First, I got to say that I just dabble in physics. I have a ton of popular books on physics; I’m fascinated by quantum physics and have tried hard to understand it. The impact of my mathematical training on my understanding of physics is nil, as far as I can tell. I’ll be waving my hands a lot here. I know I won’t be satisfied with what I come up with, but, after all, it’s not mathematics.
Heisenberg’s Uncertainty Principle is a specific example of Bohr’s Complementary Principle (you can’t measure two facets of a quantum object simultaneously). This lead to the Bohr-Heisenberg-Born Copenhagen Interpretation of quantum mechanics, which basically says that there’s no objective reality until a measurement or observation is made. (Opinions vary. Heisenberg and Bohr themselves didn’t quite agree on what it meant.) When the measurement is made, “the wave function collapses” and you’ve got the reality of the situation. This brings up the Schrodinger’s Cat thought experiment, which Schrodinger came up with in an attempt to illustrate the flaws of the Copenhagen Interpretation. Bohr’s reaction: yeah, the cat is neither dead or alive until we sneak a look.
Einstein was never satisfied with the Copenhagen Interpretation, and kept trying to come up with thought experiments like Schrodinger’s Cat to show that there was some flaw in the Copenhagen Interpretation. In 1935, he, along with Podolsky and Rosen, came up with the EPR Paradox. EPR: If you shoot two identical particles apart from each other and then measure the momentum, say, of one particle, then, according to the Copenhagen Interpretation, the wave function collapses and the momentum of the second particle, which is a certain non-zero distance away from the first particle, is instantaneously determined. (It’s a lot more nuanced than this, but whatever.) This, according to a Einstein, would violate the Special Theory of Relativity, which says that no information can travel faster than the speed of light. Einstein called it “Spooky action at a distance.”
Shoot ahead to 1982, when Aspect set up an experiment to test the EPR Paradox, and, to the surprise of nobody, showed that this sort of instantaneous communication over distance is a real thing, just as the Copenhagen Interpretation predicts. Aspect got the 2022 Nobel Prize for this.