In his reply to Michael Riordan’s Opinion piece, Pantazis Mouroulis (Physics Today, Physics Today 0031-9228 56 12 2003 13 https://doi.org/10.1063/1.1650203 December 2003, page 13 ) made statements that reflect the common viewpoint that theories can sometimes be shown to be true or factual. This is something I don’t think can happen with the most fundamental and general of our theories.
Mouroulis wrote, “But the aim of every scientific theory should be … to have its truth proven beyond doubt.” However, by the basic logic of science, it is impossible for any widely applicable theory—any of the theories that we most value—to be proven beyond doubt.
A theory is a set of statements, T, that don’t logically contradict one another and that, taken together, imply an infinite set of testable consequences {C 1, C 2, … }. In other words, T implies {C 1, C 2, … }. To test a scientific theory, then, is to test a finite subset of that infinite set of implied consequences. The theory’s statements are not directly tested. If a large number of the consequences have been tested by well-verified experiments and found to correspond to measurements within experimental errors, we can conclude only that the theory being indirectly tested hasn’t failed. In fact, while “T implies C n “ is not logically equivalent to “C n implies T,” it is logically equivalent to “not-C n implies not-T.” Consequently, although establishing the truth of a theory is not possible, we can, as Riordan suggests with his “evolutionary metaphor,” use experiments to discard theories that imply consequences inconsistent with measurements.
The issue of whether a theory is or can become a fact is different from the question of whether a specific deduction from a theory can be found to be factual. When Riordan wrote about the existence of quarks, he was referring to a deduction from a set of theories; the first few of those theories, because they lacked the concept of the color interaction, were not identical to today’s quantum chromodynamics. The original theories were not fully correct. Yet they implied the existence of quarks, which seem to be as real as any other objects that we take seriously. Were the original theories that implied quarks simply subsets of the “factual” theory of QCD? That seems unlikely. The addition of the color interaction was a boldly creative step that produced a new theory; it wasn’t just a case of simple addition. However, what matters to this argument is that the new theory of QCD can never be called a fact because it implies an infinite number of new consequences based on different initial conditions. Yet we still seem to have quarks. Of course, the quarks of QCD are not precisely those of Murray Gell-Mann’s and George Zweig’s original theories. Nor are they precisely Richard Feynman’s partons. The most complete understanding of quarks comes from the strongest theory we have (that is, the one that has best survived tests of its implied consequences) that includes quarks among its deductions. In that sense, the meaning of our concepts in science comes only from our theories, and as theories change, so do the meanings of concepts. For example, the meaning of mass in Newtonian physics is not identical to the meaning of mass in Einstein’s special theory of relativity.
In the case of a theory that hasn’t been tested or that has failed tests, the meanings given to its implied concepts are empty. That’s the difference between science and mathematics or philosophy. In science as in no other field, physical experiments allow us to objectively weed out theories. Without tests against measurements, the most wonderful-looking theory is, at best, brilliant hallucination. In our mind’s eye, we seem to see something, and the something even has meaning to us, but unless we have objective means of checking, we don’t know if the something exists.
An odd and poorly understood feature of modern science is that its most central objects—general theories—cannot be known by scientific procedure to be true or factual. That feature is a strength rather than a weakness. Without general theories, we would be limited to advancing knowledge by making lists of facts. Such a linear approach is extremely slow compared to the methods of science in which a large but finite number of theories are discarded in order to keep a few that imply an infinite number of consequences. Many people have remarked on how strange it is that this works so well, but there is little doubt that it does.
