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I would like to investigate further the concept of an explanation that provides us with understanding. Perhaps it would be useful to take an example of an issue in the natural sciences and engage in a rather extended examination of the use and nature of theory and the relationship to understanding and explanation. Let us consider gravity. Newtonian mechanics, a model the fundamental elements of which are some 400 years old, has often been heralded as the paradigm of the successful scientific theory and has indisputably yielded some of the most spectacular and useful practical applications of modern society. Yet, the fundamental elements of that model fail to achieve a meaningful theory of causation or a satisfying explanation of the how and why of the observed phenomena. See, e.g., Joseph Mazur, The Motion Paradox, p.181.

The situation has been described by a mathematician as follows: “Contrary to popular belief, no one ever discovered gravity, for the physical reality of this force has never been demonstrated. However, the mathematical deductions from the quantitative law proved so effective that the phenomenon has been accepted as an integral part of physical science. What science has done, then, is to sacrifice physical intelligibility for the sake of mathematical description and mathematical prediction. This basic concept of physical science is a complete mystery, and all we know about it is a mathematical law describing the action of a force as though it were real. We see therefore that the best knowledge we have of a fundamental and universal phenomenon is a mathematical law and its consequences.” Morris Kline, Mathematics for the Nonmathematician (1967), p.361.

In other words, we do not know how gravity actually works and, therefore, cannot even say that such a thing clearly exists (even though the consequences of what we call gravity clearly exist and can be accurately predicted by us).¹⁸ By “how it works” I am referring to what we have described as an explanatory paradigm or model—a conceptual structure that is consistent with the observed events and gives us a feeling of understanding of the type of causality that is at work. As discussed above, we tend to look for an analogy, something with which we are familiar and think we understand, that displays at least some of the characteristics of the phenomenon that we are attempting to explain.

With respect to gravity, there are two things in particular that need to be explained. First, how does the attractive force manage to operate over distances of empty space? Second, what is the reason that the attractive force between two objects is inversely proportional to the square of the distance between them? The first question arises because of our ordinary understanding that things interact through direct or indirect contact. The second question is really the question of why there is such a simple mathematical relationship between gravitational force and distance. It makes sense that distance matters, but why is the relationship based upon the square of the distance?

A Newtonian version

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.