Turing machines can emulate Boolean logic, and every process described by classical physics can be described in that logic (by way of propositional calculus etc
https://en.wikipedia.org/wiki/Propositional_calculus
Of course I know what propositional logic is, but that link is hardly sufficient to support your point. You claim that prop logic can be describe every process of classical physics.
Now this happens to NOT be true, and I'll supply the refutation in a moment.
But of course even if it was true it wouldn't matter, because classical physics is only an approximation to quantum physics, which (by current theory) is the physics of our world. So whether or not prop logic can implement classical physics is irrelevant, since the brain lives in the physical world, which is not bound by classical physics.
So I see what you mean by introducing quantum physics. The point is that whatever the brain is, it's physical, so it's bound by the laws of physics. But it's certainly not bound by the laws of classical (by which I assume you mean Newtonian) physics.
Now as it turns out, Newtonian physics can not be implemented on a computer. Say we have three bodies in space and we wish to emulate their mutual gravitational interaction over time. In other words we have the differential equations for the three body problem, and we wish to program that into a computer.
Now as you know,
the solutions to these equations involve real numbers. But computers can't store or represent real numbers, only finite approximations.
For practical calculations, our approximations are good enough. But the approximations introduce tiny errors; and over a long enough period of time, chaos theory says that the accumulated errors will end up throwing your model wildly off the mark. There's book by Ivars Peterson called Newton's Clock. It's all about chaos in the solar system. It turns out that even under the assumptions of perfectly deterministic Newtonian gravity, if we knew the exact position and velocity of every particle in the solar system, we can NOT determine whether the solar system is stable or not by using a computer!
This point is not sufficiently well known.
You can't even use a computer to perfectly model deterministic Newtonian gravity. That's how weak algorithmic computations are. Finite approximations to real numbers are not sufficient over long enough time scales.
Of course the fact that some high school kid programmed a model of the solar system into his computer does not falsify my point. Our approximations are excellent within their limits. But the approximations are not perfect and over a sufficiently long period of time, they're wildly inaccurate.
In fact the ultimate stability of the solar system under Newtonian gravity is an
open problem. We can't solve the differential equations and our computer models fail due to chaos.
https://www.amazon.com/Newtons-Clock-Chaos-Solar-System-ebook/dp/B007KLWZ00
https://en.wikipedia.org/wiki/Stability_of_the_Solar_System
The reference was to stuff like this:
http://mathworld.wolfram.com/SurrealNumber.html and complex numbers etc. We are dealing with electronic current in three dimensions - complex numbers, even quaternions, are involved.
Yes ok. When you said gaps even in the reals I thought you might mean the hyperreals or the surreals. As it happens, any model of the real numbers that contains infinitesimals (as both those systems do) can not be topologically complete. You are absolutely correct about that. But the standard reals are complete. That's why the standard reals are a better model of the continuum than the hyperreals and surreals.
By the way the complex numbers and quaternions are topologically complete. And Euclidean n-space is topologically complete.
I failed to be clear: the "numerically weighted node" is the abstraction, the physical realization of it is the model - with their connections some neurons, some transistors, are physical models of numerically weighted nodes. Your claim was that the brain contains no such things, my claim is that it does.
Yes I thought of that but probably didn't articulate my response clearly.
First an analogy.
Suppose I say that my brain/body is not an algorithm. Then you say, oh yeah? You claim your body/brain can't express algorithms or doesn't contain algorithms? No, I don't claim that. In fact if I execute the Euclidean algorithm to calculate the GCD of two integers using pencil and paper, as I did many years ago in number theory class, I am a brain using my body to execute an algorithm.
But not EVERY function of my brain and body is nothing but an algorithm. See the difference? My brain can execute algorithms. But that's not ALL my brain can do. My brain does things that are NOT algorithms. That's my claim.
So I would be perfectly happy to agree that SOME functions of the brain might be implemented as neural networks. After all, neural networks are an abstraction of neurons in the brain. I would expect that some subsets of my brain can be isolated and modeled to perfection as a neural network.
But I am saying that NOT ALL brain function can be explained as a neural network.
I hope that's clear. I did not say the brain contains NO such things. I said it doesn't contain NOTHING BUT such things.
Of course it contains much more, not only in hardware but in organizational complexity - but so do the actual machines running neural nets, at least the hardware.
Hardware itself is an abstraction. It really contains flowing currents as noted earlier in the thread. But digital electronics presents an abstraction to the software that allows the software to pretend there's Boolean logic and bit flipping.
In the same sense, brains are full of all kinds of other gooey stuff performing who knows what functions.
Now you claim the brain is implementing an abstraction layer that looks like a neural network to the mind it hosts.
I claim not. I'll allow that the brain may implement some neural networks. But that's not the ONLY thing it does.