake.)
When I was a student, the subject that gave me the most trouble was classical mechanics. That always struck me as peculiar, because I had no trouble learning more advanced subjects, which were supposed to be harder. Now I think I've figured it out. It was a case of culture shock. Coming from mathematics, I was expecting an algorithm. Instead I encountered something quite different --- a sort of culture, in fact. Let me explain.
Newton's second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right-hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left-hand side, on the other hand, has no independent meaning. Yet clearly Newton's second law is full of meaning, by the highest standard: It proves itself useful in demanding situations. Splendid, unlikely looking bridges, like the Erasmus Bridge (known as the Swan of Rotterdam), do bear their loads; spacecraft do reach Saturn.
The paradox deepens when we consider force from the perspective of modern physics. In fact, the concept of force is conspicuously absent from our most advanced formulations of the basic laws. It doesn't appear in Schrödinger's equation, or in any reasonable formulation of quantum field theory, or in the foundations of general relativity....
To anyone who reflects on it, it soon becomes clear that F = ma by itself does not provide an algorithm for constructing the mechanics of the world. The equation is more like a common language, in which different useful insights about the mechanics of the world can be expressed. To put it another way, there is a whole culture involved in the interpretation of the symbols. When we learn mechanics, we have to see lots of worked examples to grasp properly what force really means. It is not just a matter of building up skill by practice; rather, we are imbibing a tacit culture of working assumptions. Failure to appreciate this is what got me in trouble.
There's more, and I quite recommend it (a follow-up is promised in a future issue of Physics Today). As I think this shows, Wilczek does an unusually good job of writing about physics, in a comprehensible way. Indeed, the book he wrote with Betsy Devine, Longing for the Harmonies, is one of the few successful attempts to explain to non-physicists how theoretical physicists think, and why they think that way. (It is, of course, long out of print, while tripe like The Tao of Physics continues to sell; there is no justice.)
Bertrand Russell somewhere (perhaps in The Analysis of Matter?) describes the content of the second law as an injunction to describe phenomena by means of second-order differential equations in position variables. (The first law then forbids us from using second-order equations which are trivially reducible to first-order ones.) All of the physical content, he says, comes from learning what the second derivative should be. This agrees nicely with what Wilczek says here. It would be a real contribution if someone were to write a mechanics textbook making this point of view explicit.
Update, noon: Michael Nielsen writes to point to the comments thread on this post of his, where Landau and Lifshitz's Mechanics is quoted taking exactly the correct line (as one would expect of Landau).
Posted by crshalizi at November 29, 2004 11:15 | permanent link