No AI/No Supercomputers: Complexity Limits?

[Anthony]

Originally Posted by Ji ji

Not really. Sorry if I have been unclear. We can decide logically whether an undecidable proposition is true or false, something that a formal system can’t.

No we actually can't. The tools we use are not part of the formal system and are therefore not 'logical'.

[Ji ji]

Originally Posted by Anthony

You're making the assumption that we need to be able to fully comprehend a mind to create one. We don't, as long as we aren't trying to contain it within a human's memory.

No assumption, just an hypothesis. At present, we don’t know if we can or cannot create a conscious mind.

Originally Posted by Anthony

Only if you care about the external levels being able to answer questions about themselves. There are plenty of questions a human cannot answer.

True.
Still, among the things that we can do, there are some not reproducible by a formal system, as it would not be finite.

[Anthony]

Originally Posted by Ji ji

No assumption, just an hypothesis. At present, we don’t know if we can or cannot create a conscious mind.

We are reasonably certain that a conscious mind can be created without a human who is capable of comprehending said mind -- ordinary human minds are created without anyone understanding them (unless God exists and is doing so).

Originally Posted by Ji ji

Still, among the things that we can do, there are some not reproducible by a formal system, as it would not be finite.

I am not aware of any examples where that is provably true.

[Ji ji]

Originally Posted by Anaraxes

But you can't do so logically, by proof. Certainly you can simply assume that a proposition is true, arbitrarily. But then, so could a formal system built to do so.

An "undecidable" proposition isn't universally undecidable. It's only undecidable within the context of a specific formal system. And it's trivially easy to create a formal system that can decide that undecidable problem: you just add it to the list of axioms for your new system. As you said, you just arbitrarily decide that the proposition is true. (Or false, if you prefer.) Quite easily done with most AI systems. A program is itself just data, after all.

To no avail. You add the axiom to decide on undecidable propositions, and your new set of axioms has the same problem - at least one undecidable proposition, or a proposition demonstrable both true or false. At least, if your set axioms is powerful enough to encompass mathematics.
It’s the easiest attack strategy on Gödel’s theorem, and it doesn’t work.

Originally Posted by Anthony

No we actually can't. The tools we use are not part of the formal system and are therefore not 'logical'.

This is a big misunderstanding on how logic works.
Theorems are built with set of symbols and transformation rules, but the theorem meaning can’t be contained in the set.

[Anthony]

Originally Posted by Ji ji

Theorems are built with set of symbols and transformation rules, but the theorem meaning can’t be contained in the set.

You'll have to define your terms a bit better; I cannot think of a definition of 'meaning' that is both logical and impossible to express logically.

[Ji ji]

Originally Posted by Anthony

We are reasonably certain that a conscious mind can be created without a human who is capable of comprehending said mind -- ordinary human minds are created without anyone understanding them (unless God exists and is doing so).

Lol, you got me!
I reword it: we don’t know if we will be able to create one apart from the ol’ good ways.

Originally Posted by Anthony

I am not aware of any examples where that is provably true.

I was thinking to have shown it well enough. Yet you can get a much better explanation going to the sources: Gödel, Turing, Lucas, and Penrose. My advice is to go for Gödel and Lucas or Gödel and Penrose.

[Anthony]

Originally Posted by Ji ji

I was thinking to have shown it well enough. Yet you can get a much better explanation going to the sources: Gödel, Turing, Lucas, and Penrose. My advice is to go for Gödel and Lucas or Gödel and Penrose.

I have read all three. Yes, we can decide that certain problems cannot be solved from within a given formal system, but we do so from outside the system. Construct a formal system that is large enough to include us and it will include undecidable statements that we cannot demonstrate are undecidable.

[malloyd]

Originally Posted by Ji ji

Still, among the things that we can do, there are some not reproducible by a formal system, as it would not be finite.

What makes you sure of that? Can you produce any evidence that there do not exist inputs that cause human minds to lock up in infinite loops exactly like the ones you seem to think the AIs can get stuck in?

[Flyndaran]

Wouldn't that just be indecision "brain-lock" followed by simply choosing randomly? I've had cats that got stuck like that until I made some slight noise. Then they suddenly decide.
Only really simple animals lack these kinds of fail safes.
Ants can get stuck in a death-spiral where they fail to decide even to the point of death. https://en.wikipedia.org/wiki/Ant_mill

[whswhs]

Originally Posted by Ji ji

Not really. Sorry if I have been unclear. We can decide logically whether an undecidable proposition is true or false, something that a formal system can’t.

I don't think that really works. I'm aware of the history of the problem. Gödel was responding to the logicist and formalist schools of metamathematics, which assumed that you could work out all the logical implications of a given set of primitive terms and postulates. He showed that you could not. Then Turing proposed the Turing machine as a model for an idealized human logician. Then von Neumann used the Turing machine as the basis for computer design.

So it seems to me that if a human being can decide "logically" that theorem X, which is undecidable from postulates A1, A2, A3, and so on of formal system P, can be made decidable by shifting to formal system P* with different postulates, then a Turing machine, or an actual computer, can equally well shift to a different formal system. If necessary, you can program it to try new postulates at random, or in some specified order, and keep going till it gets to one that gives you formal system P* in which it can be ascertained that X is now decidable. And if for some reason you can't do that, then since a Turing machine is exactly equivalent to an idealized human logician, anything a human being does to decide X is not "logical."

Or if you are using "logical" in one case to mean only what can be done within a formal system, but in the other case to mean something more extensive, then that looks like the Fallacy of Equivocation.