You are making the same very fundamental mistake here as in your other more recent post.
Processing something program-like computationally is not at all the same thing as blindly executing it.
The busy beaver function is not computable. Therefore I cannot reason about it by computing all its values, if I am implemented on a computable substrate. But that doesn’t mean I cannot reason about it at all. (And when I think about the busy beaver function, I am not in fact computing all its values, or indeed any of them.)
Optimizing compilers—which are well understood technology and run on real computers all the time—reason about code and can do things like working out that some bit of code is an infinite loop. (There are plenty of things they can’t do, but there are surely plenty of things we can’t do too.)
Processing an English sentence computationally is not at all the same thing as … whatever too-specific thing you are imagining it is; I can’t actually tell quite what that is.
The existence of paradoxes doesn’t prove that we are not computable. How could it possibly do that? Does the fact that I can write “God cannot consistently believe this statement” disprove the existence of God? (I guess it does disprove some awfully over-specific notions of God; e.g., if you say “God knows the truth or falsity of every statement”, that presupposes that every statement has a definite truth-value, which isn’t actually true. But what we’ve refuted there is something about sentences, not something about gods, right?)
In any case, your attempted paradox—“This true (in bivalent logic) statement cannot be processed by the human brain”—doesn’t even work. What I mean is that if I say it’s false but can be processed, there’s nothing paradoxical there at all.
I think what you wanted is something more like “This statement either is false or else cannot be processed computationally”. Then a naïve analysis looks at the two possible truth values, sees that you get a contradiction from “false”, and concludes that the statement is true and hence cannot be processed computationally. But you can equally write “Either this statement is false or Santa Claus exists” and “prove” that Santa Claus exists; this kind of naïve reasoning is no good. And I know of no reason to think that no computational process can recognize this. I think you are assuming that the only possible computational process consists of doing the sort of “assume the statement false, deduce that it’s true, deduce that it’s false, deduce that it’s true …” loop you describe, and therefore that anything more sophisticated has to be “uncomputable”; that’s just plain wrong. Processing something computationally does not mean blindly doing inferences without any sort of higher-level oversight.
In each case, what you appear to be doing is to pick a particular kind of computation that you can imagine being applied (execute the program! evaluate the function! follow first-order logical inferences!), seeing that in a particular case this will fail, and saying “therefore anything that can handle the situation is non-computational”. No! Therefore anything that can handle the situation isn’t doing the particular kind of computation you thought of, that’s all.
>In any case, your attempted paradox—“This true (in bivalent logic) statement cannot be processed by the human brain”—doesn’t even work. What I mean is that if I say it’s false but can be processed, there’s nothing paradoxical there at all.
I somewhat agree. As you said you can just process it and probabilistically or empirically find out if you can process it is or not.
The statement was not precise enough:
“This true (in bivalent logic) statement cannot be deterministically processed by the human brain, but can still seen and determined to be unambiguously true by something beyond it (meaning needing no empirical confirmation).”
I was meaning to imply that, but I did not SAY that. So you are very right that it is only true in so far as that is implicitly understood, or explicitly stated.
If it is consider it could be false, you have two options, the TRUE and FALSE option with no logical option of deciding between them (regardless of your assumptions, both are logically possible a priori). But then you cannot logically decide it is false, because you would then arbitrarily reject the possibility it is true, but arbitrary rejection is not logical, it is probabilistic or assumptive, or based on free will (but again free will is free, it is not based on bivalent logic, so the truth value of the statement is independent of that).
So you get into an undecidable contradiction here logically that can only be resolved if the statement is true.
You are making the same very fundamental mistake here as in your other more recent post.
Processing something program-like computationally is not at all the same thing as blindly executing it.
The busy beaver function is not computable. Therefore I cannot reason about it by computing all its values, if I am implemented on a computable substrate. But that doesn’t mean I cannot reason about it at all. (And when I think about the busy beaver function, I am not in fact computing all its values, or indeed any of them.)
Optimizing compilers—which are well understood technology and run on real computers all the time—reason about code and can do things like working out that some bit of code is an infinite loop. (There are plenty of things they can’t do, but there are surely plenty of things we can’t do too.)
Processing an English sentence computationally is not at all the same thing as … whatever too-specific thing you are imagining it is; I can’t actually tell quite what that is.
The existence of paradoxes doesn’t prove that we are not computable. How could it possibly do that? Does the fact that I can write “God cannot consistently believe this statement” disprove the existence of God? (I guess it does disprove some awfully over-specific notions of God; e.g., if you say “God knows the truth or falsity of every statement”, that presupposes that every statement has a definite truth-value, which isn’t actually true. But what we’ve refuted there is something about sentences, not something about gods, right?)
In any case, your attempted paradox—“This true (in bivalent logic) statement cannot be processed by the human brain”—doesn’t even work. What I mean is that if I say it’s false but can be processed, there’s nothing paradoxical there at all.
I think what you wanted is something more like “This statement either is false or else cannot be processed computationally”. Then a naïve analysis looks at the two possible truth values, sees that you get a contradiction from “false”, and concludes that the statement is true and hence cannot be processed computationally. But you can equally write “Either this statement is false or Santa Claus exists” and “prove” that Santa Claus exists; this kind of naïve reasoning is no good. And I know of no reason to think that no computational process can recognize this. I think you are assuming that the only possible computational process consists of doing the sort of “assume the statement false, deduce that it’s true, deduce that it’s false, deduce that it’s true …” loop you describe, and therefore that anything more sophisticated has to be “uncomputable”; that’s just plain wrong. Processing something computationally does not mean blindly doing inferences without any sort of higher-level oversight.
In each case, what you appear to be doing is to pick a particular kind of computation that you can imagine being applied (execute the program! evaluate the function! follow first-order logical inferences!), seeing that in a particular case this will fail, and saying “therefore anything that can handle the situation is non-computational”. No! Therefore anything that can handle the situation isn’t doing the particular kind of computation you thought of, that’s all.
>In any case, your attempted paradox—“This true (in bivalent logic) statement cannot be processed by the human brain”—doesn’t even work. What I mean is that if I say it’s false but can be processed, there’s nothing paradoxical there at all.
I somewhat agree. As you said you can just process it and probabilistically or empirically find out if you can process it is or not.
The statement was not precise enough:
“This true (in bivalent logic) statement cannot be deterministically processed by the human brain, but can still seen and determined to be unambiguously true by something beyond it (meaning needing no empirical confirmation).”
I was meaning to imply that, but I did not SAY that. So you are very right that it is only true in so far as that is implicitly understood, or explicitly stated.
If it is consider it could be false, you have two options, the TRUE and FALSE option with no logical option of deciding between them (regardless of your assumptions, both are logically possible a priori). But then you cannot logically decide it is false, because you would then arbitrarily reject the possibility it is true, but arbitrary rejection is not logical, it is probabilistic or assumptive, or based on free will (but again free will is free, it is not based on bivalent logic, so the truth value of the statement is independent of that).
So you get into an undecidable contradiction here logically that can only be resolved if the statement is true.