Mathematics, Statistics, Zero and Noise

The number zero plays a fundamental role in mathematics, and by extension, theoretical physics. But the language of physics, in particular quantum theory, is not just maths but statistics. A central player in statistics, I'd argue, is not zero as such, but noise, of the fundamental variety. There remains debate as to whether fundamental noise has any physical existence. It is something that can't be mathematically defined or proved (which is why it's existence is questioned). But we can nevertheless grasp the concept of such. Fundamental noise can be described (if not defined) as the selection of a number (or numbers) for which there is no reason for selecting that number (or numbers). From a statistical point of view the infinite set of all such selections will sum to zero, because for every selection that is positive, it can be coupled with a selection of equal magnitude but negative. The couples cancel each other out.

Statistics can be regarded as a way of reconciling observables with mathematics.

Quantum theory involves both a mathematical function (the wave function) and a random function. In computer simulations the random function has to be simulated (using a pseudo-random function) because random functions are mathematically impossible.

Why a random function?

It's required to account for the observables - the particle detections. The wavefunction describes the mathematical structure of the observables but can't describe the relationship between individual detections and that structure (the so called wave function collapse problem). This is because the relationship is (propositionally) random in one direction and undefined in the other. This doesn't mean there is no relationship, just that there is no mathematical relationship. While any single detection is random, the statistics of that randomness have a structure. In much they same way that one can say the statistics of a certain type of noise add up to zero one can say the statistics of particle detections add up to a wave function.

An easy way to understand how detections end up describing the wave function is to imagine each detection as a random sampling of the wave function. This doesn't help us to get any closer to a mathematical model of the relationship but it does help in computing simulations of such.

Another way to think of it, from a mathematical point of view, is to treat observables as literally errors for which there is no cause, ie. that the observable universe (whether we are here to see it or not) is just an error.

I prefer to think of the observable universe as just a different version of the mathematical one (and vice versa), that neither is a function of the other. The former is based on noise (presence) while the latter is based on zero (absence, the void). Each can be regarded as equally fundamental.