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.
how does the laws of the universe being not axiomatizable
…I never said they are not.
relate to the brain not using uncomputable functions?
That is unspecific: Do you mean it is using external oracles? It cannot use use them because they cannot exist because they’re four-sided triangles. If you mean that it is considering uncomputable functions, then it can do so symbolically, but it cannot evaluate them, not in finite time that is: The brain can consider the notion of four-sided triangles, but it cannot calculate the lengths of those sides given, say, an area and an angle or such.
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.
…I never said they are not.
That is unspecific: Do you mean it is using external oracles? It cannot use use them because they cannot exist because they’re four-sided triangles. If you mean that it is considering uncomputable functions, then it can do so symbolically, but it cannot evaluate them, not in finite time that is: The brain can consider the notion of four-sided triangles, but it cannot calculate the lengths of those sides given, say, an area and an angle or such.