Not my point… and you know it. My point is that even if we consider that proven theorems are known facts, we still dont know if hypercomputers are infeasible. We know, however, that i’ll never write python code that decides Validity because it is not (classically) decidable. But we have no theorems on the impossibility of hypercomputation.
Right, validity is semidecidable, just like the halting problem.
We might never know for certain that any natural law is true, we might never be certain that that oracle actually solves validity. But that doesnt prevent the oracle from working. My point is that its existence is possible, not that we will ever be able to trust it.
Besides, we dont know that the physical laws we work with today are true, but we nevetheless use them for practical purpuses all the time.
Turing machines can’t exist, either.
Oh no! You got me there!
Why do you need uncountable infinities for hypercomputers, though?. I see that Martin Davis criticism has to do with that approach, and I agree this approach seems silly. But, it doesnt seem to cover all potential fronts for hypercomputers. Im not talking about current approaches to quantum computing either. What if some yet unknown physical law makes arrangements of particles somehow solve the first order logic validity problem, which is also not in R? Doesnt involve uncountable infinity at all. Again, im not saying its possible, just that theres no purely logical proof of impossibility, thats all.
A hypercomputer has its own class of unsolvable problems, I agree. That doesnt mean that a hypercomputer cannot exist.
church-turing is a a thesis, not a logical theorem. You pointed me to a proof that the halting problem is unsolvable by a Turing Machine, not that hypercomputers are impossible.
The critic Martin Davis mentioned in wikipedia has an article criticizing a kind of attempt at showing the feasibility of hypercomputers. Thats fine. If there was a well-known logical proof of its unfeasibility, his task would be much simpler though. The purely logical argument hasnt been made as far as i know and as far as you were able to show.
The diagonalization argument you pointed me to is about the uncomputability of the halting problem. I know about it, but it just proves that no turing machine can solve the halting problem. Hypercomputers are supposed to NOT be turing machines, so theres no proof of the impossibility of hypercomputers to be found there.
I know diagonalization proofs, they dont prove what you say they prove. Cite any computer science source stating that the existence of hypercomputers are logically impossible. If you keep saying it follows from some diagonalization argument without showing how or citing sources ill move on from this.
…I never said they are not.
The incompleteness theorem says that a consistent axiomatic formal system satisfying some conditions cannot be complete, so the universe as a formal system (supposed consistent, complete, expressive enough, …) cannot be axiomatized.
external oracles
What do you mean external?
The possibility of using physical phenomena as oracles for solving classically uncomputable problems in the real world is an open question. If you think this is logically as impossible as a four sided triangle you should give sources for this claim, not just some vague statements involving the incompleteness theorem. Prove this logical impossibility or give sources, thats all im asking.
Who says you cant take a first order logic sentence, codify it as a particular arrangement of certain particles and determine if the sentence was valid by observing how the particles behave? Some undiscovered physical phenomenon might make this possible… who knows. It would make possible the making of a real world machine that surpasses the turing machine in computability, no? How is this like a four sided triangle? The four sided triangle is logically impossible, but a hypercomputer is logically possible. The question is whether it is also physically possible, which is an open question.
its not a “god cant make a triangle of four sides” discussion. Disregarding the mysterious formal system that “obviously” expresses arithmetic, you always skip my question: then what? how does the laws of the universe being not axiomatizable relate to the brain not using uncomputable functions? This was always the main point of the argument and you keep avoiding giving me an answer.
I took this interpretation to the “existence of uncomputable functions” because of course they exist mathematically, but we were talking about the physical world, so another meaning of existence was probably being used.
You say you studied, but still your arguments linking incompleteness and the physical world did not make sense. To the point that you say things like the universe already is a formal system to which we can apply the incompleteness theorem. Again, expressivity of arithmetic isnt the only condition for using incompleteness. The formal system must be similar to first order logic, as the sentences must be finite, the inference rules must be computable and their set must be recursively enumerable, … among others. When I asked this, you only mentioned being able to express natural numbers. But can the formal system express them in the specific sense that we need here to use incompleteness?
Then, what do you do with the fact that you cant effectively axiomatize the laws of the universe? (which would be the conclusion taken from using incompleteness theorem here, if you could) What’s the point of using incompleteness here? How do you relate this to the computability of brain operations?
These are all giant holes you skipped, which suggest to me that you brushed over these topics somewhere and started to extrapolate unrigorous conclusions from them.
you mention a lot of theory that does exist, but your arguments make no sense. You might want to study the incompleteness theorems more in depth before continuing to cite them like that. The book Godels proof by Nagel and Newman is a good start to go beyond these youtube expositions.
No,
Ok. So nothing you said backs the claim that “logic” implies that the brain cannot be using some uncomputable physical phenomenon, and so be uncomputable.
I’m not sure about what you mean by “cause and effect” existing. Does it mean that the universe follows a set of laws? If cause and effect exists, the disjunction you said is implied by the incompleteness theorem entails that there are uncomputable functions, which I take to mean that there are uncomputable oracles in the physical world. But i still find suspicious your use of incompleteness. We take the set of laws governing the universe and turn it into a formal system. How? Does the resulting formal system really meet all conditions of the incompleteness theorem? Expressivity is just one of many conditions. Even then, the incompleteness theorem says we can’t effectively axiomatize the system… so what?
Adequate in which sense?
I dont mean just architecturally, the turing machine wouldnt be adequate to model the brain in the sense that the brain, in that hypothetical scenario, would be a hypercomputer, and so by definition could not be simulated by a turing machine. As simple as that. My statement there was almost a tautology.
You say an incompleteness theorem implies that brains are computable? Then you consider the possibility of them being hypercomputers? What is this?
Im not saying brains are hypercomputers, just that we dont know if thats the case. If you think that would be “supernatural”, ok, i dont mind. And i dont object to the possibility of eventually having AI on hypercomputers. All I said is that the plain old Turing machine wouldn’t be the adequate model for human cognitive capacity in this scenario.
No, you misread what I said. Of course humans are at least as powerful as a turing machine, im not questioning that. What is unkonwn is if turing machines are as powerful as human cognition. Who says every brain operation is computable (in the classical sense)? Who is to say the brain doesnt take advantage of some weird physical phenomenon that isnt classically computable?
Its a definition, but not an effective one in the sense that we can test and recognize it. Can we list all cognitive tasks a human can do? To avoid testing a probably infinite list, we should instead understand what are the basic cognitive abilities of humans that compose all other cognitive abilities we have, if thats even possible. Like the equivalent of a turing machine, but for human cognition. The Turing machine is based on a finite list of mechanisms and it is considered as the ultimate computer (in the classical sense of computing, but with potentially infinite memory). But we know too little about whether the limits of the turing machine are also limits of human cognition.
im not so sure the devs have fault in any of this though
I see, so you see it not an actual state we may achieve, but rather the negation of present authorities and systems.
So as long as an organization is truly democratic, it can be considered anarchist?
For example, if one person likes to make coca cola but as a side effect he pollutes a river that the rest of the group wants to keep clean. The group may decide democratically to force him to not make coca cola. I would call this a goverment-like organization, even though it does not need to have a leader to fulfill its goal.
excuse my ignorance, but ive always wondered this about anarchism: Seems to me that people gather and organize themselves to reach common goals. How can these organizations not become governments? is that actually possible?
Never said AGI would be unable to.