In this post, I discuss some philosophical questions concerning uniqueness and validity of theories in physics.
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In his seminal essay On the Unreasonable Effectiveness of Mathematics in the Natural Sciences, Wigner mentioned the interesting topic of the uniqueness of physical theories, stating that “[a] proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.” While he deliberately did not expound much on the subject, he did point out that different theories developed to explain different phenomena could ultimately be incompatible, since each such theory would introduce new assertions beyond the simple statements of the phenomena. (We generally expect physical theories to introduce “new physics” beyond just mathematical equivalence to the observations, and we expect to be able use this new physics to produce additional testable predictions that can’t be derived just from knowing the initially observed phenomena.) Historically, when this has happened, it’s been an impetus for physicists to develop an even larger and more encompassing theory that rectifies the incompatibilities, so that the previous incompatible theories are just approximations of the new one. But Wigner asked the question: if the incompatibility were to be demonstrated in such an abstract and generalized manner that it would make developing an encompassing theory prohibitive, how should theoretical physicists then react?
There is another question that I had thought about as I had gone through some of my undergraduate physics classes. Let’s say we have a set of phenomena and a theory that explains (logically implies) them, and we build evidence for the theory by finding further testable predictions of it. How many predictions should we find, and of what nature should those predictions be, in order for us to be “generally confident” of the theory? How can we consider a set of predictions to be “well-representative” of the theory, in terms of “covering it adequately?” This question leads to an idea: can we develop mathematically formalized frameworks for judging our level of confidence in a theory based on its predictions (assuming that those predictions pan out in experimental testing?) How would our currently accepted physical theories fare under such frameworks? I have written down some responses to this that I will publish and discuss on my site in an exploration, but for this post we’ll focus on a separate philosophical point.
Before I discuss that philosophical approach, let’s consider a third question, one that has been around since the beginning of the study of physics and science: given a physical theory that explains a set of phenomena, how do we know that that theory is the “best” explanation? Typically, people mention something like Occam’s razor to resolve this. However, if we are looking for more precise (e.g. mathematically rigorous) criteria, the razor is fraught with problems — for example, it doesn’t on its own precisely define what it means for an explanation to be the “simplest.” And actually, even if we agree to forego rigor, there are still some opposing arguments for the razor. The Wikipedia article on Occam’s razor has a more comprehensive and detailed discussion of both supporting and opposing arguments. (I have also been investigating ways in which we can formalize Occam’s razor in a mathematically rigorous manner, and then how we can use these formulations to evaluate currently accepted physical theories. I will publish these writings in an exploration as well.)
All these ideas are useful to provide better and better ways to evaluate and judge the quality of our theories. However, at a more basic level than quality, we do not need to be concerned about the validity of our theories even if they provide fewer accurate predictions. Philosophically, rather than treating a physical theory itself as knowledge, I would prefer to treat it as a suggestion of knowledge, based on assertions that it implies. When we test these assertions, or the predictions of the theory, and when those assertions pass our experimental bars, then we can accept them as new knowledge that we can rely on in applications to engineering and technology. Then, it’s not as important for all of a theory’s statements to be “fully” accurate, as the implications we would actually use would be tested anyway before they are used and applied.
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EDIT (2025): The philosophical framework I introduced for validity, where theories are suggestions of knowledge rather than knowledge themselves, seems inadequate based on how theories are commonly treated and discussed. I’m still unclear then on what number/level/kind of predictions are needed for a physical theory to be accepted — for example, if I try to test Galileo’s assertion of uniform gravitational acceleration without ever varying the height or kind of object I dropped, how can I say I’ve truly tested the assertion properly?
Hamming’s response to Wigner’s essay could shed some light on this. He claims that physicists often engage in what he calls “scholastic reasoning” to come up with their theories. This reasoning may not be say rigorous by mathematical standards — we can in fact argue it can’t be as these assertions are falsifiable and have even turned out to be wrong when tested in edge cases — but it could provide better understanding for the intuition and motivation behind physics theories.
Personally, I’ve realized that this is what I’ve been missing — the motivation. While I did my physics major, I took the laws on faith, almost as if I were just doing math on top of them. At some point when I have bandwidth I’d be curious to go back and try to learn the motivation / gain better intuition for physics theories, which would help me understand them better beyond just a “memorize the laws as axioms and do math” level.
