Is there a proof that we can't prove a physical theory?

Question

I am thinking of physical theories (e.g. Newtonian Mechanics) as axiomatic systems. We have a list of axioms and from there we can derive theorems, make predictions etc. If the prediction don't agree with the observations we disregard the theory or we modify it. We usually say that a theory can never be proved correct but I can't quite get it.

For example suppose that all the observations that have happened and will happen agree with the theory. Then couldn't this prove the theory correct? It seems that if the problem is the huge amount of observations etc then in principle we could evaluate a theory as correct or wrong.

Even if we were capable to observe everything and check if it agrees with the theory then there would be also the possibility that the "root" axioms of nature are different from the axioms of the theory. For example we can predict something from the theory using the axioms. We end up with a statement "If P then Q" where P stands for the axioms (which are assumed to be true) and Q the statement we derive. If there is another set of axioms (another theory) that can derive the stament Q then there is no reason to believe that the axioms of the first theory are the same with the "root" axioms of nature. But if they both (the first and second theory) make the same predictions then we can say that the theories are equivalent. So is it only the fact that we are unable to observe everything that leads to the conclusion that a theory can never be proved correct?

Suppose we build a theory using only one axiom. This axiom is:

Every car in UK is red.

The quantifier goes all over the set of cars (in the time of formulation of the axiom, so there is no need to check past or future cars) in UK. If the amount of cars in UK was small say 10 (so we could check it) and all the cars were red, wouldn't this account as a "proof" of the theory? What prevent us in this case to say that "the theory has been proved correct"?

Answer 1

Summary

A. The Example

The difference is that your statement is not a theory. To be a theory it must make predictions as in an “A implies B” form, and be generalizable.

The University of California, Berkley, defines a theory as

A broad, natural explanation for a wide range of phenomena. Theories are concise, coherent, systematic, predictive, and broadly applicable..

https://www.livescience.com/21491-what-is-a-scientific-theory-definition-of-theory.html

B. Theories in physics in general

The interesting question is whether any physics theory could even be true - hotly debated in philosophy of science as discussed below. Theories usually introduce mechanisms and notions about physical reality on top of their predictions.

Secondly, showing it will never be falsified is not only impossible, but we can’t even claim to have high confidence. We continue to get better at both measuring and at creating extreme conditions, and we have observed other theories fail under new conditions. There are even theories that the laws of physics vary through space and time. One example is bubble universes, and another is inflation.

---

The meaning of “a true theory”

Assume we somehow can show it will never face a direct counter-example. We’d still need to know what it means for a theory to be true. In logic, “A” is the same as “A is true”. But for physics, some (e.g. below) say predictive capacity is not the same as true representation.

For example, electric fields are used extensively, but we do not know that they exist per se, nor whether the question even has meaning. We know charges exert forces by Coulombs Law. That is indisputable and arguably primary. An infinite variety of charge distributions can result in a certain net force on a unit charge at a point. For tractability, the magnitude of an electric field at that point expresses the same information.

So the question is whether Maxwell’s equations are “true”. If we say that electric fields are real and that theories using them as primitives are “true”, then what about electric potential? That’s the energy in this field that we now say exists. Does that exist? Potential is the energy (as work) it would take to bring a unit-charge to that location. These and other questions are debated in philosophy of science - as is the whole notion of physical theories and truth.

The same type of thinking applies to gravity. From Forces and Fields by Mary B Hesse

There is a physical difference between a gravitational field ... and the velocity field of a fluid. In the latter case the field function is an actual property of material at every point of the field, but in the gravitational case the potential function V is 'potential' in the sense that it does not necessarily describe a material property of the field ... it describes a potential property, namely, the force that would be exerted if a small mass were introduced into the field at that point.

Despite her claim that the case is simpler, debate has sprung up about the reality of the stream function. We would at least need one fluid particle to travel the entire streamline for it to be real. As hydrodynamics has advanced, the shape of the water molecule itself is being taken into account, such as here: https://pubs.acs.org/doi/10.1021/acs.jpcb.6b01012 ( The basics of including molecular dynamics into hydrodynamics: https://www.redlandsusd.net/site/handlers/filedownload.ashx?moduleinstanceid=16224&dataid=10931&FileName=Water%20Properties%20Activity.pdf

No particle follows the streamline: I would say that means we have realized streamlines don’t actually exist - depending again on what we mean by existing, even though models using them do work.

Inability to Predict future counterexamples (direct falsifications)

You alluded and another answer alluded to this: Our capacity to measure has been advancing and our capacity to create extreme situations has too. Things like capturing images the infant universe, and the Haldron Collider, are good examples respectively. Perhaps e.g. at new power densities .. the theory no longer works.

Because you ask about proving, the relevant supposition would instead be, “Suppose that all the observations that have happened and will happen agree with the theory and we know this fact now”. Which reformulation implies its own answer.

Answer 2

Suppose we build a theory using only one axiom. This axiom is:

Every car in UK is red.

The quantifier goes all over the set of cars (in the time of formulation of the axiom, so there is no need to check past or future cars) in UK.

The claim you are actually making is "At time state t, every car in the UK is red." This claim is not falsifiable because if the current moment in time is not t, then you cannot make an observation that proves the statement wrong. Iff a claim is not falsifiable, then it is not scientific.

What happens if you drop the temporal aspect from the claim? The claim is now "At all time states, every car in the UK is red." This is a falsifiable claim because observing a single time state where there is a non-red car in the UK will prove the claim false. However, the only way to prove it true is to observe every car in the UK at every time state (i.e. in the past, present, and future). As a finite being, this is not something you can do. Therefor, you can never prove your claim correct.

To put it simply, it is impossible to confirm, "that all the observations that have happened and will happen agree with the theory".