Comment by streetfighter64
8 days 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 [1] https://en.wikipedia.org/wiki/Chinese_room
> 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.
> 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?
> 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_Mo...
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.
> 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?
7 replies →
No
HEH. Well... the only other option is supernatural. Is that what you mean?
2 replies →
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?