Uniqueness in Nature and the tacit physicist method of "solving problems by uniqueness"

This was a question I asked on Physics SE which was, unfortunately, not well-received. I'm asking it here in the hopes that someone can talk about the nature of the logic/assumptions used in the sorts of strategies I describe below. I've also included an example from physics, but can remove it if that's not appropriate here.

It seems to me that it is a fairly oft-used (although rarely mentioned explicitly) strategy in physics textbooks (and perhaps the research literature) to solve problems in the following manner:

1. Make a postulate (likely based on good physical intuition and/or observation).
2. Compute the solution given the postulate.
3. Verify that the solution confirms the postulate.

But of course, strictly speaking, we are begging the question in doing this. Is it therefore fair to say that in all physics problems, it is acceptable to use that nature converges on a unique solution to solve problems? Can we prove that this is a valid strategy? Because one could conceive of the possibility (and now roughly speaking) that by making a postulate as per 1), we are "led down the wrong path" and so obtain the wrong result in 3) which nevertheless agrees with what we guessed in 1). That is to say, we can derive absurdities by assuming absurdities.

In case the general commentary I make above is not clear, below I attach an example of such a strategy from Example 6.3 of Zangwill's Modern Electrodynamics. NB that we:

1. Assume the form of σP (as σP = σP(θ) so that we can use a result derived earlier for a surface density of this type),
2. Compute σP, and
3. Observe that σP = σP(θ).

It's not clear to me that resolving this particular example's use of the strategy above is as simple as saying we're using the uniqueness of Maxwell's equations here (a particular physical theory) seeing as those fields are uniquely determined given a fixed functional form of the sources, but here we do not a priori know that functional form.

• "Postulates" do not necessarily lead to unique solutions. Uniqueness of solutions often follows in typical models from mathematical theorems about differential equations, hence this is often used, but not always, and cases of non-uniqueness are also well-known (non-Lipschitz forces, for example). But yes, physics textbooks often skip full mathematical treatment and rely on intuition, which may lead to mistakes. Commented Jan 11, 2023 at 18:05
• @Conifold What is an example of such mistakes? There seems to be an "unreasonable effectiveness" to such methods, so I wonder if it's possible to reasonably defend the position that such methods have validity (more reasonably than my saying they are ostensibly effective).
– EE18
Commented Jan 11, 2023 at 19:00
• A simple example of non-unique solution in Newtonian mechanics is Norton's dome. The standard justification of validity is that in most models the forces do satisfy Lipschitz continuity and uniqueness follows, there is electromagnetism uniqueness theorem for your type of example too (but without uniqueness for the source distribution). Commented Jan 11, 2023 at 19:47
• Your formula messed up. It seems the author already confirmed from application 4.3 the charge density as defined as a flux is determined to be proportional to cosine of the angle (due to the background electric field) which has to obey divergence theorem additionally . What’s exactly is your circularity confusion here? Commented Jan 11, 2023 at 20:30
• We can never know... maybe the result happens to satisfy what we saw in nature only by chance. However, physics is self-correcting by design, and there will be more observations, which will confirm or infirm the result. The strategy of postulating/calculating/verifying is not universal in physics either. Commented Jan 12, 2023 at 3:47

First things first... Nature doesn't have problems. Humans have problems, which derive from the human desire to understand the way nature is. Such human problems are strictly cognitive, and the general assumption is that solutions to these problems will converge to some formulation that represents the natural world well. This is the essence of empirical testing (and what seems to be missing from your presentation). We:

1. Make a postulate based on intuition and/or observation.
2. Calculate a solution given that postulate, as things should occur in theory.
3. Verify that solution we calculated in theory agrees with results we achieve in practice through experimentation.

The example you've given is a pedagogical problem, not a scientific one. Its goal is not to establish that electrical fields operate in this manner, but to teach a student how to do analyses of this sort, given the presumption that electrical fields operate in this manner. As a student one is trained into the reigning paradigm for a given field, and it's only after that training that one can begin to question the paradigm: i.e., do experiments that might contradict what the paradigm says.

Is it therefore fair to say that in all physics problems, it is acceptable to use that nature converges on a unique solution to solve problems? Can we prove that this is a valid strategy?

Alright, you have some muddle we need to demuddle.

First and foremost, doing textbook physics problems is not exploring the nature of nature. In no way do the artificially created mathematical problems that comprise the end of a mechanics or optics chapter reflect nature, the philosophy of science, or even the philosophy of physics. They're pedagogical artifacts, and more specifically, they are oversimplified, domain-specific models generally presented as mathematical diagrams (SEP) imbued with deductive mathematical methods. From a pedagogical perspective, one might essentially say that they are cognitive scaffolding targeting a specific proximal zone of development in a student. You've digested a bunch of pre-calc and calc, you have a highly abstracted, shortly written context, and you are picking out salient features in the semantics of physics to set up and solve some math to come to a conclusion. So, that convergence you note on a single way that results in a unique solution for the problem? Yeah, that's the textbook author's design, and not a facet of nature. In fact, one of the seminal and controversial ideas between philosophers of science, is one of the question of underdetermination. Radical instrumentalists will insist that no amount of explication will ever prove a theory adequately, though most philosophers take a heavily realist approach, and believe in positions that advocate, like some generalized version of Cummins's functional analysis, that additional iterations of empirical process will resolve any ambiguity that inheres regarding how data maps to justified theory. You state the third step in problem solving is:

Verify that the solution confirms the postulate.

In math class, sure. But in science, the notion of verification (and confirmation) have been abandoned by philosophy of science thinkers, though obviously not in practice; Karl Popper is famous for his rejection of verificationism and confirmationism with his proposal of falsifiability. In a textbook, you verify an answer when you model the problem correctly (often using derived formulas that are part of the canonical theory) by calculating, checking your math, and then checking the solution manual. Again, this activity isn't Nature. It's the publisher, and a means for an evidenced-based evaluation of a students performance for grading, much in the same way researchers use shallow citation metrics to measure their impact or contributions to science (which is to say, largely a useless cultural artifact used for political reasons, with the exception of rare papers that are tremendously influential) to keep their jobs in publish or perish environments. The problem solving you're focusing on might be seen as perpetuating normal science (SEP), under Kuhn's theory put forth in The Structure of Scientific Revolution.

The practice of physics is much broader than textbook problems, and if you're interested in what physics is more broadly, at least through the lens of an empiricist who prides himself on creating and testing variables, take a look at Martin Krieger's Doing Physics (GB). In great detail, he explicates how the scientific apparatus is designed to create the math necessary to isolate and prove scientific hypotheses. One you get beyond textbook problems aren't really physics in the sense of an empirical methodology, then you're ready to begin exploring the philosophy of physics and the philosophy of science more broadly. For the former, I'd recommend *Philosophy of Physics: A Very Short * (GB), and for the latter I'd recommend A Companion to the Philosophy of Science (GB). I poked around the SEP and found Experiment in Physics, which while not my cup of tea, does looked like a solid introduction into the empirical nature of physics. Exercises for the classical mechanics, E&M, and modern physics (usually taught as a three-course, calculus based sequence) are just to help you develop the bare bones necessary for understanding what physics is.

(PS And Nature, Natural Kinds, and the Laws of Nature have largely been deprecated as concepts, but that's another Q&A response entirely.)

Regarding the derivation of absurdities by assuming absurdities, it is an essential skill for every working physicist to know how to detect an absurdity in his or her answer to a problem, and correctly identify the absurd assumption. In fact, this is a common practice in physics because it enables the practitioner to flag a bad assumption to the rest of the community, and then to identify the specific thought or conceptual error responsible for making the assumption invalid. This is one way progress is made in the field.

It has been said that the objective of a course of study in physics is to learn how to solve every class of problem which has already been solved in the field. Those then furnish the basis for the problems posed in textbooks. Unless a physics student knows how to solve the problems which have already been solved, it is unlikely that they will be able to furnish valid solutions to problems which have not yet been solved. This is just the way things work in the teaching and subsequent practice of physics.

Regarding the issue of underdetermination of theory by evidence, this is a rhetorical trick akin to a "divide by zero" instruction in a computer program in that it can be used to invalidate ("blow up") any hypothesis at all regardless of its truth content, and as such is useless for assessing whether or not any hypothesis is in fact valid. As an aside, you haven't lived until you have had an engineering manager play the underdetermination card on you as a way of making you (and the problem you rode in on) go away.