So given how for example we can’t know an electron’s position and velocity at the same time and other small scale observations and measurements involve uncertainty and randomness. Is there any solid proof that we’re not just wrong? I know all science is inherently possibly wrong and is a model of best fit but this part of science in particular seems so arbitrary to actually be a good model.
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You may be misunderstanding the uncertainty principle. It doesn’t say that we can’t know/measure position and velocity simultaneously. It says, as a consequence of the wave nature of matter, that the two properties are intrinsically linked and in fact together do not have arbitrarily definite values. You can’t know position and velocity the way you can’t make a square with angles adding up to 400 degrees. The more precisely defined one is, the less precisely defined the other is. It’s a mathematical consequence of quantum mechanics.
Quantum mechanics could be wrong, but all experimental evidence has shown it to be right. Actually, something has to be wrong with current physics because general relativity and quantum mechanics can’t be reconciled on some fundamental issues, but that’s beyond ELI5 and actually beyond what I can explain at all. They’re wrong in the way Newtonian mechanics is wrong: right enough to be useful to us, but still missing something fundamental about how the universe works.
Your question is basically is about the buzz around the universe not being locally real from some time ago, as well as other propositions like the uncertainty principle. The uncertainty principle isn’t about measurement or even any specific physical thing, but about the mathematics of waves. Since reality is constructed of waves (e.g., photons, electrons, protons show wave-particle duality), it applies to them.
The opposing side usually appeals to the same arguments made many moons ago ([https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates](https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates)), but the accumulated experimental evidence is that they’re wrong.
Could we still be wrong? Sure, I guess. At the very least, our understanding is still incomplete. But we’re about as right as we can be about it. Also you may want to adjust your expectations of what is arbitrary for physical reality haha. Do you feel a deep easy to understand reason for why the rest mass of an electron is 9.1093837015 × 10−31 kg? Couldn’t there have been a universe where it is slightly different? Sure, I guess. But it isn’t.
Generally we consider that it’s not that we “don’t know” a particle’s position and velocity to perfect precision at the same time, so much as that the question doesn’t make sense. The concepts of “position” and “velocity” as we understand them for large classical objects, like rocks, just don’t work the same way at a quantum level. Asking for the “position” or “velocity” of an electron, especially under complicated circumstances like the two-slit experiment, is simply asking the wrong question. It’s like an alien reading about countries, and then asking what the “population” or “governmental structure” of a single person is.
A good example of this is the [double slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment). We’ll make a wall, cut two slits in it, shoot things through the slits, and look at where they end up on the other side. If you shoot BB’s, it’s obvious – you get a pile of BB’s behind the first slit, and a pile of BB’s behind the second slit. If you do this with waves, though, something different happens – the waves go through both slits, and the pattern on the back is big where the waves add together, and small where the waves cancel out.
So now, send not a BB, but an electron through. Do this a bunch of times so you can build up the pattern, but only ever send one electron at a time. Now what happens? You get the wave pattern, not the BB pattern. The only way that can happen is if somehow the electron went through both slits at the same time. You can’t think of it as “well, the electron went through one or the other slits, I just don’t know which one it was”, because then you’d end up with the BB distribution. If you say “well, I’m going to watch very carefully what happens, so I can tell which slit the electron went through”, you indeed see the electron go through one or the other. But when you do that, the wave pattern goes away, and you’re back to the BB pattern. That’s what we’re talking about with uncertainty, it’s not that something has a definite position but we don’t know it. Instead, quantum things are really truly spread out over a finite size. We call this uncertainty because if I go and force a particle to give me an exact position, it will (at the price of giving up information about its velocity), but it didn’t have that exact position until I forced it (the technical term is “collapsing the wavefunction”)
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