Agreed. A Turing machine operates in a discrete countable state space, whereas the human brain requires real numbers for a complete state description. Cantor showed (with the diagonal argument) that real numbers are uncountable - so there are (infinitely many!) real numbers that are unreachable using a Turing machine. My suspicion is that AGI, consciousness, perhaps even the supernatural soul you allude to, can be found only in this unreachable state space. There exists a Cardinality Barrier!
Every single neuron has many levels of excitement with different ramp-up, cool down, refractory periods just to name a few non-discrete variables. Not accounting for the rest of chemical soup and meyham happening in the nervous system. Thinking that nervous system is discrete and finite is very, very reductionist and wrong, perhaps as if to compare a novel and an alphabet ("why would you need infinite set space to describe all the possible novels? There are only 30 letters in the alphabet").
Exactly this. To be more precise with your analogy, given a finite alphabet and a finite length for any novel, then in fact the set of all novels is countable (and computable) - but when novels can have (countably) infinite length (or are written using an infinite alphabet?), then the set of novels is uncountable (and indeed by construction correspond to the reals, if my understanding is correct).
If the set of all novels is an infinite subset of all finite symbol strings, it may not be computable. This is because the set of all halting programs is not computable, even though each halting program is a finite symbol string. So, we could have a set of novels that enumerate the halting programs (best sellers, those!), and since this subset of novels is not computable, then the set of all novels is not computable.
Ah - but though a subset (bestsellers/halting programs) may not be computable (recursively enumerable) as you say, this does not preclude all novels/finite programs from being enumerated. Taking the alphabet as just the uppercase letters, I can list all novels as A, B, C.. AA, AB, AC... ... and hence eventually reach any novel you may supply. But this tells me nothing of whether they are bestsellers (or halting programs)!
Yes, of course, enumerating all symbol strings is computable.
What I mean is the novels as a whole correctly label the halting machines, an unreasonable assumption to be sure. That would prevent the novels from being computable.
If real novels somehow exhibit an equivalent characteristic, then that would prevent real novels from being computable. Perhaps the logical consistency of good novels would be such a characteristic. We probably cannot depend on novels being perfectly consistent, but still consistent to a greater degree than we can achieve by randomly generating bitstrings.
The objection would be that a finite set of axioms can generate consistent novels, so there is also some other criterion that is necessary to make the novels uncomputable. I would say this second criterion is novelty, defined as an increase in Kolmogorov complexity.
Using a single set of axioms to enumerate an infinite set of novels will hit a novelty plateau, as per the proof of uncomputability of Kolmogorov complexity.
So, based on these two criteria, one can derive a hand wavy argument that novels in general are uncomputable.
It is? Is this at the quantum level? To be honest my understanding of quantum physics is cursory at best, though I had thought that the notion of a 'clockwork universe' had been otherwise discredited.
Maybe the quantum waveform probabilities are irrational numbers. That's the closest thing I know of that might not be finite and discrete. I'm including fractions in my 'finite and discrete' definition.
My understanding is quantum physics disproves the 'clockwork universe' because it is probabilistic instead of deterministic. Nothing to do with using real numbers.
Also, even if the universe made use of real numbers, the universe can still be computable in an asymptotic sense, which is still inadequate to violate the law of information conservation.
Violation of said law requires essentially non mathematical entities, such as halting oracles and non stochastic processes (neither deterministic, random, nor combination thereof). None of which are dealt with in the STEM fields.
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