experimental physics - Is the principle of Conservation of Energy empirically verifiable?

Before I am inundated by myriad and vociferous claims that conservation of energy is the single most well-attested and experimentally verified principle in all of science, let me say that I am well aware of the ever-growing body of evidence which seemingly bears this principle out. However there seems to be, in my view at least, an abiding problem inherent to a principle of conservation with respect to how we could ever truly verify such a thing empirically. That no humanly constructed device can be perfectly accurate or precise is an incontrovertible and obvious truth. As a result, I am unsure as to how we could ever be sure that energy (or any other conserved quantity) is not being lost or gained (in defiant violation of physical principles) at length and/or time scales orders of magnitude below the resolution of our best instruments. I mean, how could we ever know? Ontologically speaking, it's almost inconceivable that energy could not be conserved. But epistemologically speaking, I am at a loss as to how we might ever verify—via experimentation or measurement—that energy is in fact being conserved to the precision guaranteed by a conservation law.

Answer

I will try to give a short introduction into the ideas of scientific truth as I understand them.

In mathematics, the world is beautifully simple. We have axioms that the set to be true, and from these we can deduce a plethora of statements to be undoubtedly true - given that the axioms are true. There may be undecidable statements about which we cannot say anything, but within axiomatic system everything is either true, false or undecidable.

Now, in reality, we are not in as comfortable a situation: We don't know the axioms the world is grounded upon, we try to guess them. The axioms we guess are what we call laws of physics. Now, given two different hypotheses (i.e. guesses for the laws of physics) $H_1,H_2$, we may look at some situation and, by logic alone, deduce that $H_1 \implies O_1$ and $H_2 \implies O_2$. Then we perform the experiment. If we observe $O_2$ not to be realized in reality, then, by the law of contraposition, we may conclude $\neg O_2 \implies \neg H_2 $, so the second hypothesis is clearly and unambiguously false.

But, suppose that, within the errors of our observation, $O_1$ holds. Then we cannot say that $H_1$ holds because $(H_1 \implies O_1) \implies (O_1 \implies H_1)$ is not a valid form of argumentation. All we can say is that our observation is compatible with $H_1$ being a law of nature, but it is not true in any rigorous sense. No statement about reality can ever be true in the axiomatic sense because of this reasoning.

Now, of course there could be a third hypothesis $H_3$ also predicting $O_1$. Then, until we perform experiments for which the hypotheses predict different things, $H_1$ and $H_3$ are equally true within our heuristic approach. And the more experiments we perform, the more often $H_1$ survives this process of constant enquiry, the more certain we grow that this is actually a good description of the world.

This is what science is all about: Taking the vast landscape of possible ideas about how the world works - take as a simple example Aristoteles' idea that everything wants to be at rest vs. Newton's idea that everything continues uniform motion unless acted upon - looking at the predictions these different ideas about the laws of nature make and then performing experiments eliminating those ideas that are false. Thus, we make a gradual progress towards the underlying truth, which is empirically inaccessible. The longer we perform experiment, and think of new hypothesis, the more refined will our laws of nature be, the greater our confidence can be that these are really good approximations to the way the world really works.

To say "We cannot prove it, we really don't know" is to vastly underestimate the power of falsificationism, and shows a callous disregard for the scientific method, whose success is reflected in every bit of technology around us.

[One may also find my answer to Why should a (physical) principle be applicable to different systems in different positions in space and time? to be relevant in this context.]