On Interpretation is the English translation of the Latin title of Aristotle's book Περὶ Ἑρμηνείας (Peri Hermeneias), which means, literally, "about meaning".
It's a book about language, going into precise detail about nouns and verbs, and the propositions that can be formed by affirmation and denial, ultimately presenting us with the "square of opposition" that defines how various kinds of positive and negative statements relate to each other and the world we live in.
What we mean by what we say is a hard problem, and the more abstract the language we use the harder the problem becomes.
In the sciences, some abstractions are harder to deal with than others. I confess a small degree of envy for geologists: tectonic plates may not be precisely visible to the naked eye, but there is relatively little wiggle room about what we mean by them. They are great slabs or lighter, more brittle, rock that float atop denser, more plastic layers beneath.
Biology has it a little worse. It's a science whose foundational tome is The Origin of Species, but there is a vigorous and ongoing debate about what a species is. The "biological species concept" has fuzzy edges, particularly in botany, where hybridization between plants is very significant. So not only can't we see a "species" we don't have a great definition of the abstraction.
Where concepts have fuzzy edges we typically fill them with other concepts. For example, we have a concept of "ocean" and a concept of "land" but the boundary between them is far from precise, so we have multiple additional concepts associated with the fuzzy region, including "beach", "littoral", "splash zone", "drying rocks", and so on.
Biologists of a philosophical bent are still arguing about how best to do this with regard to "species", or were the last time I looked.
Physics, though, is the worst. Physicists deal with abstract representations of things we cannot see and which may not even exist.
When Newton introduced the concepts of "force" and "mass" they were purely abstract, mathematical, concepts, whose meaning was not clear. You can't see gravity, only its effects. Nor is it easy to perceive mass, except via its effect as weight. Their development into coherent concepts took decades, building on a foundation of centuries. It will be interesting to see if human children raised in orbital habitats where they are routinely get to interact with the world in low-gravity environments develop a more natural, visceral, concept of mass than our gravity-bound selves.
And those are the simple concepts that were created the old fashioned way, by reasoning from phenomena--the fall of apples, the motions of the wandering stars--to the mathematical language that defines the abstractions. The concepts at least to some extent preceded the mathematics.
But once we had the mathematical language, it was reasonable to ask "What does it mean?", which in more concrete terms translates to "How would the world have to be for this description of it to be correct?"
This is a perfectly reasonable question. We ask it all the time about other human beings, although we aren’t very good at answering it. Actors read scripts that describe what a character says and does, and ask themselves, "What kind of person would behave this way?" Those interpretations then inform their performance.
Such things are far more elastic than is often appreciated: one of my favourite exercises as an actor or director is to see how far a character can be played "against type". That is, consider the "obvious" interpretation of a character and see how far can they be pushed in the opposite direction.
The answer is often: a long way. A decisive Hamlet, a compassionate Lady MacBeth, an honest Iago... while these interpretations might ultimately do violence to the integrity of the play as a whole, in individual scenes they can be made to work, and work well. Being aware of that as an actor can result in a richer, more nuanced, portrayal, and I’m told there was a London production of “The Taming of a Shrew” some years ago that made the whole thing into some kind of sexy, consensual D/s game rather than the tsunami of gaslighting and abuse that a literal reading of the script might imply.
The range of interpretations open to a given behaviour, especially in isolation of the whole context of a person's life, is broad enough that insisting that one of them must be the sole correct one is a recognized cognitive fallacy, sometimes called "the fundamental error of attribution", which is the idea that if a person behaves badly they must be a bad person, rather than just distracted or confused or having a momentary lapse for some other reason.
The same is true of the abstract language of physics: there is rarely just one interpretation of any given bit of mathematics, and the ontological commitments involved in different interpretations--the claims we make about how the world must be given how our equations describe it--can even be contradictory.
This irritates people—including me—and in the middle of the twentieth century there was an attempt by the logical positivists to do away with the problem by reducing "meaning" in physics to pure operationalism. Everything we could not see was to be defined by some concrete set of operations on matter: a particular experimental setup. This was supposed to banish "metaphysical speculation" from physics.
It didn't work.
The difficulty is that to choose what set of operations were canonical you had to make some claims about what it was you were defining, which got you right back in the soup. The only justification for the claim that "force is defined by this set of operations on a spring balance" was the claim that "forces are the result of continuous fields or potentials," which was exactly the kind of "metaphysical" claim the positivists were trying to get away from. These claims about the underlying, invisible, structure of the world were the only thing that gave any of the operational procedures meaning.
Furthermore, operational or procedural definitions are far too complex for thinking about, and the whole point of our hierarchies of abstraction is to make things easier to think about, so strict operationalism failed the most basic test of cognitive utility.
That said, operationalism does provide a useful touchstone to ensure that our concepts don't float completely free from reality. So while operational definitions turn out to be useless in the general case, being able to reduce a concept to an operational demonstration can be a very useful tool for clarifying meaning, even if it can't define it.
This leaves us roughly where we are today, where physics has a handful of descriptive things that arose out of purely mathematical deduction whose meanings are not at all clear.
The quantum mechanical wavefunction is the biggest deal to me in this regard--about which I'll have more to say in future--but the metric tensor of general relativity is also a problem because we have to know what it is to go beyond classical relativity and find a gravitational theory that can accommodate quantum mechanics. The various approaches to quantum gravity differ precisely in what interpretation is put on the metric tensor at the start of the process.
This is the problem of interpretation: purely mathematical thinking won't get us a deeper understanding of the world we live in, any more than an actor learning their lines absolutely perfectly will give them a deeper understanding of the character they're playing (although in both cases deep familiarity will help with more meaningful thinking). It's only by asking what these things mean that we have a hope of gaining deeper insight into the world.
In one case asking what a thing meant led to the discovery of a new phenomenon: the Aharonov-Bohm effect (which inevitably is not named after its original discoverers.
In keeping with Newtonian mechanics, Maxwell's equations describe electromagnetic fields, which were held to be the cause of the motion of charged particles. Ehrenberg and Siday (and later Aharonov and Bohm) realized that quantum interference phenomena allowed a wider view of "motion" that included the shifting of an interference pattern, and that these shifts could be shown to occur purely as a result of electromagnetic potentials by creating a setup where the fields were zero in the region of space an electron beam passed through, and the hitherto "purely mathematical" electromagnetic potential shifted the interference pattern created by the beam as it moved around a shielded, magnetized, metal whisker.
The way a thing is causes what it does, so showing potentials can cause a change in the world we can observe means they are "real" in the usual sense of the term, although we are still left with some deeply confusing questions about them, because the nature of their reality is hidden behind the quantum veil, beyond which it is very difficult to see.
Which fact might itself have meaning.