“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ”
Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies
Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?
Certainly Turing and Godel showed that computation is not universal (complete). Looking at one element of number theory in isolation seems unproductive? Arithmetic space is staggeringly complex, but structured, layered. IMHO number theory is like a hall of mirrors without a defined interface with physics. See Yang Mills mass gap.
