>>27
You're stuck in old perspective of viewings such that mathematics should form the foundation of everything.
In a way, but PA and computation are expressible in each other, thus I consider computation the absolute minimum (along with logic) of math that is likely 'true' (consistent, physically realizable, if not equivalent, if Physical Church Turing Thesis (PCCT) is true). I have no idea if non-computable math (uncountable ordinals, real numbers, ...) makes sense or is physically realizable.
In a way, I don't think our thinking is too different, so I'll just list my current positions:
- 90% confidence in PA being true, and thus computational processes operating on
unbounded, but finite naturals make sense (an ultrafinitist will insist a bound exists, while I think the induction schema makes sense and while any machine can have bounds, there can always exist a machine with higher bounds than any you've previously considered (why am I even thinking about this? if PCCT is true, you have to consider a Computational Universe Hypothesis(CUH) which includes the set of all computational universes (similar to Tegmark's Mathematical Universe Hypothesis/Ultimate Ensemble, but restricted to discrete/computable math)).
- 30% confidence of infinitary set theories, hypercomputation making sense/being physically realiziable, which is to say that I don't think it is and I would be mildly surprised if it turns out it is the case (but it's impossible for us to ever tell apart a computational universe from a non-computational universe because we ourselves are quite finite and our substitution level is presumably quite high (that is, if you were to perform a functional replacement of one's own brain cells, you would become able a digital abstraction which would function more or less the same as our biological brain, to put it more simply, I conjecture that our brains do not exploit any hypercomputational processes (if our universe would have such properties(unlikely)) and thus such a substitution would work)). Penrose would assign a much higher probability to this - in a way, his hypothesis "solves" one problem by making it intractable (when considering antorphic reasoning within the CUH, there's the "White Rabbit" problem, or why our history appears so consistent; what is the probability that one would find oneselves experiencing this particular moment, in this particular universe? what are the totality of structures that support oneself and why is our past/feature so lawful?)
- If CUH is true, 35% confidence in the set of computational universes being finite, that is, while each universe can be finite, the set of all universes cannot, this is because I consider mathematical induction schema axiom valid (a ultrafinitist would assign a much higher confidence to this, they may also believe in a single universe, or merely refuse to think about whatever meta-rules the universe is based on).