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To be clear, I'm speaking of Penrose specifically. Penrose goes no further than Godel. He doesn't add anything to Godel's argument about formal systems and stat

To be clear, I'm speaking of Penrose specifically. Penrose goes no further than Godel. He doesn't add anything to Godel's argument about formal systems and statements. All Penrose is saying is that Turing machines can not be the basis for modeling the human mind. He is not saying that the human mind is not governed by physics. He is not saying that the universe is not governed by physics. And he is not resorting to 'quantum woo' like some here are saying. Penrose' argument is that there is something non-algorithmic going on in the human brain and that this non-algorithmic process is governed by physical laws. He then goes on to discuss objective collapse theories for quantum mechanics and how that may be the source of non-algorithmic behavior in the human brain. To be clear, objective collapse theories have as of yet no experimental evidence. But, crucially, they are not inconsistent with QM as some above have said. Also, the precise biological mechanism that Penrose postulates for how the brain operates with objective collapse is highly controversial ie, microtubules.

mcguire 9y ago
Models of computation have some interesting properties. One is that all of the "reasonable" ones are less powerful than Turing machines (FSMs, stack machines, etc.) or equivalent to Turing machines (recursive functions, combinators, TMs with multiple tapes, etc.). It's effectively a closed category. There are models that are more powerful than TMs, but the ones I know about have "unreasonable" powers. The one that comes to mind came from A.K. Dewdney (I think); it does an unbounded amount of work in a single time step by having an unbounded number of computational units. And that is just not kosher. So there is a certain amount of circumstantial evidence for the definition of "computable" being something like a Turing machine. And this definition of computable has limits, as in Godel (never thought I'd miss my phone's keyboard) and Turing. As you describe, Lucas, Penrose, and others such as Searle want to walk a very fine line. They want to claim that the human mind is not limited in the way that computable functions are limited, which I understand; it certainly doesn't feel like I am so limited. On the other hand, they want to avoid going off the edge into full-blown mysticism: dualism and ye immortal soul. Unfortunately for those like me (and I know I am) who take materialism somewhat seriously, their arguments feel more like the post-Darwin arguments that, yes, animals and stuff do evolve like that, but people are different. The closed set of computational models of TM-equivalence do seem to capture "computable functions", and those computable functions do seem to cover what is possible from any system built on biology, chemistry, and ultimately physics. Their arguments seem to be either making a distinction that doesn't make a difference, or sneaking in supernatural powers. (You can see the truth or falsehood of a statement without a proof? Reliably?) So arguments for a non-algorithmic middle ground between TM-equivalence and "quantum woo" (great phrase, btw) are giving me problems. If I am a mixed bag of incomplete and inconsistent, with limited rationality that is not well described as a formal system but is weaker than a Turing machine, as I suspect I am, then the appearance of a non-algorithmic process is simply a misunderstanding. On the other hand, if the non-algorithmic goings-on are correct, "quantum woo" feels like the most reasonable description of their powers. I'd really like some strong, unarguable evidence before I accept that step.
- manyoso 9y ago
  Let's take the question of the human mind and whether it can be modeled by a non-quantum Turing machine out of the equation. Penrose's contention can be summed up without it. He basically thinks that gravity will be discovered to play a role in quantum wave function collapse. And that this role will render non-quantum Turing machines incapable of accurately modeling quantum gravity in some very serious way. Another way of saying it is that he believes when a real theory of quantum gravity is developed we'll find that non-quantum Turing machines will not be able to accurately model the physics of a black hole. Note that this says nothing whatsoever about the human mind or microtubes or formal systems, etc. This is about his subject of expertise: gravity, mathematical physics, and quantum gravity. He's one of the world's leading experts in these fields. And if he is right, he conjectures it will have strong implications for non-quantum Turing machines as a basis for Strong AI.
  - mcguire 9y ago
    That's a good point. I'm tempted to point out that quantum computers can currently be emulated by classical computers, but that may be beside the point.
    - manyoso 9y ago
      Yes, quantum supremecy has not been demonstrated, but the question of whether non-quantum Turing machines can accurately model all physical laws has not. Of course, we don't even have a quantum theory of gravity for them to model at this point...
    - deleted 9y ago
      [deleted]

mcguire 9y ago

Models of computation have some interesting properties. One is that all of the "reasonable" ones are less powerful than Turing machines (FSMs, stack machines, etc.) or equivalent to Turing machines (recursive functions, combinators, TMs with multiple tapes, etc.). It's effectively a closed category. There are models that are more powerful than TMs, but the ones I know about have "unreasonable" powers. The one that comes to mind came from A.K. Dewdney (I think); it does an unbounded amount of work in a single time step by having an unbounded number of computational units. And that is just not kosher. So there is a certain amount of circumstantial evidence for the definition of "computable" being something like a Turing machine. And this definition of computable has limits, as in Godel (never thought I'd miss my phone's keyboard) and Turing. As you describe, Lucas, Penrose, and others such as Searle want to walk a very fine line. They want to claim that the human mind is not limited in the way that computable functions are limited, which I understand; it certainly doesn't feel like I am so limited. On the other hand, they want to avoid going off the edge into full-blown mysticism: dualism and ye immortal soul. Unfortunately for those like me (and I know I am) who take materialism somewhat seriously, their arguments feel more like the post-Darwin arguments that, yes, animals and stuff do evolve like that, but people are different. The closed set of computational models of TM-equivalence do seem to capture "computable functions", and those computable functions do seem to cover what is possible from any system built on biology, chemistry, and ultimately physics. Their arguments seem to be either making a distinction that doesn't make a difference, or sneaking in supernatural powers. (You can see the truth or falsehood of a statement without a proof? Reliably?) So arguments for a non-algorithmic middle ground between TM-equivalence and "quantum woo" (great phrase, btw) are giving me problems. If I am a mixed bag of incomplete and inconsistent, with limited rationality that is not well described as a formal system but is weaker than a Turing machine, as I suspect I am, then the appearance of a non-algorithmic process is simply a misunderstanding. On the other hand, if the non-algorithmic goings-on are correct, "quantum woo" feels like the most reasonable description of their powers. I'd really like some strong, unarguable evidence before I accept that step.

manyoso 9y ago
Let's take the question of the human mind and whether it can be modeled by a non-quantum Turing machine out of the equation. Penrose's contention can be summed up without it. He basically thinks that gravity will be discovered to play a role in quantum wave function collapse. And that this role will render non-quantum Turing machines incapable of accurately modeling quantum gravity in some very serious way. Another way of saying it is that he believes when a real theory of quantum gravity is developed we'll find that non-quantum Turing machines will not be able to accurately model the physics of a black hole. Note that this says nothing whatsoever about the human mind or microtubes or formal systems, etc. This is about his subject of expertise: gravity, mathematical physics, and quantum gravity. He's one of the world's leading experts in these fields. And if he is right, he conjectures it will have strong implications for non-quantum Turing machines as a basis for Strong AI.
- mcguire 9y ago
  That's a good point. I'm tempted to point out that quantum computers can currently be emulated by classical computers, but that may be beside the point.
  - manyoso 9y ago
    Yes, quantum supremecy has not been demonstrated, but the question of whether non-quantum Turing machines can accurately model all physical laws has not. Of course, we don't even have a quantum theory of gravity for them to model at this point...
  - deleted 9y ago
    [deleted]