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I still don't understand what is formally meant by "computational". I assume it means "can be expressed algorithmically (by a series of rules)". If there are na

I still don't understand what is formally meant by "computational". I assume it means "can be expressed algorithmically (by a series of rules)". If there are natural laws, then by definition isnt all of physics computational?

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LegionMammal978 2y ago
Supposing that there are natural laws, it might still be that there is no algorithm that can apply them in their entirety, if they depend on one or more undecidable problems. E.g., some people postulate that hypercomputation is physically possible, which by definition can't be expressed by an algorithm. But while the existence of such things can't be ruled out, I'd say we don't have any real evidence in favor of them. Alternatively, the natural laws might simply contain an infinite amount of information, which could never be stuffed into a finite algorithm. E.g., if the ratio between the electron mass and one of the quark masses is an uncomputable real number. Or if there are infinitely many laws, governing deeper and deeper structure ad infinitum past the subatomic level. The latter has some philosophical arguments against it (why would we apparently be near the top of the infinite structure?), but the former is consistent with the apparent continuum that physical constants reside in.
- scottaino 2y ago
  Do you have any references or further readings about finite vs. infinite number of laws? Sounds interesting.
  - LegionMammal978 2y ago
    Nothing in particular, sorry. The closest would be the idea of a recursively axiomatizable theory in mathematics, which is one that can be fully evaluated, enumerated, etc. using a finite algorithm. Usually, such a theory is presented as a finite list of axioms and axiom schemata: a theory that is not recursively axiomatizable could not be presented in such a simple way.
- andoando 2y ago
  Even if there is an infinite scale, wouldnt it be computational given infinite amount of time/energy. I.e each step itself is still a computational process.
  - LegionMammal978 2y ago
    Suppose that infinitely many independent laws are active simultaneously in some region of spacetime. Then we can perform as many computational steps as we want, but our simulation will never proceed: it will just get bogged down in law after law for all eternity, assuming we have some way to enumerate them in the first place.

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LegionMammal978 2y ago

Supposing that there are natural laws, it might still be that there is no algorithm that can apply them in their entirety, if they depend on one or more undecidable problems. E.g., some people postulate that hypercomputation is physically possible, which by definition can't be expressed by an algorithm. But while the existence of such things can't be ruled out, I'd say we don't have any real evidence in favor of them. Alternatively, the natural laws might simply contain an infinite amount of information, which could never be stuffed into a finite algorithm. E.g., if the ratio between the electron mass and one of the quark masses is an uncomputable real number. Or if there are infinitely many laws, governing deeper and deeper structure ad infinitum past the subatomic level. The latter has some philosophical arguments against it (why would we apparently be near the top of the infinite structure?), but the former is consistent with the apparent continuum that physical constants reside in.

scottaino 2y ago
Do you have any references or further readings about finite vs. infinite number of laws? Sounds interesting.
- LegionMammal978 2y ago
  Nothing in particular, sorry. The closest would be the idea of a recursively axiomatizable theory in mathematics, which is one that can be fully evaluated, enumerated, etc. using a finite algorithm. Usually, such a theory is presented as a finite list of axioms and axiom schemata: a theory that is not recursively axiomatizable could not be presented in such a simple way.
andoando 2y ago
Even if there is an infinite scale, wouldnt it be computational given infinite amount of time/energy. I.e each step itself is still a computational process.
- LegionMammal978 2y ago
  Suppose that infinitely many independent laws are active simultaneously in some region of spacetime. Then we can perform as many computational steps as we want, but our simulation will never proceed: it will just get bogged down in law after law for all eternity, assuming we have some way to enumerate them in the first place.