Duhem, Brody, Hubble: Approximation and the Scope of Theories

In this essay I'd like to highlight some similarities between the ideas of two physicists who have written extensively on the philosophy of physics. The first is Pierre Duhem, author of The Aim and Structure of Physical Theory. The second is Thomas Brody, author of The Philosophy Behind Physics. In reading these two books I have been struck by some remarkable similarities that I think are worth pointing out. This is somewhat surprising since Duhem was primarily a phenomenalist (though he makes concessions to realism with his idea that science approaches a "natural classification") while Brody seems to be a realist (but one who makes concessions to phenomenalism in his insistence that science is necessarily approximate). It is also surprising because Brody makes no reference to Duhem's work, even though Duhem's book was published in 1906 (in French, an English translation has been available since at least 1954) and Brody 's philosophical work was published mostly in the 1970's and 1980's. These ideas have also given rise to some thoughts about some of Edwin Hubble's work, which I have been studying recently (Hubble's The Realm of the Nebulae is a good introduction to his work).

One similarity between these two is their emphasis on approximation in physics. Both Duhem and Brody recognize that all measurements are approximate and all theoretical predictions are approximate as well. Duhem makes this point the basis for his famous thesis that any experimental measurement outcome is necessarily consistent with an infinite number of theories and any theoretical prediction is consistent with an infinite number of experimental measurement outcomes (a thesis, often known as underdetermination, which was later expanded by W. V. O. Quine, but with somewhat different emphasis). Brody emphasizes the fact that approximations are generally valid in some circumstances in invalid in others. The approximate nature of scientific theories then becomes the basis for his idea of the scope of a theory. The scope of a theory is the range of phenomena for which the theory is valid. The theory is not expected to be valid for phenomena that lie outside its scope. Brody argues that one of the main goals of science is to delimit the scope of theories (as well as to create new theories).

The concept of a limited scope for physical theories is common to Duhem and Brody. Indeed, they both find it quite acceptable for a physicist to use two completely incompatible theories in the course of her work. Duhem quotes Poincare to state that one can use logically incompatible theories as long as one takes care not to mix them or to "get to the bottom of things." Brody presents a somewhat subtler view based on his idea of scope. It is acceptable even to mix theories that are logically incompatible provided that one doesn't use any theory to describe phenomena that are outside of its scope. He cites as an example molecular physics in which the nuclei are treated as classical Newtonian point masses, while the electrons are treated as relativistic quantum particles. Perhaps an even better example he uses is that of studying the influence of the Moon's gravity on a pendulum by first calculating the Moon's orbit (treating Earth as a Newtonian point mass for this purpose) and then treating the gravitational force between the Moon and the pendulum bob as a perturbation on the pendulum's "normal oscillation" (treating Earth as an infinite plane and Earth's gravitational field as uniform). Here within a single problem the physicist uses to logically incompatible models of the same object, but for different phases of the problem. Perhaps Poincare would not consider this "mixing" the two models - but the main point is that each model is used to predict a phenomenon that is within the scope of that particular model. I'm well acquainted with this type of work, since my own research has largely focused on the interaction of quantum particles with oscillating classical electric fields, so I mix classical electrodynamics and quantum physics all the time.

An important consequence of the fact the scientific theories are approximate and have limited scope is that scientific theories are not about truth.
Both Duhem and Brody insist that scientific theories cannot be evaluated on a logical basis. Theories are neither true nor false in a logical sense. The concept of a theory of everything (a theory that explained all phenomena) would be meaningless for both Brody and Duhem. Duhem would view such a theory as a "cosmology" (in the ancient meaning of this term) and thus not a scientific theory at all. Indeed, his chief goal in Aim and Structure was to separate science from, and make it independent of, cosmology.

One more similarity between Duhem and Brody is their insistence on the evolutionary nature of science. Duhem seems to disdain the very idea of scientific revolutions. In part this is based on his extensive historical study of medieval physics which illustrate the origins of many of the ideas that eventually reached maturity in Newton's physics. It should be noted that Duhem wrote his book around 1905, so he was unaware of the coming quantum and relativistic "revolutions" (though I doubt these would have changed his views). Brody seems to accept the idea of revolutions, but rejects Kuhn's idea that between revolutions physicists only solve problems using an established paradigm. He points to the extensive development of mechanics after Newton, pointing out that Newton might very well be unable to understand things like Hamilton-Jacoby theory and the geometrical mechanics of Poincare even though these are supposedly the result of "problem solving" within the paradigm that Newton himself created. I intend to write more about evolution versus revolution in science at a later date.

For now what I'd like to do is apply the framework of a theories scope to an idea that is a key part of Edwin Hubble's work, an idea he refers to as "The Principle of the Uniformity of Nature." Now this phrase is often used to denote an essentially metaphysical statement that is supposed to justify induction, but I don't think that's what Hubble means by it. He means something more like a methodological principle, and I think it can be clearly explained in terms of Brody's idea of scope. What Hubble is saying is this: when an empirical law has been found, we should assume the widest possible scope for this law. For example, Henrietta Leavitt found an empirical law relating the apparent brightness (and thus, essentially, the intrinsic brightness) and the period of Cephied variables in the Large Magellanic Cloud. Hubble applied the Principle by assuming that ALL Cepheid variables (as identified by the shape of their light curves) follow this law, even those in distant galaxies and in various parts of our own galaxy. This turned out to cause problems because it led to inconsistent results. But Hubble was never trying to say that these empirical laws really did have universal scope, but only that we should assume that they do until we have reason to think otherwise (i.e. until that empirical law leads to contradictions with another empirical law, or with directly observed data). When such contradictions occur the scope of one of the laws involved must be reduced. In the case of Cepheids, the contradictions were resolved by proposing that there are two types of Cephieds with different period-luminosity relations (one type resides in the galactic plane, the other type in the halo).

Of course, if two empirical laws contradict it may be hard to determine which one should have its scope reduced. In some cases we may be able to carry out an experiment or observation that will clearly favor the modification of one law over the other. But in many cases we may need to guess, and our guess will be guided by how the modification fits with all of our other theories. This view actually ties in well with Duhem's other famous thesis: that we never test a theory in isolation, but rather we test the entire system of current theories. When a predictions is contradicted by a measurement we never know which theory (or assumption, etc.) is to blame, but we must make a choice of what to modify. That choice will be made with a consideration for the impact it will have on our system of theories and its fit to all previously known data. For example, we will be unlikely to modify a foundational theory that explains a wide range of phenomena. Instead, we will probably choose to modify (or delimit the scope of) a theory which if of lesser importance to the entire structure of our theoretical system. This is essentially Lakatos' idea of modifying the "protective belt" rather than the "hard core" of our theoretical system.

Note that the Principle of the Uniformity of nature is a methodological assumption with no logical basis. Logically we have no reason to suppose that the scope of an empirical law or theory extends beyond the data already known to fit it. It is interesting, though, that Hubble's methodological assumption can be recast in terms of Popper's fundamental methodological assumption to always choose the most falsifiable theory. Certainly, we make any theory more falsifiable by assuming it has a universal scope rather than a limited scope. The difference in Hubble's proposal is that he suggests limiting the scope of the empirical law rather than discarding it as Popper (at least in his early work) would have it. I think Hubble's perspective on astronomy fits in very well with the scheme that seems common to both Brody and Duhem.

An analogy that Brody uses can help make sense of all this. He says that science is rather like a map. A map is always approximate. The idea of a map that depicted its subject exactly down the finest detail (i.e. showing blades of grass in Central Park, and the ant crawling on the blade of grass, and the crumb of bread in the ants mandibles, etc.) is ridiculous. Not only that, but such a map would be useless. Moreover, as we make our way around a city we may use multiple incompatible maps. For example, we may have a street atlas, a subway map, and a restaurant guide. These maps are not logically compatible because they will indicate different relative distances between supposedly identical locations (subway maps, in particular, are always schematic and do a poor job of depicting geographical relations between stations). However, in going from a hotel on Fifth Avenue (why not?) to the Statue of Liberty we might make use of a street map to find the nearest subway stop, the subway map to get us to the station closest to the ferry terminal, the street map again to find the ferry terminal, then the ferry map to make sure we get on the correct route. None of these maps embodies the "truth" of New York City, but they all provide a useful depiction of certain structural relations within the "real" New York City. They are incompatible in a logical sense, and yet we can use them together to get where we want to go. In a similar way, none of our scientific theories embody the "truth" of the world, but they do provide useful depictions of certain structural relations within the "real" world. We can use incompatible scientific theories to solve problems and make predictions about the physical world, provided we know which structural relations are accurately depicted by a given theory and which are not.