Universal Modeling
Our understanding of the structure of the physical universe starts from the model that we settle upon to describe it. The universe has a form, and models are what we might call 'formalizations': a way of specifying a form that is clear and unambiguous.
There are, of course, a multitude of possible models that we could choose from, and not all of them are equivalent for any given purpose. The central axis across which the models can be broken down is the fundamental modeling tradeoff →, namely that models strike a balance between complexity and accuracy. The most accurate model, i.e. the perfect one, is the one with the same amount of complexity as the real thing. Which, since we're talking about the physical universe here, is to say that ititself is its own perfect model.
This is an unattainable degree of complexity, if for no other reason than because we, the modelers, are encompassed within it, and so its total complexity is at least the sum of the model itself [1].
[1]I'll note that this is generally true for models of any system. The only exception that comes to mind are quines ⬈, computer programs that output their own source code. This is something of a degenerate case though; it works because the entire system exists for the sole purpose of modeling itself, and has no other capability. I'm generally not inclined to think that the physical universe is a quine, though.
Any sufficiently ambitious model of the universe will eventually run up against this hard limitation, and most do so very quickly. You see, the universe is big, to put it lightly, and there's a lot going on out there [2]. Fortunately, I don't actually want to create a model; my purpose is understanding, not simulation. I'm not here to predict the future states of the universe, nor to reconstruct its past [3], rather just to consider how one might seek to model it.
[2]Usually when someone wants to model the universe, they simplify it to make it tractable in one of two ways: either by the elimination of details they don't care about (like the motions of individual fundamental particles, replacing them with macroscopic objects with the same stochastic properties), or by making their quantum of time coarser [2-1], so that they model the world second-by-second, or day-by-day, rather than at the Planck scale.
[2-1]Almost all physical models I've encountered formalize the universe as time-slices, and processes to iterate from one time slice to the next (video games are the most obvious such models to do this).
Even if such time-slicing could be extended to that scale, I doubt that this time-slice model could be used for a perfect model of the universe. Specifically, since distance is the time it takes a photon to travel between two points, if you were comparing the distance between points at a fixed point in time, their distance would only be defined if the points were the same, or if the points were entangled (in both cases, yielding a distance of zero). Otherwise, since a photon cannot travel some distance in no time at all, distance must be physically undefined in any time-slice; not exactly the result you were expecting!
[3]Eventually, we may want to take a crack at the beginning and ends of the universe, but that's still aways-aways out, and not on my radar at this time.
To assemble our perfect model of the universe, first lets figure out the terms in which we'll measure the model. Back in 2019 the SI base units were redefined ⬈ so that (paraphrasing extensively), there is only one base unit of measure, the second (of time), and all other units are constructed as constant factors of relationships between other units [4]. This is especially interesting to us as universal modelers, because it means that our model only needs one unit [5]: time. The rest is all just constant factors [6].
[4]
Namely,
- a second (time) is defined by the number of oscillations of a caesium-133 atom in its ground state.
- a meter (distance) is the a second times the speed of light in a vacuum.
- a kilogram (mass) is a second over meter-squared times the Planck constant.
- an ampere (current) is the elementary change of an electron times a second.
- (other units elided as being not particular to this discussion, but the general pattern of definition in terms of previously-defined units holds)
[5]Another curious corollary to this observation is that time can expand or contract without us being capable of noticing.
[6]Which we could normalize, if we wanted to. Some physicists do so, resulting in what are called natural units ⬈.