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… our brains are physical systems which themselves can be modeled by mathematics.. How do you know this? What is the model? Can an AI come up with the Incomp

… our brains are physical systems which themselves can be modeled by mathematics.. How do you know this? What is the model? Can an AI come up with the Incompleteness Theorems? It can be proven ZFC that PA is consistent. Can an AI or Turing Machine or whatever do the same? EDIT: I’m equating “our brians” with consciousness.

adastra22 2y ago
Unless you're a Cartesian dualist, the mind maps to the brain which is governed by physical laws that are, in principle, fully modeled by mathematical theory that can be simulated on a computer.
- gedpeck 2y ago
  ..physical laws that are, in principle, fully modeled by mathematical theory that can be simulated on a computer. Consciousness so far appears to be something that can’t be modeled mathematically. Can a Turing Machine conclude that while it can’t prove the consistency of the axiomatic system it works under if one embeds that system in a larger system then it could be possible to prove consistency? Aren’t you assuming super determinism? What if consciousness is not “computable” in any meaningful way? For example suppose the theoretically most efficient mathematical model of consciousness requires more variables than the number of particles in the observable universe.
  - Terr_ 2y ago
    > Consciousness so far appears to be something that can’t be modeled mathematically. That doesn't tell us anything though: Almost every single physical phenomenon we can start to model mathematically looked impossible at some point in history. Just because something's hard is not a reason to expect it's magic. If consciousness is somehow un-model-able, that will most likely be for a different reason, where the premise itself is somehow flawed, like how we can't bottle phlogiston or calculate the rate of air turning into maggots.
    - gedpeck 2y ago
      ...something's hard is not a reason to expect it's magic. Something that is impossible to accurately be modeled mathematically is not magic.
  - adastra22 2y ago
    I think you are very confused about consciousness, which is nothing more than what an algorithm feels like from the inside. There is nothing mysterious about it. But we are now way off topic.
    - gedpeck 2y ago
      You are claiming to be able to mathematically model consciousness or that it is theoretically possible. So what’s the model or proof that it can theoretically be modeled. Your statements regarding were lacking.
      - adastra22 2y ago
        Dennett’s Consciousness Explained would be a good resource if you want more worked out theory and explanation.
        gedpeck 2y ago
        https://en.wikipedia.org/wiki/Chinese_room https://en.wikipedia.org/wiki/Chinese_room
        adastra22 2y ago
        More than adequately covered in Dennett. There's a whole chapter devoted to the Chinese room IIRC.
        gedpeck 2y ago
        It’s not a settled issue in the sense that the experts overwhelmingly agree on one side vs. the other. My link was posted to show a contrarian view. Lots of research papers arguing on either side have been written about the Chinese Room. Your argument is similar to a Christian telling an atheist it’s a settled issue since the book “The Case for a Creator” covers the issue.
lisper 2y ago
> How do you know this? I don't. But it seems like a plausible hypothesis, and I see no compelling evidence to the contrary. > Can an AI or Turing Machine or whatever do the same? Can you?
- gedpeck 2y ago
  Yes. It’s a basic result proven in any mathematical logic course at the senior or beginning graduate level. If a Turing Machine can’t prove that it can’t prove the consistency of its axiomatic system from within that system but that it could from within a larger system but I can then this is evidence against your belief. At least as I see it. I have the minority view that the Incompleteness results (the proof of them) are a limitation of artificial intelligence.
  - lisper 2y ago
    > Yes. It’s a basic result proven in any mathematical logic course at the senior or beginning graduate level. OK, but note that you've moved the goal posts here. Your original question was: > Can an AI come up with the Incompleteness Theorems? There is a difference between coming up with those theorems, and being able to reproduce them after having been shown how. There can be no doubt that an AI can do the latter, it's not even speculative any more. ChatGPT can surely recite the proof of the consistency of PA within ZFC. > I have the minority view that the Incompleteness results (the proof of them) are a limitation of artificial intelligence. Yeah, well, there's a reason this is the minority view. How do you know that the incompleteness results don't apply to you? Sure you can see that PA can be proven consistent in ZFC, but that is not the same thing as being able to see the consistency of the formal system that governs the behavior of your brain. You don't even know what that formal system is. It's not even clear that it's possible for you to know that. It's possible that your brain contains all kinds of ad-hoc axioms wired in by millions of years of evolution, and it's possible that these are stored in such a way that they cannot be easily compressed. Evolution tends to drive towards efficient use of resources. So even if you had the technology to produce a completely accurate model of your brain, your brain might not have the capacity to comprehend it. History is full of people making predictions about how humans will ultimately prove to be superior to computers. Not a single one of those predictions has stood the test of time. Chess. Go. Jeopardy. Writing term papers. Generating proofs. Computers do all these things now, and they've come to do them in the span of a single human lifetime. I see absolutely no reason to believe that this trend will not continue.

adastra22 2y ago

Unless you're a Cartesian dualist, the mind maps to the brain which is governed by physical laws that are, in principle, fully modeled by mathematical theory that can be simulated on a computer.

gedpeck 2y ago
..physical laws that are, in principle, fully modeled by mathematical theory that can be simulated on a computer. Consciousness so far appears to be something that can’t be modeled mathematically. Can a Turing Machine conclude that while it can’t prove the consistency of the axiomatic system it works under if one embeds that system in a larger system then it could be possible to prove consistency? Aren’t you assuming super determinism? What if consciousness is not “computable” in any meaningful way? For example suppose the theoretically most efficient mathematical model of consciousness requires more variables than the number of particles in the observable universe.
- Terr_ 2y ago
  > Consciousness so far appears to be something that can’t be modeled mathematically. That doesn't tell us anything though: Almost every single physical phenomenon we can start to model mathematically looked impossible at some point in history. Just because something's hard is not a reason to expect it's magic. If consciousness is somehow un-model-able, that will most likely be for a different reason, where the premise itself is somehow flawed, like how we can't bottle phlogiston or calculate the rate of air turning into maggots.
  - gedpeck 2y ago
    ...something's hard is not a reason to expect it's magic. Something that is impossible to accurately be modeled mathematically is not magic.
- adastra22 2y ago
  I think you are very confused about consciousness, which is nothing more than what an algorithm feels like from the inside. There is nothing mysterious about it. But we are now way off topic.
  - gedpeck 2y ago
    You are claiming to be able to mathematically model consciousness or that it is theoretically possible. So what’s the model or proof that it can theoretically be modeled. Your statements regarding were lacking.
    - adastra22 2y ago
      Dennett’s Consciousness Explained would be a good resource if you want more worked out theory and explanation.
      - gedpeck 2y ago
        https://en.wikipedia.org/wiki/Chinese_room https://en.wikipedia.org/wiki/Chinese_room
        adastra22 2y ago
        More than adequately covered in Dennett. There's a whole chapter devoted to the Chinese room IIRC.
        gedpeck 2y ago
        It’s not a settled issue in the sense that the experts overwhelmingly agree on one side vs. the other. My link was posted to show a contrarian view. Lots of research papers arguing on either side have been written about the Chinese Room. Your argument is similar to a Christian telling an atheist it’s a settled issue since the book “The Case for a Creator” covers the issue.

lisper 2y ago

> How do you know this? I don't. But it seems like a plausible hypothesis, and I see no compelling evidence to the contrary. > Can an AI or Turing Machine or whatever do the same? Can you?

gedpeck 2y ago
Yes. It’s a basic result proven in any mathematical logic course at the senior or beginning graduate level. If a Turing Machine can’t prove that it can’t prove the consistency of its axiomatic system from within that system but that it could from within a larger system but I can then this is evidence against your belief. At least as I see it. I have the minority view that the Incompleteness results (the proof of them) are a limitation of artificial intelligence.
- lisper 2y ago
  > Yes. It’s a basic result proven in any mathematical logic course at the senior or beginning graduate level. OK, but note that you've moved the goal posts here. Your original question was: > Can an AI come up with the Incompleteness Theorems? There is a difference between coming up with those theorems, and being able to reproduce them after having been shown how. There can be no doubt that an AI can do the latter, it's not even speculative any more. ChatGPT can surely recite the proof of the consistency of PA within ZFC. > I have the minority view that the Incompleteness results (the proof of them) are a limitation of artificial intelligence. Yeah, well, there's a reason this is the minority view. How do you know that the incompleteness results don't apply to you? Sure you can see that PA can be proven consistent in ZFC, but that is not the same thing as being able to see the consistency of the formal system that governs the behavior of your brain. You don't even know what that formal system is. It's not even clear that it's possible for you to know that. It's possible that your brain contains all kinds of ad-hoc axioms wired in by millions of years of evolution, and it's possible that these are stored in such a way that they cannot be easily compressed. Evolution tends to drive towards efficient use of resources. So even if you had the technology to produce a completely accurate model of your brain, your brain might not have the capacity to comprehend it. History is full of people making predictions about how humans will ultimately prove to be superior to computers. Not a single one of those predictions has stood the test of time. Chess. Go. Jeopardy. Writing term papers. Generating proofs. Computers do all these things now, and they've come to do them in the span of a single human lifetime. I see absolutely no reason to believe that this trend will not continue.