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.

History of Astronomy with Errors

This blog post will be a bit unusual. I just wrote a letter to the editor of APS News pointing out a few errors in a historical piece on Edwin Hubble that was in the January 2008 edition (this will be accessible only to APS members until the next APS News comes out, and then it will be available to all). What I plan to do here is print my letter and give some additional comments. I have no idea if my letter will be published in APS News, but here it is:

Dear Editor,

I always enjoy reading “This Month in Physics History” and the January installment on Hubble’s discoveries was no exception. However, I would like to point out a few minor errors in that piece. Most astronomers in the early 20’s favored the theory that spiral nebulae were “island universes” and in fact believed the Milky Way to be much smaller than we now know it to be. Shapley and a few others favored the idea of a much larger Milky Way which contained the spiral nebulae, but Shapley’s letters indicate that he knew he was in the minority on this issue. Also, it was Henry Norris Russell who presented (on behalf of Hubble) the data on Cepheids in Andromea at the AAS meeting in January 1925. Most importantly, it is untrue that “Hubble didn’t discuss the implications of what he had found” in his 1929 PNAS paper. In the final paragraph of that paper he says “the velocity-distance relation may represent the de Sitter effect”, referring to the model of the Universe presented by Willem de Sitter in 1917. This model was originally interpreted as a static model, but did predict a redshift that increased with distance because of scattering and an apparent slowing down of distant atomic vibrations. So in 1929 Hubble did not interpret his data as indicating an expanding Universe, but rather as supporting de Sitter’s static model. It was only later realized that de Sitter’s model was equivalent via a coordinate transformation to expanding models such as that proposed by Georges Lemaitre in 1927 (Lemaitre’s model was unknown to Hubble and most astronomers until 1930). A detailed account of this history is given in Robert W. Smith’s The Expanding Universe (Cambridge U Press, 1982).

Now let me add a few comments:

My pointing out the second error may be me just being picky. It WAS Hubble's data on Cephieds in Andromeda that was presented at the AAS meeting, even if it Russell presented it for him. The piece in APS news implied that Hubble presented it himself, but the wording could be interpreted to fit the facts (though I doubt many readers would interpret it that way). The other errors are more problematic in that they serve to glorify Hubble at the expense of historical accuracy. I seriously doubt that this was the conscious intent of the person who wrote the piece (or the APS News editors), but there it is. Most astronomers were already convinced that there were other galaxies long before Hubble's Cepheid discovery. That discovery, though, put the nail in the coffin. It was a MAJOR discovery, but ultimately what it indicated was what most astronomers thought already. It did when over the few dissenters, some of whom were very important astronomers like Harlow Shapley. The discovery of Cepheids in Andromeda was of immense importance because up to that point the evidence for the extra-galactic nature of the spiral nebulae was circumstantial and conflicting. Hubble found the smoking gun, and subsequently got rid of the conflicting evidence by dismantling Adrian van Maanen's work on the rotation of spiral nebulae (and interesting story in its own right).

It is the third error that I found most surprising. Hubble clearly proposes in his 1929 paper that the velocity-distance relation could be evidence that favored de Sitter's model of the Universe (which was a static model). Hubble did not at that time think that he had found evidence for an expanding Universe. In fact, Hubble continued to resist the idea of a non-static Universe for years. I'm guessing that this is where the statement in the APS News article came from. In later years Hubble did refuse to comment on the interpretation of the velocity-distance relation. But this was after de Sitter's model had been invalidated (mainly because the mean density of the Universe was too high for his model to be relevant) and new non-static models (actually old models that nobody had paid attention to, like Lemaitre's and Friedmann's) had become the focus of the discussion. Hubble apparently did not believe the the redshifts he observed were genuine Doppler shifts, due to actual recessional motion. He did not withhold his opinion because he thought interpretation should be left to others (after all, he was quite ready to support de Sitter's model and in fact his work was likely an attempt to test that model directly). But when the only options up for discussion were expanding models he did not want to side with any of them.

Again, the importance of Hubble's (and Humason's) work on the velocity-distance relation can hardly be overstated. We NOW recognize it as a crucial piece of evidence for the expansion of the Universe. But it was not recognized as such in 1929 (certainly not by Hubble). I don't intend to fault Hubble for this - after all, he was an observational astronomer and an incredibly good one. And in 1929 astronomers were essentially unaware of the existence of expanding models like Lemaitre's. Given what he had to work with, Hubble made a reasonable suggestion that his data supported de Sitter's model. This turned out to be wrong and from that point on Hubble was reluctant to throw his support behind any particular model. All of this is entirely reasonable behavior on his part. But let's not try to hide the fact that Hubble backed the wrong horse.

The errors in the APS News piece were innocent enough. But unfortunately I suspect that such errors are made in many similar cases. They serve to produce an alternate history of science in which our greatest scientists made no mistakes. But this dehumanizes them and makes their accomplishments seem out of reach. Even the greats stumble on occasion. And the achievements of the greats are inevitably built on the work of many who came before (even Einstein was preceded by Lorentz, Fitzgerald, Poincare, etc.). A more accurate history of science might actually be more interesting and might help us to see that science really is, of necessity, a community enterprise. Even the great ones need others to lay the groundwork, catch their few mistakes, and follow up on the leads they leave open. We certainly wouldn't want incorrect physics in such a publication - let's try to keep incorrect history out as well.