Lagrangians, Hamiltonians, and Scientific Realism

Up at Mormon Philosophy, Clark has a couple of posts concerning the interpretation of physical theories, with the concrete example of Newtonian mechanics. Probably most of my readers are aware that there are different versions of the theory which are mathematically equivalent, but very different in intuitive terms. Thus the traditional formulation in terms of Newton’s Laws describes the world as elemnts of matter, with the intrinsic property of mass, acting on each other by forces and responding to forces by accelerating; the more “global” Lagrangian formulation says that a system will behave in such as way as to minimize the difference between the potential energy and the kinetic energy across different possible behaviours, and the Hamiltonian formulation gives equations for how position and momentum treated as independent quantities will vary depending on the total energy of the system. In his posts Clark states his belief in scientific realism, defined as the assumption that “the entities we discuss in science correspond in some way to entities in the real world”. Then he ponders:

Now when we consider this statement of realism with the three formulations of mechanics we quickly see the problem. There are three different, yet ultimately incompatible, sets of entities mechanics is framed in terms of. Now I'm sure some will jump up and contest this, arguing for momentum, potential and kinetic energy, mass and so forth. But if we take Sellars to be saying that is the ultimate entities the theory is framed in terms of that we have epistemic justification to believe exist, then we see the problem. We can have multiple mathematically equivalent theories in this sense of theory. This is often called having the theory underdetermined.

An other way of looking at the issue is to talk about empirical adequacy. In this view the acceptance of a theory isn't to necessarily accept the entities postulated by the theory. It is just to accept that the theory accounts for the kinds of phenomena we encounter or would potentially encounter. So, for instance, we might be unable to say if mechanics really is about minimize the difference in kinds of energy or is due to forces between masses. We can just say that whatever's going on underneath, the theory explains the kinds of phenomena we encounter.

This, roughly is the distinction in science between realism and anti-realism.

He thinks that the existence of alternative formulations for mechanics, or other physical theories, is an argument against realism and for anti-realistic instrumentalism; and though he reiterates that he is still a realist, he does not provide any solution for the problem. I will try to suggest one for him.

In a nutshell, I see the problem as a manufactured one due to trying to translate mathematical language into ordinary language. The three formulations of mechanics, being mathematically equivalent, must be saying the same thing “about reality” (if we choose to interpret them realistically). The appearance that they say reality is made of “three different, yet ultimately incompatible, sets of entities” cannot be correct, because equivalent statements cannot be incompatible, and this logical fact is independent of whether we interpret the statements in a realistic or instrumentalistc way. The problem arises only because we are not satisfied with a mathematical equation as a description of reality, but we want an explanation of “what it means”, which amounts to a translation of the statement from mathematical language into non-mathematical language. The translation of Newton’s laws seems to be “objects exert forces and accelerate in response to them” and the translation of a Lagrangian action principle seems to be “systems try to minimize action”, and at first view these descriptions seem hardly compatible, especially if the second is interpreted teleologically. But insofar as there is a problem with the compatibility of the translations, this is only a reason to suppose that the translation has been done carelessly and must be corrected. It is as if we started from two Spanish sentences with equivalent meaning, tried to translate each literally into English and got two English sentences differing in meaning because two Spanish words have a shared possible meaning which is not shared by the most literal English equivalent of each of them.

The three formulations of mechanics, then, say the same thing about reality: that there exist certain quantities (given the names of mass, energy, momentum, etc.) which are related mathematically in ways given by the equations of the theory. It makes little or no sense to ask which of these quantities or equations are “fundamental” and which are merely “derived”. I am skeptical even of the possibility of distinguishing “definitions” from “empirical laws” among these mathematical equations; p = mv (momentum = mass x velocity) is a “definition” in the Newtonian formulation and a “dynamical equation” at equal level with F = ma (force = mass x acceleration) in the Hamiltonian one. The ordinary language translations (“objects exert forces”; “systems try to minimize action”) are just pedagogical or heuristic devices, useful and perhaps indispensable for actual scientific practice but not related to the austere philosophical question “What does the theory say about the world?”

There are theories for which “interpretation” is a serious question: quantum mechanics is the primary example. But the “interpretations” of quantum mechanics are a very different thing from the “interpretations” of classical mechanics we have been discussing, and raise very different issues. The equivalent kind of question would be a contrast between, for example, the Heisenberg and the Schroedinger pictures, asking whether reality consists in “states evolving in time and timeless operators” or “timeless states and operators evolving in time” (both descriptions are related by a simple mathematical equivalence). And these alternative formulations, just like those of classical mechanics, raise no philosophical problems at all.