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
> 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.
> 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.
> 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?
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.
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?
PLEASE just stop complaining about "moving the goalposts" when the issue is your own lack of clarity of both expression and reasoning. WHAT is the distinction between "simulate" and "predict"? You said neither by the way, you said "evaluate".
> ANY physical system can be evaluated to ANY DEGREE of accuracy by a computer
It's completely SENSELESS to claim that they are distinct, because in order to EVALUATE or SIMULATE the physical system you will need a FUNCTION which COMPUTES the STATE of the system at a given point in time. The only POSSIBLE distinction between SIMULATING and PREDICTING would be the time taken for the computation, but that is COMPLETELY IRRELEVANT as long as it is finite.
Again, your own source says:
> We CERTAINLY don’t have ANYTHING CLOSE to a complete UNDERSTANDING of how the basic laws actually play out in the real world
> You said neither by the way, you said "evaluate".
You are entirely correct. I was sloppy in my first comment. I should have said simulate. My sincere apologies if that's been the crux of our dispute.
> The only POSSIBLE distinction between SIMULATING and PREDICTING would be the time taken for the computation
No. The distinction is in determining which "you" is you. When simulating the wavefunction, every you is simulated.
> We CERTAINLY don’t have ANYTHING CLOSE to a complete UNDERSTANDING of how the basic laws actually play out in the real world
It's very understandable if you include his following sentence:
> But these are manifestations of the underlying laws, not signs that our understanding of the laws are incomplete
He's saying we don't understand emergent behavior produced by the laws - not that the laws themselves are incomplete. E.g. we don't know how/why a bag of neurons turns into a person.
Re: Collatz - you've moved the goalposts. Answering Collatz requires solving a halting problem. I didn't claim that I could find the end of an infinite computation. I claimed to be able to simulate a finite one.
And - you should reply to my comments rather than edit your old ones.
Ur computer is a physical system and some aspects of it uncomputable. You cant approximate or predict everything. Picture it as billiard balls modeling a circuit, you can’t tell whether it halts, meaning you don’t know where the balls will be.
Or a program that knocks out a bottle when it finds a counter example for whatever . U can’t tell whether it will spill the liquid.
And you have as well the problems of approximating the continuous with bits, you’ll always lose information
A human can't tell if it halts either, friend. A simulation of that human will follow the same infinite step calculation and they will behave identically. Physics can be simulated in finite time to any degree of accuracy.
I hear how passionately you want to believe these things. Unfortunately they're not true. I suggest you look into it.
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?
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