One way to highlight what we are missing is to discuss examples of what an explanatory theory might look like. Richard Feynman set out a hypothesized theory of gravity in his 1964 lectures at Cornell University. He said, suppose that all bodies are subject to continuous bombardment of particles from all directions, which particles would “push” those bodies except that they are exerting pressure from all directions at once. Then imagine two bodies close together. Each would block particles coming from behind it toward the other body, with the result that the “normal” balance of pressures would be disrupted so that the two bodies would be pushed toward each other by the particles that are not being blocked.
One interesting prediction from this scenario is that the forces pushing the bodies toward each other would be inversely related to the distance squared (as in the mathematical formula). The reason is that the areas being blocked (where no particles reach each of the bodies to push it) would get smaller the farther away the two bodies are from each other and the diminution in areas would be a function of the distance squared. Id., pp.31–33. (Note, however, that the mathematical formula relates gravity to the masses of the objects, not their size or diameters. Therefore, we would need to say that the blockage of particles is a function of the mass, not the volume, of each body. That refinement undermines the rather neat geometrical calculation supporting the inverse squared relationship.)
Now, we have not said anything about what the particles are, where they come from or how they are generated. But, assuming such particles, we have set out a model or theory of gravity that in some important way “explains” it. The model speaks to our intuition and is capable of visualization, like a good analogy. Feynman goes on to note that this model does not work in fact, because it leads to predictions that are incorrect. For example, a moving body would be impacted by more particles on its front, from the direction toward which it is moving, than on its back, where the particles would be “chasing” it, which phenomenon should tend to slow down the forward movement of the body, contrary to the law of inertia. Id., p.33.19
Einstein’s version
Newton’s theory has been subsumed into (or refined by) Einstein’s theory of gravity, which will be discussed below in the section on General Relativity. Einstein’s theory proposes that the observed phenomenon occurred as a result of “curvatures” or distortions to the fabric of space itself or, more precisely, of space-time.20 The object with greater mass creates greater warp in space-time, causing objects with lesser mass to move towards it. Thus, gravity is not a force that operates across empty space; it is a phenomenon that occurs in space and is part of the nature of space (or space-time). There is no need for the hypothetical “graviton” to convey gravitational attraction; the fabric of space (or, more accurately, space-time) is the medium through which gravity is understood to operate. Greene, The Hidden Reality, pp.14–15.
The theory was, of course, propounded in an elaborate and innovative mathematical structure, based upon what have been called Einstein Field Equations. Id., p.16. Einstein’s General Theory of Relativity has enabled highly accurate calculations of the motions of planets and other heavenly bodies by calculating the presumed curvature of space-time caused by matter (mass plus energy).
However, in order to try to convey a feeling of what is occurring, Einstein and subsequent physicists have used an analogy to a trampoline (or tightly stretched sheet) with two heavy balls placed on it, giving us another example of an “explanatory” model of gravity. See, e.g., Greene, The Elegant Universe, pp.67–71. Each ball would tend to create a depression in the surface. At a sufficient distance, each depression would have no discernible effect on the other ball. But, as the balls were placed closer together, the depression caused by one would “reach” toward the other, tending to cause the surface to tilt toward the first ball, and vice versa. The expected consequence would be that the balls would begin to move toward each other, with more force, the closer together they were. Also, the mass of each body would determine the degree of depression and, therefore, the amount of “tilt” that would occur on the surface near the other body. Thus, we have a “model,” in the form of a physical analogy that maybe said to allow us to visualize and “understand” the force that we have called gravity.21
The physicists and mathematicians can undertake to explore whether the observed phenomena can be generated from mathematical representations of the assumed “surface” (the trampoline or sheet) and the effects of bodies of different masses. If they are successful, we would conclude that we have a working model, at least temporarily.
But, we have said nothing about the nature of the surface that we have used as a central feature of our explanation. In addition, we have no explanation of why the curvature or indentation occurs (it cannot be gravity “pulling” downward on the body on the surface, causing the surface to sag, since it is gravity that we are trying to explain). Id., p.71. Similarly, our initial reaction that the indentation of the surface would explain the tendency of the two balls to move towards each other in a manner consistent with Newton’s formula is incorrect. In the analogy, it is the force of gravity that causes a ball to roll down the slope (like a ball on an inclined plane)—absent gravity, the ball would just “float” where it is, whether the surface on which it appears to rest (absent gravity, it would not actually be resting on the surface) is level or sloping.
Finally, this analogy is in only two dimensions, while gravity obviously operates in (at least) three dimensions. One can clearly conceptualize a three dimensional space, but then what does it mean to “curve” or “warp”? Id., pp.72–3. The sheet or trampoline analogy is easy. In contrast, imagine a bowling ball dropped into a tank of water. The ball displaces water, pushing water in all directions. But, how could the ball cause a curvature of the three dimensional body of water? In short, into what dimension does the distortion caused by the object of mass occur?
These are significant questions about the analogy. Nonetheless, we may still tend to believe that we had achieved something more than we had with just the mathematical formula. The mathematics seems to be clear, internally consistent and powerful. Perhaps the problem is simply the limits of the human imagination, that we just cannot “see” what is taking place. Perhaps the analogy brings us closer to understanding what it is.
The nature of understanding
Alternatively, one could say that these types of explanation are really just appeals to familiar phenomena, which do not really “explain” anything either; they just make the observed relationships seem familiar or normal to us so that we feel that there is no pressing need for further explanation. But, how satisfactory is that? Before being too critical, we should ask, what is a good explanation of a causal relationship? Do we not always face the question of what caused that which we have identified as a cause?
There is also the matter of the “physical mechanism” by which the phenomena is caused, calling again for an analogy to things familiar to us in ordinary life. See id., p.71 (Newton’s theory calls up the image of a force of attraction, like magnetism; Einstein’s theory of gravity pictures objects taking the shortest paths through a curved or warped space-time, in which, because of the curvature, the shortest path will be a curve rather than a straight line—yet, we have no identification of the mechanism by which the curvature is caused by the object with mass or, in fact, the mechanism causing the object to take the “shortest” route).22
Richard Feynman has claimed to paraphrase Newton’s reported response to the accusation that his theory did not really tell us anything as follows: “‘It tells you how it moves. That should be enough. I have told you how it moves, not why.’” The Character of Physical Law, p.31. Should it be enough?
I do not have a good or satisfying response. In part, there is a factual question of whether a particular theory is just an analogy or, in some way, actually reflects the mechanism by which the phenomenon occurs. If the latter, then I think it is fair to say that we have at least a first step closer to an explanatory theory than we would have simply with a mathematical formula. On the other hand, in much of current physics, especially with respect to the very small (quantum