Peter Soderqvist

TO EXCREATIONIST

Soderqvist1: In a consistent arithmetical system when every statement is true, the proof sequence is incomplete, according to Godel's incompleteness theorem, because the theorems in the system are incomplete! I am reading David Deutsch book, the Fabric of Reality, I am on the chapter 5; Universality and the Limits of Computation; theoretically, a universal virtual-reality machine can rend every possible physical environment into virtual-reality, but that is only an infinitesimal subset of all mathematical logical possible environments, which this machine cannot rend into virtual-reality, because it doesn't have any such program in its repertoire! It reminds me about chocolate cakes Incompleteness Theorem! A book by Casti regarding Godel's life, (A Life of Logic) according to the book; that there is an infinite amount of logical possible chocolate cakes, which not have any recipes, and for the same reason, these "Platonic-cakes" cannot be baked, because recipes or theorems or algorithms are incomplete! Godel's incompleteness theorem is created by his mind or neuronal network; neither symbolic up down, nor neuronal down-up approach can change this fact, namely, that there is something beyond mere computations, which algorithmic approach cannot cope with!

excreationist

Peter Soderqvist:
Earlier you quoted Susan Greenfield:
I would just like to indicate why biological brains are currently not like current artificial systems. First we have non algorithmic processes, that is to say we have commonsense and intuition, we don't necessarily think in a step by step algorithmic process.
Well I am saying that neural networks develop "hunches" or intuition. e.g. if show it examples of male and female faces and tell it which one is which, you can give it a new picture and it will determine the best match (whether it is male or female). And you can see how "certain" it is by the strengths of its outputs. Initially you might have taught it in terms of 100% or 0% but if it has a new input its output could be 100% for both sexes, or 50% for both, or 34.6542% male and 10.3% female... then the final decision would be whatever has the highest value.
Anyway, if things are merely "common-sense" or intuition, they aren't adequately proven as far as logical proofs go. Sometimes the facts can go against our intuitions because of our lack of knowledge. e.g. to ancient peoples, they would have thought that it was common sense that the earth wasn't moving and the Sun travelled around the earth. So I think that "intuition" can describe (in a non-scientific way) neural network inferences and predictions (classifications of unseen inputs, etc).

In a consistent arithmetical system when every statement is true, the proof sequence is incomplete, according to Godel's incompleteness theorem, because the theorems in the system are incomplete!
Human beings would draw on real world experience about physical quantities, cause and effect, etc, to justify their ideas...
Can humans logically "prove" all of those Godel problems? Or do they sometimes need to rely on commonsense/intuition. If the mathematicians use some common-sense/intuition then they are a bit like simple neural networks - though their brains would have spent many years learning... (most of that learning would be self-motivated - by instinctual drives - seeking coherence, newness, etc) BTW, people begin with instincts like reflex behaviours and tastes/sights/sensations they should seek and avoid, etc. So they begin with a "grounding". I guess they are like initial axioms...

I am reading David Deutsch book, the Fabric of Reality,
I'm a fan of his although I haven't read much of his work (e.g. books, etc). Unfortunately I don't have enough time to do much more than skim read some things.

that there is an infinite amount of logical possible chocolate cakes, which not have any recipes, and for the same reason, these "Platonic-cakes" cannot be baked, because recipes or theorems or algorithms are incomplete!
Humans can't prove/disprove those theorems either can they? They might be able to use "common-sense" though.

Godel's incompleteness theorem is created by his mind or neuronal network; neither symbolic up down, nor neuronal down-up approach can change this fact, namely, that there is something beyond mere computations, which algorithmic approach cannot cope with!
In order to respond to that statement properly, I was wondering, could you give examples (or an example) of theorems, etc, that the algorithmic approach can't deal with, but people can?
Another thing to consider is that people can recognize dead-ends and temporarily give-up after a while... e.g. if the theorem involved numbers going to infinity, they'd forget about dealing with explicit numbers after a while... they'd try another approach... But it takes years for people to learn good problem solving strategies. And if those strategies aren't working people can try and modify old strategies and solutions to problems in order to come up with new strategies. (it would usually be an automatic process though)

cobrashock

[QUOTE]Originally posted by excreationist


[In a consistent arithmetical system when every statement is true, the proof sequence is incomplete, according to Godel's incompleteness theorem, because the theorems in the system are incomplete![/b]

Human beings would draw on real world experience about physical quantities, cause and effect, etc, to justify their ideas...

I believe this is correct. And futher more any AI created using this method would soon become self aware. Then it would ask, is there a God?

I was wondering, could you give examples (or an example) of theorems, etc, that the algorithmic approach can't deal with, but people can?

How about concepts? ie- other realitys, dimensions, after-lifes, rational concepts such as justice or free press?
The question then becomes could a AI be taught these concepts?
Indeed, do we deal with them properly? (that's a concept too)

cobrashock

excreationist

I meant the things that Godel's incompleteness theorem was talking about though... those other things are pretty different - i.e. they're not really about Godel-style logical questions... (sorry I'm trying to stay out of discussions that go off on too many tangents)

sodium

You've gone from,

For each Turing machine, there is an undecidable statement.
to
There is a statement that is undecidable by any Turing machine.

The error is of the same as going from:

Every man has a shoe that fits him.
to
There is one shoe that fits every man.

By the way, can you tell me about the truth value of the following proposition:

Peter Soderqvist will conclude that this statement is false.

Peter Soderqvist

TO SODIUM

Soderqvist1: I don't understand what you are trying to say!
But these statements appears truth to me anyway! For instance, if philosophers are mortals are decidable by every Turing Machine, but Cretans are liars is a statement that is undecidable by any Turing Machine! This is a decidable Proposition: All humans are mortals, all philosophers are humans, therefore; philosophers are mortals too! But this is an undecidable Proposition: All Cretans are liars, I am a Cretan; therefore I am a liar! If it is truth that all Cretans are liars, I am not a liar because I have told you the truth about it! If it is a lie that all Cretans are liars, I have still told you the truth that I am a liar, thus I am not a liar! If Cretans are liars or not is undecidable by any Turing Machine! The Cretan paradox is Bertrand Russell's brainchild!

Soderqvist1: Syllogism cannot be framed in that way!
But it can be framed in this way, every man has a shoe that fits him, I am a man, therefore; I have shoe that fits me, but my shoe doesn't necessarily fits every man! This is an undecidable self-referential loop; "Peter Soderqvist will conclude that this statement is false." :banghead:

cobrashock

I disagree- Godel's proofs are intended as precurser to analytic philosophy. Those questions I asked are indeed subject matter to that discipline, and THAT's why such subjects in a math based logic system are incomplete, according to Godel.
Godels proofs when applied a math based machines show incompletness too, but the fact that they apply to machine technology has not been well established. We just assume by majority that it does, or so I'm reading. So, as I'm seeing it right now, Godels proofs as they apply to machines that use math, are just a rationalism that could possibly be true. It's just possible that a machine could figure out the proofs of the algorithms it runs on! We do such things on math with supercomputers all the time now. The difference is we have to command the computer to work out the problem. With a AI it would have to be self motivated and that implys conscience.
cobrashock

John Page

The Cretan paradox apparently dates back around 2,500 years. IEP Link Here
Cheers, John

sodium

For any Turing machine, you can construct a question that it can't answer, similar in nature to the question I gave which you can't answer. But you can't construct any questions that have an answer, and yet aren't answerable by any Turing machine. So, the "computationally undecidable" statements don't exist.

Now, it might seem that we humans have at least the advantage that you can't construct Godel-based questions that we will find unanswerable. But this is only true if our internal problem solving doesn't correspond to any algorithm. And that's exactly what we're debating.

John Page

Ho sodium:

A couple of comments. The first may seem trite but has a serious point. Are these decidable computationally: 1. You can take a Lego set and build a computer with it, but the instructions won't necessarily tell you how. 2. You can take a bunch of non-organic material and build artificial consciousness, but the instructiond for Visual Basic won't necessarily tell you how.

My second comment is about an article in todays Washington Post. I haven't copied it because I respect copyright - but here's a small snippet:
and here's the link
Cheers, John