5 ms·

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 accountin

by arpa 7y ago

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").

somewhereoutth 7y ago
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).
- yters 7y ago
  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.
  - somewhereoutth 7y ago
    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)!
    - yters 7y ago
      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.

by arpa 7y ago

somewhereoutth 7y ago

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).

yters 7y ago
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.
- somewhereoutth 7y ago
  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)!
  - yters 7y ago
    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.