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Well, I only know how to define computability in terms of Turing machines. For math I don't have a fix definition, but it's surely a bit more specific than tha

by gf000 8d ago

Well, I only know how to define computability in terms of Turing machines. For math I don't have a fix definition, but it's surely a bit more specific than that (e.g. I wouldn't consider the computation that prints a 0 at the same place for infinity math) - but of course I do see the circularity in my argument: a Turing machine is a mathematical object in and of itself. Though being able to talk about something doesn't necessarily change which is "bigger". As for the other direction, this gets a bit more into the philosophy behind math itself. Constructive math's territory is "easy" - but I am on the opinion that if humans (or any intelligent physical entity) are at most Turing-complete [1], then any non-constructive math "steps" or thoughts must also be at most computable. Well, unfortunately I can't prove whether math done by transcendent entities are also computable, though. In any case, I am no mathematician, so whatever I think regarding this topic may not have much relevance to anyone, only done CS course with quite a bit of math, but that's obviously not the same. [1] I believe religion is an escape hatch here from an argument perspective

streetfighter64 8d ago
> 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 8d 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 8d 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 8d 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 8d 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?

by gf000 8d ago

streetfighter64 8d ago

> 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 8d 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 8d 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 8d 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 8d 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?