>>25
You're stuck in old perspective of viewings such that mathematics should form the foundation of everything.
It's because you think nature operates on some different set of rules aside from computation at the most fundamental of levels. This is not true. See the Holographic Principle and the AdS/CFT Correspondence. Once you accept that the Universe is capable of at least Universal Quantum Computation, and thus simulating any other Universal Quantum or Turing equivalent machine (if the Universe wasn't, it would be impossible to build our modern Von Neumann/Register machines), everything else follows.
Mammalian brains are nothing more than emergent, vastly complex computational networks. Peano Arithmetic doesn't mean anything in the abstract sense without a human mind (or non-human mind with similar capabilities), or capable computational device to interpret and reason about it. That doesn't mean Peano Arithmetic is nonsense, it's just merely an illusion, a facet of the underlying computatiob of our own deliberative thought processes. That's why it appears that computation is equivalent to Peano Arithmetic, or rather, it should be stated that Peano Arithmetic is equivalent to computation.
I agree with the rest of what you say, the reason we can recognize and reason about things like infinity, non-computable reals, etc. is because we have constructed different rules and axioms (hint: algorithms) when thinking about them. There's a lot of implicit piece-wise logic going on. It's quite evident when you look at the source code for theorem provers and equation solvers, for example.