A brief description of physics as I understand it.
At the astronomical level, regarding galaxy cluster formation, Hubble expansion, and other bizarre ideas with Lovecraftian names like “dark energy”. Despite the prevailing metaphysical assumption of our culture that we are nigh the “end of history (of science)”, we truly don’t understand a lick of what’s going on at this level. For all we know, we’d need to invent a different sort of mathematics altogether.
(Barnard 68. WTF is that and why does it exist?)
Zooming in to the level of orbital mechanics, concerning the motion of stars and planets, everything is mere geometry. Orbits are elliptical, merely tracing out some predetermined section of a gravitational light-cone (a sort of isotherm of angular kinetic energy), and objects are tiny spherical points whose only interesting quality is their mass. It is very beautiful to a logical, mechanistic sort of mind, though one wonders where are the strings that hold this beautiful marionette show together.
One begins to wonder if all gravity-warped mass-time-space is merely one large, geometric shape for which we need only some grand formula.
If you zoom in a little closer, to the level of everyday life, the resolution doesn’t just improve- it turns out there’s a lot of weird detail. No object available to the human perspective is spherical- even the earth and sun are slightly oblong, and dust particles are asymmetric craggy meteors. There are only two sorts of math that allow us to understand anything at all in this perspective, statistics and differential equations. We have faith in the central limit theorem, and hope that air particles act enough like spheres that their emergent behavior in mass is more or less geometrical.
This, despite that the collisions are surely more akin to a tumbling forest of trees whose branches sometimes catch and break off and sometimes stick back on. We hope that by fitting curves to these statistics, we can collect enough cross sections of the grand formula that we can guess its shape.
The way we guess grand geometric shapes is by solving differential equations. More to the point: partial differential equations. That’s because a partial differential equation, as written, is literally just a verbal description of one of these cross-sections. E.g. The heat equation.
Zoom in way closer, and the math changes again. I’ll admit that I’m a little hazy at this point, but it seems to me that the statistical behaviors we see are created by the behavior of extremely complex systems. The interactions of these systems are forever and always hidden behind a veil called Heisenberg uncertainty. Thus particle physicists and quantum mechanics are in the position of applying voltages to the leads of a motherboard and observing the other leads, and trying to guess how the transistors are laid out on the CPU. I think it is unlikely human civilization can exist at a high enough level of technology for long enough to unravel mysteries lower than may be viewed by electron tunneling microscopes, but ultimately time will tell.
(Do not talk to me about hand-wavy “true randomness”. I have a history of becoming very upset about that particular sort of stupidity. Pseudorandom radioactive decay has not invalidated causality – moreover, there is no reason to suppose that it has. The unfortunate belief that it has is a consequence of the fact that apparently no one in the world understands how probability works except me, and it works just fine for this, thank you. It would be just as senseless to suppose that Einstein’s relativity has overthrown the conservation of energy principle on the grounds that most people have difficulty with calculus.)
So in brief, we have physical behaviors which build up to each other, following the correspondence principle, according to this hierarchy:
??? -> Complex systems -> Statistics -> Differential equations -> Geometry -> ???