Limits of Science?. John E. Beerbower
that the physical world—reality—really does exist and that it is accessible to rational inquiry. Id. Furthermore, he says that the aim of science is explanation which enables us to understand the external world and that the use of theories permeates man’s mental activities.17 The dramatic progress in understanding has been the result of the development of a “tradition of criticism,” which has been embraced by post-Enlightenment intellectuals and scientists. Criticism has enabled the differentiation between good and bad explanations. “Bad explanations” (like myths) tend to have a flexibility that accommodates new evidence and experience without changing the substance of the explanation. As a result, bad explanations thwart progress. (In contrast, a good explanation is “hard to vary, because all of its details play a functional role.” Id., p.24.)
Proper scientific methods allow for the detection and correction of errors in theories. However, scientific theories, providing explanations, originate as conjectures or guesses; the discovery of theories is an act of intellectual creation by man. Id., pp.1–30. Finally, he concludes, scientific progress is always (and will always be) a work in progress, as existing explanations get replaced by better explanations which will themselves be replaced by better explanations in the endless quest for the best explanation. Id. Deutsch provides a somewhat simplistic view of the philosophy of science developed by Karl Popper and others in the twentieth century. But, as noted below, the philosophy has a tendency to be forgotten when Deutsch discusses scientific theories themselves.
As we shall discuss below, developments in quantum theories in the second half of the twentieth century gave rise to an increasing emphasis on the predictive successes and testability of the mathematical models being employed, as opposed to the concepts of explanation and understanding. However, it would appear that there are among the current generation of leading physicists some who seem to see the possibility of explanatory models that can be grasped by the intelligent layman—or, at least, the marketability of books that purport to present such explanations.
For example, Columbia University Professor of Physics and Mathematics Brian Greene recently wrote: “There’s a difference between making predictions and understanding them. The beauty of physics, its raison d’être, is that it offers insights into why things in the Universe behave the way they do. The ability to predict behavior is a big part of physics’ power, but the heart of physics would be lost if it didn’t give us a deep understanding of the hidden reality underlying what we observe.” The Hidden Reality, p.271. Albert Einstein, we can assume, would have wholly agreed.
Interestingly, there is a similar contrast or tension in the philosophy of mathematics among types of proof. Ian Hacking, Why Is There Philosophy of Mathematics at All?, pp.21–40. At one end, there are long, detailed proofs that appear to establish a proposition and can be carefully checked step by step. These proofs are often identified with Leibniz (1646–1716). At the other end, there are proofs that can be absorbed as a whole and which are compelling in their simplicity. This type of proof is associated with Descartes. The point of interest here is that the Cartesian proofs are thought to impart understanding—“they enable one to see not only that something is true, but why it is true. They give a feeling of understanding of the fact proven.” Id., p.32.
But, what does it mean to “make sense” of the observed regularities, to create a sense of insight or understanding? Natural scientists and many social scientists seem to aspire to the ideal of the deductive theory, and much of modern science appears in the mathematical forms to which a deductive theory lends itself. However, what we mean by an explanatory theory in the sense here is not necessarily or even primarily a mathematical model of the phenomena under investigation.
Toulmin argues that the notion of explanation “…involves appeal to some principle of regularity or ideal of natural uniformity.” Foresight and Understanding, pp.41–42. And, he acknowledges that those “notions” will reflect the paradigms and ideals of the persons involved. Toulmin attributes to Copernicus the objective of a scientific theory as “consistent with the numerical data” while being “absolute” and “pleasing to the mind.” Id., pp.41, 115. In more contemporary language, a satisfactory explanatory theory will be based upon circumstances that the observer considers to be natural or self-evident, circumstances that require no further explanations or raise no questions. See, id., p.41.
Toulmin refers to “ideals of natural order,” which are perceptions of circumstances or events—such as motion or change—that are in the natural order of things and that, therefore, require no explanation. The goal is to explain new data or observations of events that differ from the natural order in terms that make sense of the differences. Id., pp.44–82. Of course, man’s perception of what is the natural order of things is a historical/cultural phenomenon that can and does vary among men and over time. Again, those perceptions necessarily shape the questions that are asked (i.e., the observations that “need” explanation) and the nature and content of the theories that are developed to provide the explanations. This notion clearly introduces a significant subjective element into a field of intellectual inquiry (or “creation”) that we like to think of as “objective.” Hacking says, “[O]thers talk of the proof explaining the fact that is proved. Such words—understand, why, explain—are sound, but do little more than point to a satisfying phenomenon that is experienced, rather than one that can be well defined.” Why Is There Philosophy of Mathematics at All?, p.32. I return to this issue below.
An example: gravity
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