If cognition magically exists outside of math and science, then sure, all bets are off.
We don't even know if the flow of water in a river can always be represented by a mathematical function - this is one of the Millennium Problems. And we've known the partial differential equations that govern that system since the 1850's.
We are far, far away from even being able to write down anything resembling a mathematical description of cognition, let alone being able to say whether the solutions to that description are in the class of Lebesgue-integrable functions.
There was, past tense, no reason to believe cognition could be represented as a mathematical function. LLMs with RLHF are forcing us to question that assumption. I would agree that we are a long way from a rigorous mathematical definition of human thought, but in the meantime that doesn't reduce the utility of approximate solutions.
The Navier-Stokes equations are a set of partial differential equations - they are the problem statement. Given some initial and boundary conditions, we can find (approximate or exact) solutions, which are functions. But we don't know that these solutions are always Lebesgue integrable, and if they are not, neural nets will not be able to approximate them.
This is just a simple example from well-understood physics that we know neural nets won't always be able to give approximate descriptions of reality.
"Neural networks are universal approximators" is a fairly meaningless sound bite. It just means that given enough parameters and/or the right activation function, a neural network, which is itself a function, can approximate other functions. But "enough" and "right" are doing a lot of work here, and pragmatically the answer to "how approximate?" can be "not very".
A lot of people who argue that cognition is special to biological systems seem to base the argument on our inability to accurately model the detailed behavior of neurons. And yet kids regularly build universal computers out of stuff in Minecraft. It seems strange to imagine the response characteristics of low-level components of a system determine whether it can be conscious.
But GP specifically says neural nets should be able to do it because they are universal approximators (of Lebesgue integratable functions).
I'm saying this is clearly a nonsense argument, because there are much simpler physical processes than cognition where the answers are not Lebesgue integratable functions, so we have no guarantee that neural networks will be able to approximate the answers.
For cognition we don't even know the problem statement, and maybe the answers are not functions over the real numbers at all, but graphs or matrices or Markov chains or what have you. Then having universal approximators of functions over the real numbers is useless.
Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.
-- Erwin Schrödinger
- Carl Sagan
Sagan, while he did a little bit of useful work on planetary science early in his career, quickly descended into the realm of (self-promotional) pseudo-science. This was his fanciful search for 'extra-terrestrial intelligence'. So it's apposite that you bring him up (even if the quote you bring is a big miss against a philosophical statement), because his belief in such an 'ET' intelligence was a fantasy as much as the belief in the possibility of creating an artificial intelligence is.
We don't know if physics is the fundamental substrate of being, and given Agrippa's trillemma we can't know.
That's a pretty simplistic view. How do you know we can't determine whether an arbitrary program will halt or not (assuming access to all inputs and enough time to examine it)? What in principle would prevent us from doing so? But computers in principle cannot, since the problem is often non-algorithmic.
For example, consider the following program, which is passed the text of the file it is in as input:
function doesHalt($program, $inputs): bool {...}
$input = $argv[0]; // contents of this file
if (doesHalt($input, [$input])) {
while(true) {
print "Wrong! It doesn't halt!";
}
} else {
print "Wrong! It halts!";
}
Your argument doesn't disprove my assumption *. In which case, what's the point of it?
* - I don't necessarily believe this assumption. But I do dislike bad arguments.
Can you tell me if a program which searches for counterexamples to the Collatz conjecture halts?
Turing's entire analysis started from the point of what humans could do.
It doesn't matter what the algorithmic doesHalt function returns - it will always be incorrect for this program.
The point is we currently have very little understanding of what gives rise to consciousness, so what is the point of all this pontificating and grand standing. Its silly. We've no idea what we are talking about at present.
Clearly, our state of the art models of nueral-like computation do not really simulate consciousness at all, so why is the default assumption that they could if we get better at making them? The burden of evidence is on conputational models to prove they can produce a consciousness model, not the other way around.
The function we are trying to compute is undecidable. Sure we as humans understand that there's a dichotomy here: if the program halts it won't halt; if it doesn't halt it will halt. But the function we are asked to compute must have one output on a given input. So a human, when given this program as input, is also unable to assign an output.
So humans also can't solve the halting problem, we are just able to recognize that the problem is undecidable.
Note: whatever algorithm is implemented in the doesHalt function will contain a bug for at least some inputs, since it's trying to generalize something that is non-algorithmic.
In principle no algorithm can be created to determine if an arbitrary program will halt, since whatever it is could be implemented in a function which the program calls (with itself as the input) and then does the opposite thing.
> What makes you certain there is an algorithmic analog for all human reasoning?
(Maybe) not for ALL human thought but at least all communicatable deductive reasoning can be encoded in formal logic. If I give you an algorithm and ask you to decide if it does halt or does not halt (I give you plenty of time to decide) and then ask you to explain to me your result and convince me that you are correct, you have to put your thoughts into words that I can understand and and the logic of your reasoning has to be sound. And if you can explain to me you could as well encode your though process into an algorithm or a formal logic expression. If you can not, you could not convince me. If you can: now you have your algorithm for deciding the halting problem.
And while the human brain might not be a bio-computer, I'm not sure, its computational prowess are doubtfully stronger than a quantum turing machine, which can't solve the halting problem either.
func main() {
var n = 4;
OUTER: loop {
for (var i = 2; i < n/2; i++) {
if (isPrime(i) && isPrime(n-i)) {
n += 2;
continue OUTER; // Goldbach’s conjecture
}
break;
}
}Many things are non-algorithmic, and thus cannot be done by a computer, yet we can do them (e.g. love someone, enjoy the beauty of a sunset, experience joy or sadness, etc).
There's an interesting article/podcast here about what computers can't do: https://mindmatters.ai/2020/08/six-limitations-of-artificial....
Moreover, are you sure that e.g. loving people in non-algorithmic? We can already make chatbots which pretty convincingly act as if they love people. Sure, they don't actually love anyone, they just generate text, but then, what would it mean for a system or even a human to "actually" love someone?
Maybe. When computers solve it then we'll know.
I don't see how it could be reasoned otherwise.
What do you mean? Is cognition a set of weights on a gradient? Cognition involves conscious reasoning and understanding. How do you know it is computable at all? There are many things which cannot be computed by a program (e.g. whether an arbitrary program will halt or not)...