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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.