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.
You say an incompleteness theorem implies that brains are computable?
No, I’m saying that incompleteness implies that either cause and effect does not exist, or there exist incomputable functions. That follows from considering the universe, or its collection of laws, as a logical system, which are all bound by the incompleteness theorem once they reach a certain expressivity.
All I said is that the plain old Turing machine wouldn’t be the adequate model for human cognitive capacity in this scenario.
Adequate in which sense? Architecturally, of course not, and neither would be lambda calculus or other common models. I’m not talking about specific abstract machines, though, but Turing-completeness, that is, the property of the set of all abstract machines that are as computationally powerful as Turing machines, and can all simulate each other. Those are a dime a gazillion.
Or, see it this way: Imagine a perfect, virtual representation of a human brain stored on an ordinary computer. That computer is powerful enough to simulate all physical laws relevant to the functioning of a human brain… it might take a million years to simulate a second of brain time, but so be it. Such a system would be AGI (for ethically dubious values of “artificial”). That is why I say the “whether” is not the question: We know it is possible. We’ve in fact done it for simpler organisms. The question is how to do it with reasonable efficiency, and that requires an understanding of how the brain does the computations it does so we can mold it directly into silicon instead of going via several steps of one machine simulating another machine, each time incurring simulation overhead from architectural mismatch.
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.
entails that there are uncomputable functions, which I take to mean that there are uncomputable oracles in the physical world.
It means that there are functions that are not computable. You cannot, for example, write a program that decides, in finite time, whether an arbitrary program halts on a particular input. If you doubt that, have an easy-going explanation of the proof by diagonalisation.
We take the set of laws governing the universe and turn it into a formal system. How?
Ask a physicist, that’s their department not mine. Also I’d argue that the universe itself is a formal system, and lots of physicists would agree they’re onto the whole computability and complexity theory train. They may or may not agree to the claim that computer science is more fundamental than physics, we’re still working on that one.
Does the resulting formal system really meet all conditions of the incompleteness theorem?
Easily, because it will have to express the natural numbers. Have a Veritasium video on the whole thing. The two results (completeness and in computability) are fundamentally linked.
he turing machine wouldnt be adequate to model the brain in the sense that the brain, in that hypothetical scenario, would be a hypercomputer,
If the brain is a hypercomputer then, as already said, you’re not talking physics any more, you’re in the realms of ex falso quodlibet.
Hypercomputers are just as impossible as a village barber who shaves everyone in the village who does not shave themselves: If the barber shaves himself, then he doesn’t shave himself. If he shaves himself, then he doesn’t shave himself. Try to imagine a universe in which that’s not a paradox, that’s the kind of universe you’re claiming we’re living in when you’re claiming that hypercomputers exist.
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.
Dude any uni you go to likely has lectures that are worse than the Arsdigita ones. If you want to save face, act less like a philosopher next time and don’t assume that you know things better than someone who actually studied it. I ended up linking veritasium because I realised that you have no idea about the mathematical fundamentals or you wouldn’t say silly things such as
entails that there are uncomputable functions, which I take to mean that there are uncomputable oracles in the physical world.
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.
To the point that you say things like the universe already is a formal system to which we can apply the incompleteness theorem.
And that is contentious, why? If the laws of the universe are formalisable, then the universe is isomorphic to that formalisation and as such also a formal system. We’re not talking being and immanence, here, we’re talking transcendent properties.
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?
How do you express them in ways that do not trigger incompleteness? Hint: You can’t. It’s a sufficient condition, there’s equivalent ones, if I’m not mistaken an infinite number of them, but that doesn’t matter because they’re all equivalent.
These are all giant holes you skipped
These are all things you would understand if I didn’t have to remind you of basic computability and complexity theory literally every time you reply. As said: Stop the philosophising. The maths are way more watertight than you think. We’re in “God can’t make a triangle with four sides” territory, here, just that computability is a wee bit less intuitive than triangles.
If you want to attack my line of reasoning you could go for solipsism, you could come up with something theological (“god chooses to hide certain aspects of the universe from machines” or whatever). I’m aware of the limits. I didn’t come up with this stuff yesterday and my position isn’t out of the ordinary, either.
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.
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, I’m saying that incompleteness implies that either cause and effect does not exist, or there exist incomputable functions. That follows from considering the universe, or its collection of laws, as a logical system, which are all bound by the incompleteness theorem once they reach a certain expressivity.
Adequate in which sense? Architecturally, of course not, and neither would be lambda calculus or other common models. I’m not talking about specific abstract machines, though, but Turing-completeness, that is, the property of the set of all abstract machines that are as computationally powerful as Turing machines, and can all simulate each other. Those are a dime a gazillion.
Or, see it this way: Imagine a perfect, virtual representation of a human brain stored on an ordinary computer. That computer is powerful enough to simulate all physical laws relevant to the functioning of a human brain… it might take a million years to simulate a second of brain time, but so be it. Such a system would be AGI (for ethically dubious values of “artificial”). That is why I say the “whether” is not the question: We know it is possible. We’ve in fact done it for simpler organisms. The question is how to do it with reasonable efficiency, and that requires an understanding of how the brain does the computations it does so we can mold it directly into silicon instead of going via several steps of one machine simulating another machine, each time incurring simulation overhead from architectural mismatch.
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?
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.
It means that there are functions that are not computable. You cannot, for example, write a program that decides, in finite time, whether an arbitrary program halts on a particular input. If you doubt that, have an easy-going explanation of the proof by diagonalisation.
Ask a physicist, that’s their department not mine. Also I’d argue that the universe itself is a formal system, and lots of physicists would agree they’re onto the whole computability and complexity theory train. They may or may not agree to the claim that computer science is more fundamental than physics, we’re still working on that one.
Easily, because it will have to express the natural numbers. Have a Veritasium video on the whole thing. The two results (completeness and in computability) are fundamentally linked.
If the brain is a hypercomputer then, as already said, you’re not talking physics any more, you’re in the realms of ex falso quodlibet.
Hypercomputers are just as impossible as a village barber who shaves everyone in the village who does not shave themselves: If the barber shaves himself, then he doesn’t shave himself. If he shaves himself, then he doesn’t shave himself. Try to imagine a universe in which that’s not a paradox, that’s the kind of universe you’re claiming we’re living in when you’re claiming that hypercomputers exist.
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.
Dude any uni you go to likely has lectures that are worse than the Arsdigita ones. If you want to save face, act less like a philosopher next time and don’t assume that you know things better than someone who actually studied it. I ended up linking veritasium because I realised that you have no idea about the mathematical fundamentals or you wouldn’t say silly things such as
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.
And that is contentious, why? If the laws of the universe are formalisable, then the universe is isomorphic to that formalisation and as such also a formal system. We’re not talking being and immanence, here, we’re talking transcendent properties.
How do you express them in ways that do not trigger incompleteness? Hint: You can’t. It’s a sufficient condition, there’s equivalent ones, if I’m not mistaken an infinite number of them, but that doesn’t matter because they’re all equivalent.
These are all things you would understand if I didn’t have to remind you of basic computability and complexity theory literally every time you reply. As said: Stop the philosophising. The maths are way more watertight than you think. We’re in “God can’t make a triangle with four sides” territory, here, just that computability is a wee bit less intuitive than triangles.
If you want to attack my line of reasoning you could go for solipsism, you could come up with something theological (“god chooses to hide certain aspects of the universe from machines” or whatever). I’m aware of the limits. I didn’t come up with this stuff yesterday and my position isn’t out of the ordinary, either.
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.