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> if humans (or any intelligent physical entity) are at most Turing-complete This is a bit of a strange assumption to make. I do agree that a human, if it had

> if humans (or any intelligent physical entity) are at most Turing-complete This is a bit of a strange assumption to make. I do agree that a human, if it had infinite memory, would be an universal machine, i.e. capable of computing any given Turing machine [0]. But would that be the limits of its capabilities? It's far from certain. You'll get into the philosophy of free will (funnily enough, a sort of inverted Turing test), i.e. for a given human with infinite memory, is there a Turing machine that exactly replicates the behavior of that human? Is our behavior governed entirely by rules? Would that imply that a human themselves is a kind of Chinese room [1]? > any non-constructive math "steps" or thoughts must also be at most computable. What does it mean for a "thought" to be computable? Compare to Gödel's incompleteness theorem. Clearly the act of stating the thought, or writing down the theorem, is computable. But proving it to be true or false may very well be impossible. [0] https://en.wikipedia.org/wiki/Universal_Turing_machine https://en.wikipedia.org/wiki/Universal_Turing_machine [1] https://en.wikipedia.org/wiki/Chinese_room https://en.wikipedia.org/wiki/Chinese_room

qarl 10d ago
> But would that be the limits of its capabilities? It's far from certain. Do you agree that humans are physical systems? My understanding is that any physical system can be evaluated to any degree of accuracy by a computer, no?
- streetfighter64 10d ago
  > any physical system can be evaluated to any degree of accuracy by a computer That's an interesting hypothesis, but I don't know why you'd assume it to be true at face value. It's a bit unclear how you would even define "evaluated", given that we don't yet have a mathematical model of all of physics as we know it. [0] And then consider unknown unknowns. > Do you agree that humans are physical systems? Do you consider humans _with infinite memory_ as physical systems? Do you consider computers _with infinite memory_ as physical systems? [0] https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Mo...
  - gf000 10d ago
    I agree that physics being simulated is not that easy to handwave away. That's why I mention that brains probably don't "depend" on some quantum-level behavior and a more macro view of physics could be enough. What I mean here is that while there are obviously quantum effects in play at the atomic/molecular levels, if we take the cells as a black box and replace them with statistical processes, we would probably still get a human intelligence as a result - but of course I can't prove it. As a hunch, the 100 billion neurons and their 100 trillion connections, and of course their environment (glial cells are important)'s proper Simulation is enough. As for the infinite memory, Turing machines have this nice property that they can only visit a finite amount of memory after finite steps, no matter what. A Turing machine running for a finite time (we got this) will surely use a finite space, so being "a bit short" on infinite space is not a problem, I believe.
  - qarl 10d ago
    > given that we don't yet have a mathematical model of all of physics as we know it Yes... but that's in the area of the big bang and black holes. My understanding is that the chemistry of the brain is very well modeled. So, unless we find unknown physics, and unless that physics behaves differently than every other known physics, humans are computable? Do I have that right?
    - streetfighter64 10d ago
      Let me illustrate with an example. Are you familiar with with the Collatz conjecture? It's an example of a system with only one variable, and two simple rules. Are you certain that there exists a computer program that in finite time can compute where any given integer ends up? Now consider throwing a ball in the air. Can you even write down the rules that each of the ball's subatomic particles obeys? How can you be certain there exists a computer program that in finite time can predict where any of the particles, for any ball, ends up? > the chemistry of the brain is very well modeled There are models, but the fact of those models is that they do not apply to "any degree of accuracy", as you claim. Consider the ball thrown in the air again. Is the ball affected by what happened 100 years ago, inside of a black hole 100 light years away? Why would it not be affected by that? Or if you grant that it is affected by that, do we then need a model to predict those effects before we can "evaluate" them? EDIT regarding the below linked blog post: Did you read the rest of my comment? Did you even read the blog post you linked to? > We certainly don’t have anything close to a complete understanding of how the basic laws actually play out in the real world — we don’t understand high-temperature superconductivity, or for that matter human consciousness Can you try to consider my central point before replying: Are the rules governing physical reality simpler or more complex than the Collatz conjecture? Does there exist a (theoretical) computer that can "evaluate the Collatz conjecture to any degree of accuracy"? EDIT 2: I'm not the one moving goalposts. On what grounds are you classifying the question whether a given number ends at 1 or not for the Collaz conjecture as an "inifite" computation? It's a simple boolean question, yes or no. All you have to do is build a computer that can answer yes or no for each integer. Isn't that simpler than answering the position of each atom in the ball after the throw? Each is just a function, what makes one more infinite than the other? Also, regarding determinism, just read this article by the same guy you linked: https://preposterousuniverse.com/blog/2011/12/05/on-determinism/ https://preposterousuniverse.com/blog/2011/12/05/on-determin... > For everyday-life purposes, we can’t get around the fact that quantum mechanics makes it impossible to predict the future robustly.
- rsrsrs86 9d ago
  No
  - qarl 9d ago
    HEH. Well... the only other option is supernatural. Is that what you mean?
gf000 10d ago
> What does it mean for a "thought" to be computable? Well, given our scientific knowledge it's a molecule-level (only important to disregard quantum physics to make the case easier) physical/chemical process, that we should in principle be able to simulate on any other medium, including a Turing machine. Nonetheless, I can accept the definition of math where it's about "truths" and truths can obviously exist without being computable.
throw310822 10d ago
Does that mean that humans could produce mathematical proofs that are entirely logical and verifiable by other humans, but that cannot be formalised in any automatically verifiable language such as lean?

qarl 10d ago

> But would that be the limits of its capabilities? It's far from certain. Do you agree that humans are physical systems? My understanding is that any physical system can be evaluated to any degree of accuracy by a computer, no?

streetfighter64 10d ago
> any physical system can be evaluated to any degree of accuracy by a computer That's an interesting hypothesis, but I don't know why you'd assume it to be true at face value. It's a bit unclear how you would even define "evaluated", given that we don't yet have a mathematical model of all of physics as we know it. [0] And then consider unknown unknowns. > Do you agree that humans are physical systems? Do you consider humans _with infinite memory_ as physical systems? Do you consider computers _with infinite memory_ as physical systems? [0] https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Mo...
- gf000 10d ago
  I agree that physics being simulated is not that easy to handwave away. That's why I mention that brains probably don't "depend" on some quantum-level behavior and a more macro view of physics could be enough. What I mean here is that while there are obviously quantum effects in play at the atomic/molecular levels, if we take the cells as a black box and replace them with statistical processes, we would probably still get a human intelligence as a result - but of course I can't prove it. As a hunch, the 100 billion neurons and their 100 trillion connections, and of course their environment (glial cells are important)'s proper Simulation is enough. As for the infinite memory, Turing machines have this nice property that they can only visit a finite amount of memory after finite steps, no matter what. A Turing machine running for a finite time (we got this) will surely use a finite space, so being "a bit short" on infinite space is not a problem, I believe.
- qarl 10d ago
  > given that we don't yet have a mathematical model of all of physics as we know it Yes... but that's in the area of the big bang and black holes. My understanding is that the chemistry of the brain is very well modeled. So, unless we find unknown physics, and unless that physics behaves differently than every other known physics, humans are computable? Do I have that right?
  - streetfighter64 10d ago
    Let me illustrate with an example. Are you familiar with with the Collatz conjecture? It's an example of a system with only one variable, and two simple rules. Are you certain that there exists a computer program that in finite time can compute where any given integer ends up? Now consider throwing a ball in the air. Can you even write down the rules that each of the ball's subatomic particles obeys? How can you be certain there exists a computer program that in finite time can predict where any of the particles, for any ball, ends up? > the chemistry of the brain is very well modeled There are models, but the fact of those models is that they do not apply to "any degree of accuracy", as you claim. Consider the ball thrown in the air again. Is the ball affected by what happened 100 years ago, inside of a black hole 100 light years away? Why would it not be affected by that? Or if you grant that it is affected by that, do we then need a model to predict those effects before we can "evaluate" them? EDIT regarding the below linked blog post: Did you read the rest of my comment? Did you even read the blog post you linked to? > We certainly don’t have anything close to a complete understanding of how the basic laws actually play out in the real world — we don’t understand high-temperature superconductivity, or for that matter human consciousness Can you try to consider my central point before replying: Are the rules governing physical reality simpler or more complex than the Collatz conjecture? Does there exist a (theoretical) computer that can "evaluate the Collatz conjecture to any degree of accuracy"? EDIT 2: I'm not the one moving goalposts. On what grounds are you classifying the question whether a given number ends at 1 or not for the Collaz conjecture as an "inifite" computation? It's a simple boolean question, yes or no. All you have to do is build a computer that can answer yes or no for each integer. Isn't that simpler than answering the position of each atom in the ball after the throw? Each is just a function, what makes one more infinite than the other? Also, regarding determinism, just read this article by the same guy you linked: https://preposterousuniverse.com/blog/2011/12/05/on-determinism/ https://preposterousuniverse.com/blog/2011/12/05/on-determin... > For everyday-life purposes, we can’t get around the fact that quantum mechanics makes it impossible to predict the future robustly.
rsrsrs86 9d ago
No
- qarl 9d ago
  HEH. Well... the only other option is supernatural. Is that what you mean?

gf000 10d ago

> What does it mean for a "thought" to be computable? Well, given our scientific knowledge it's a molecule-level (only important to disregard quantum physics to make the case easier) physical/chemical process, that we should in principle be able to simulate on any other medium, including a Turing machine. Nonetheless, I can accept the definition of math where it's about "truths" and truths can obviously exist without being computable.

throw310822 10d ago

Does that mean that humans could produce mathematical proofs that are entirely logical and verifiable by other humans, but that cannot be formalised in any automatically verifiable language such as lean?