One of the things I learned later on in my physics classes was that physical laws don’t necessarily guarantee uniqueness. Of course, when written out, this sounds obvious — why would we expect a particular set of laws to always guarantee only one satisfying physical state? — but intuitively I didn’t recognize this until later on.
Part of the reason is that physicists by nature want to narrow down the list of predicted physical states as much as possible. This is in fact the very goal of physics: to predict the state (at any given time.) The ideal set of laws would be one that guarantees uniqueness. Thus, it would be natural for physicists to continually evangelize the “one” set of laws that predicts one state, and in common discourse it becomes an intuitive assumption that whenever we work with a set of physics laws, we can predict the physical state uniquely.
This assumption gave me a bit of confusion in my physics major, when I realized that for example Maxwell’s equations don’t on their own guarantee uniqueness, without other conditions (the “initial conditions” — as an offshoot, see my post on laws and initial conditions in physics.) In the case of these equations, we can use gauges and certain initial conditions to provide uniqueness for most cases, but in general this isn’t an assumption that is worth making, especially once the physics curriculum path transitions to theories like quantum mechanics.
A consequence of this is that when people claim that something like Newton’s laws implies determinism, this is not actually fully true — Newton’s laws really just constrain what the predicted physical variables can be, and for many situations other than the most basic ones, they don’t necessarily guarantee uniqueness. Thus, the laws don’t “fully” predict the physical state at any time, but rather place restrictions on what it can be.
