Did Godel disprove the idea of artificial intelligence?

[jay gw]

Has artificial intelligence been discredited by Godel's incompleteness theorem?

Hao Wang A logical journey: From Godel to philosophy, 1997

Godel's Incompleteness Theorem states that in any consistent formal system which is adequate for arithmetic there is a true but unprovable sentence.

Second Theorem

If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent. No consistent system can be used to prove its own consistency.

Therefore, in order to establish the consistency of a system S, one needs to utilize some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.

The theorem does not imply that every interesting axiom system is incomplete. The theorem only applies to systems that allow you to define the natural numbers as a set.

Minds, Machines and Gödel

Gödel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met.

The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics
by Roger Penrose

For decades, proponents of artificial intelligence have argued that computers will soon be doing everything that a human mind can do. Admittedly, computers now play chess at the grandmaster level, but do they understand the game as we do? Can a computer eventually do everything a human mind can do?

Roger Penrose--physicist, and Stephen Hawking - puts forward his view that there are some facets of human thinking that can never be emulated by a machine.

Penrose examines what physics and mathematics can tell us about how the mind works, what they can't, and what we need to know to understand the physical processes of consciousness. He is among a growing number of physicists who think Einstein wasn't being stubborn when he said his "little finger" told him that quantum mechanics is incomplete, and he concludes that laws even deeper than quantum mechanics are essential for the operation of a mind.

[drkitten]

Originally posted by jay gw
Has artificial intelligence been discredited by Godel's incompleteness theorem?


Simply put, no. Assuming for the sake of argument that humans are considered to be "intelligent," then we must accept that a hypothetical perfect artificial copy of the human cognitive system would also be "intelligent."

Godel's theorem simply states that a sufficiently powerful formal system must be either incomplete or inconsistent. Humans are known to be both incomplete and inconsistent. Therefore, insofar as as human reasoning would be modeled by a formal system, the formal system itself would be both incomplete and inconsistent, entirely compatible with Godel.

Actually, this question (and many, many more implications for AI than I have the room to address here) were done to death by Doug Hofstadter in Godel, Escher, Bach : An Eternal Golden Braid, to which I recommend you. But the Godelian argument is explicitly refuted (he attributes this argument to the philosopher Lucas) in detail.

[phildonnia]

On the contrary, I think Goedel's theorem suggested inherent limitations to human minds.

Machines can never be "truly intelligent" in the sense of being able to know all truth. If one takes the view that Mind is a product of Brain, then the same argument applies, and we must conclude that human minds can never be "truly intelligent" either.

[patnray]

I voted "No".

For an alternative view point try "Godel, Escher, Bach : An Eternal Golden Braid" by Douglas R Hofstadter, a delightfully witty book in which he argues that intelligence emerges from a complex multilevel system.

"My belief is that the explanations of 'emergent' phenomena in our brains -- . . . [including] finally consciousness and free will -- are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being determined by the bottom level. In other words, a self-reinforcing 'resonance' between different levels . . . ."

The book, itself, is a complex, multilevel synthesis of "top down" and "bottom up" approaches to underestanding intelligence...

Damn. new drkitten beat me to it...

[drkitten]

Originally posted by jay gw

The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics
by Roger Penrose

[...]
Penrose examines what physics and mathematics can tell us about how the mind works, what they can't, and what we need to know to understand the physical processes of consciousness. He is among a growing number of physicists who think Einstein wasn't being stubborn when he said his "little finger" told him that quantum mechanics is incomplete, and he concludes that laws even deeper than quantum mechanics are essential for the operation of a mind.

I'd feel much more confident about Penrose's explanation of either physics or psychology (or indeed, of computer science) if he actually knew anything about any of these disciplines. Unfortunately, Penrose is a mathematician (and an exceptionally good one), but not a polymath. The Emperor's New Mind, in particular, is not a particularly well-thought out book, and is fundamentally full of holes in how it presents stuff. Check out some of the excerpts from the reviews on amazon.com:


Right. Take a deep breath here. For it's a scary thing for a mere mortal (with a decidedly ordinary bachelor's degree in the humanities) to say something like this about the one of the cleverest men on the planet, but I can't see any way around it: In this book Roger Penrose completely, totally, misses the point. Insofar as it's considered an entry on the Consciousness/AI debate, The Emperor's New Mind - all 583 pages of it - is all but worthless.

Here's where I think he goes wrong. Firstly, his attempt to undermine the AI position is founded on purely mathematical reasoning. Pure mathematics is a closed logical system. Its truths aren't falsifiable, so by themselves have no explanatory force. Mathematical statements (such as "1+1=2") are necessarily true for all time and all universes so, ipso facto, they can't - by themselves - tell us anything about any particular universe. Yet that is just what Penrose asks them to do. He invokes Gödel's theorem of undecidability, perhaps to counter the argument I have just made, but it isn't convincing - being logically unable to prove all truths in a particular set (even though you know they are true) is very different from being able to falsify them. Without that power, you have no explanatory traction in the outside world. Penrose's entire attack on Strong AI is based on an unfalsifiable, and therefore non-content carrying, argument.

Ultimately, when Penrose says "quantum theory explains consciousness" he is really saying no more than "something magic happens!" or even "THEN A MIRACLE OCCURS".

Mr Penrose, I think you should be more explicit here in step two.



or this


His arguments are unclear, weak and largely dependent on philosphers like Lucas and Searle, while his idea of quantum effects is improbable and surely in the end irrelevant (cannot computers tap into quantum effects?) and his knowledge of computer science deeply, deeply suspect.
For example, I quote here from the final sections which actually have something to do with his ostensible reason for writing the book:
"Neverthless, one still might imagine some kind of natural selection process being effective for producing approximately valid algorithms. Personally, I find this very hard to believe, however."
The entire flourishing, commercially succesful field of evolutionary computing begs to differ here, Mr. Penrose. SUch bonehead errors compells me to point out that this mathematician has no clothes.


From a more formal context, here's the abstract of a recent paper on Penrose's work, by professional philosophers working on AI ( Selmer Bringsjord, Hong Xiao, published in JETAI 2000).


Having, as it is generally agreed, failed to destroy the computational conception of mind with the Godelian attack he articulated in his The Emperor's New Mind, Penrose has returned, armed with a more elaborate and more fastidious Godelian case.... The core argument... is enthymenmatic, and when formalized, a remarkable number of technical glitches come to light. Over and above these defects, the argument, at best, is an instance of either the fallacy of denying the antecedent, the fallacy of petitio principii, or the fallacy of equivocation. More recently, writing in response to his critics in the electronic journal Psyche, Penrose has offered a Godelian case designed to improve on the version presented in [Shadows of the Mind]. But this version is yet again another failure. In falling prey to the errors we unconver, Penrose's new Godelian case is unmasked as the same confused refrain J. R. Lucas initiated 35 years ago.


(I should note that the author of this article (Bringsjord) is himself an opponent of AI, so this isn't simply an instance of "preaching to the choir.")

Penrose, and more generally Godel via Lucas, has nothing useful to contribute to the AI discussion. Penrose in particular has this fundamental problem of an lack of understanding of the material under discussion --- when he's correct, he's not interesting. When he's interesting, he's unfortunately not correct. It's a bad combination that I urge you to eschew.

[MESchlum]

I just read the topic of the poll - after voting according to the topic of the discussion.

Bait and switch.

Annoying.

And what does quantum mechanics being incomplete (or not) have to do with abstract logic, exactly?

[Robin]

Silly question really. The human mind is a machine so one machine already thinks as a human mind does. No other machine will ever be able to think exactly as the human mind does (and why would we want to try?).

(I am assuming that 'machine' is used in its general sense rather than mathematical as in 'Turing machine').

But once we have properly understood the mechanisms involved I see no reason why the same principles should not be applied to other machines.

I totally fail to see what Godel's theorems have to do with the case. The human mind is not a system of logic.

[patnray]

Good point MESchlum. I voted "No" to the question "Did Godel disprove the idea of artificial intelligence?"

[Soapy Sam]

Define "Artificial intelligence."

Do we mean conscious awareness or simply complex response to stimuli?

Wire up a motion sensor to a servo operated machine gun. Hook in a pc and write the obvious "track and fire" software. Now put a burglar over the fence. Turing test anybody?

If it's smart enough to kill you, how much more intelligent does it have to get?


The first AI probably had the intelligence of a fundamentalist. We passed that stage a few years ago. We are probably up to bacterial levels now, at least. That's a billion years of organic evolution in about a century. Give things a chance. They'll be here soon enough.

[phildonnia]

Originally posted by Robin
I totally fail to see what Godel's theorems have to do with the case. The human mind is not a system of logic.

I think the idea is: human minds are based on physical brains, and physical brains are subject to the laws of physics, and the laws of physics can be modeled by logic.

Therefore, if the mind is capable of discerning truth, then so is the logical system that can ultimately model it.

[T'ai Chi]

As far as I am aware, the human brain does not work like + and * with the numbers {0,1,2,...}, so I too fail to see where Godel's theorem comes in.

[Jorghnassen]

I don't see how Godel's theorem comes in either (though pure math isn't my forte). I don't think it is required in a discussion on whether artificial intelligence is achievable or not.

[Jorghnassen]

Originally posted by phildonnia
I think the idea is: human minds are based on physical brains, and physical brains are subject to the laws of physics, and the laws of physics can be modeled by logic.


What a reductionist argument...

[Robin]

Originally posted by jzs
As far as I am aware, the human brain does not work like + and * with the numbers {0,1,2,...}, so I too fail to see where Godel's theorem comes in.

Philidonnia's reply is clearly not what Penrose had in mind and I have not read the book, but here is a quote from a review of the book by John McCarthy:

The Penrose argument against AI of most interest to mathematicians is that whatever system of axioms a computer is programmed to work in, e.g. Zermelo-Fraenkel set theory, a man can form a Gödel sentence for the system, true but not provable within the system.

If McCarthy has accurately represented Penrose then he appears to be assuming that a machine must work with a specific system of axioms. In which case the argument appears to be irrelevant on a number of levels.

[phildonnia]

Originally posted by Jorghnassen
What a reductionist argument...

Um, yeah. Ok, glad we got that observation out of the way.

Originally posted by jzs
As far as I am aware, the human brain does not work like + and * with the numbers {0,1,2,...}, so I too fail to see where Godel's theorem comes in.

As far as we can tell, the physical world does work like + and *. If mind can be entirely explained in terms of its physical substrate¹, then it must be possible for number theory to produce anything that mind can.

That is, mind cannot produce anything that number theory cannot. This is where Goedel's theorem comes in.

Originally posted by Robin
Philidonnia's reply is clearly not what Penrose had in mind...

Well that's good, since I would be arguing against Penrose.

¹ For frantic and desperate attempts to refute this idea, search this forum for posts by "Interesting Ian".

[drkitten]

Originally posted by phildonnia

As far as we can tell, the physical world does work like + and *.

As far as we can tell, the physical world is not described by a finite set of axioms. That's part of the legacy of quantum mechanics and the intellectual death of God-the-Watchmaker.

If the physical world is not axiomatizable, then a) it doesn't work like + and *, which are defined by their axioms, and b) is entirely outside the auspices of Godel's theorem, which is a theorem about formal axiomatic systems.

In fact, even Penrose recognizes the first paragraph above -- that the physical world is not axiomatizable. That's part of why he claims that artificial intelligence is not possible, because in his unbelievably naive view of computers, they are axiomatic systems, while realistic (quantum) effects are not. Ergo, the mind, being non-axiomatic, must be related somehow to quantum.

[phildonnia]

quote:
--------------------------------------------------------------------------------
The Penrose argument against AI of most interest to mathematicians is that whatever system of axioms a computer is programmed to work in, e.g. Zermelo-Fraenkel set theory, a man can form a Gödel sentence for the system, true but not provable within the system.
--------------------------------------------------------------------------------

This is the classic "Goedel" argument for the impossibility of AI. Two holes in this argument:

We cannot be really sure that a man always can form a Goedel sentence for a given axiomatic system. We've only really tried this on systems that are simple enough for our minds to understand. As you might guess, there is no way to formalize this process.

We cannot be sure that it is impossible for a machine to create a Goedel sentence for a human mind. This would require a computer capable of analyzing the entire underlying brain structure, which obviously is not now feasable.

[phildonnia]

Originally posted by new drkitten
As far as we can tell, the physical world is not described by a finite set of axioms. That's part of the legacy of quantum mechanics and the intellectual death of God-the-Watchmaker.
...Ergo, the mind, being non-axiomatic, must be related somehow to quantum.


I see your point that if the mind were "non-axiomatic", then quantum mechanics would be one possible explanation. However, there is no indication that the mind is "non-axiomatic", nor that quantum mechanics plays any role in the workings of the brain or mind.

[Jorghnassen]

Originally posted by phildonnia


That is, mind cannot produce anything that number theory cannot. This is where Goedel's theorem comes in.


How does number theory produce anything?

[Dr Adequate]

The Penrose argument against AI of most interest to mathematicians is that whatever system of axioms a computer is programmed to work in, e.g. Zermelo-Fraenkel set theory, a man can form a Gödel sentence for the system, true but not provable within the system.
And so, of course, could the computer. Only it would do it a million times faster, and would be much less likely to make a mistake.

The incomleteness of first-order axiomatisations of the arithmetic of the integers has nothing to do with the hardware or wetware trying to make use of such an incomplete axiomatisation. Consider the following equally valid --- or rather, equally invalid --- argument:

"The argument against human intelligence of most interest to mathematicians is that whatever system of axioms a human is asked to work in, e.g. Zermelo-Fraenkel set theory, a computer can form a Gödel sentence for the system, true but not provable within the system."

Convincing, huh?

[phildonnia]

Originally posted by Jorghnassen
How does number theory produce anything?

Following the rules of number theory produces "theorems". The question was whether these theorems represented "truth". Goedel's theorem showed that not all truths were represented by theorems (assuming that no falsehoods were represented).

The question is whether human minds are also subject to this limitation.

If minds can be modeled by number theory, then they are subject to this limitation. If not, then other systems that can be modeled by number theory (such as AI) are necessarily less able to recognize truth.

At our current level of understanding, it seems to be something of a point of faith one way or the other, and I suspect that some measure of anthropocentric pride may get involved.

[phildonnia]

Originally posted by Dr Adequate
"The argument against human intelligence of most interest to mathematicians is that whatever system of axioms a human is asked to work in, e.g. Zermelo-Fraenkel set theory, a computer can form a Gödel sentence for the system, true but not provable within the system."

Very nice.

I would like to add that:

Dr Adequate does not believe this sentence.

Since this sentence is obviously true to everyone else, and we have no trouble believing it, this demonstrates the inadequacy of Dr. Adequate with respect to the truth of his beliefs.

[Jorghnassen]

Originally posted by phildonnia
Following the rules of number theory produces "theorems". The question was whether these theorems represented "truth". Goedel's theorem showed that not all truths were represented by theorems (assuming that no falsehoods were represented).

The question is whether human minds are also subject to this limitation.

If minds can be modeled by number theory, then they are subject to this limitation. If not, then other systems that can be modeled by number theory (such as AI) are necessarily less able to recognize truth.

At our current level of understanding, it seems to be something of a point of faith one way or the other, and I suspect that some measure of anthropocentric pride may get involved.

I was under the impression that the human mind produced number theory and its theorems, axioms, corrolaries, etc. My point is that number theory doesn't exist outside of the human mind. I don't think number theory can model the mind (as it only deals with abstract notions), because it does not deal with the very physical aspect of the mind: the human brain.

My quarrel with AI, in the sense of a machine being able to think like a human does, is that it is not purely a software problem, and that trying to come up with AI from a purely axiomatic approach is bound to failure. A better approach (that is of course, already being worked on) would also consider reverse engineering the humain brain...

[drkitten]

Originally posted by phildonnia
I see your point that if the mind were "non-axiomatic", then quantum mechanics would be one possible explanation. However, there is no indication that the mind is "non-axiomatic", nor that quantum mechanics plays any role in the workings of the brain or mind.

.... which is part of why the Penrose argument is a load of tosh.

[jay gw]

I don't think number theory can model the mind (as it only deals with abstract notions), because it does not deal with the very physical aspect of the mind: the human brain.

Can the physical brain be mapped entirely? Can the map be expressed as numbers?

[Robin]

Originally posted by jay gw
Can the physical brain be mapped entirely? Can the map be expressed as numbers?

That is not really the point. To my knowledge the point of AI is to understand the processes by which the brain produces thought and to replicate this process using a machine.

To my knowledge there is no-one doubting that this can be done and that there will be one day computers that not only pass the Turing test but that will replace current call-centres and that we probably will not be able to tell the difference.

The debate that Penrose appears to be entering is whether a machine that appears in every respect to think, reason, use common sense, solve problems like a human can truly be said to be a 'mind'.

[Jorghnassen]

Originally posted by jay gw
Can the physical brain be mapped entirely? Can the map be expressed as numbers?

It takes more than a map to know how the brain works (besides, brain mapping has its own problems), and to build an artificial brain. Can an omelet be mapped? Can number theory make an artificial omelet (or rather, how does one use number theory to make an artificial omelet)? You see there is a huge difference between description (math can be used to describe reality, but in a limited way), and implementation.

/stealing and misusing material from a lecture on A.I. I once attended...

[AWPrime]

Why does the poll question differ from the title question?

[Interesting Ian]

Originally posted by new drkitten
[B]Simply put, no. Assuming for the sake of argument that humans are considered to be "intelligent," then we must accept that a hypothetical perfect artificial copy of the human cognitive system would also be "intelligent."



We must accept no such thing, I certainly don't accept it.



Godel's theorem simply states that a sufficiently powerful formal system must be either incomplete or inconsistent. Humans are known to be both incomplete and inconsistent. Therefore, insofar as as human reasoning would be modeled by a formal system, the formal system itself would be both incomplete and inconsistent, entirely compatible with Godel.


We human beings can recognise truths that do not appear to have been derived from an algorithmic process.

[Robin]

Originally posted by new drkitten
[B]Simply put, no. Assuming for the sake of argument that humans are considered to be "intelligent," then we must accept that a hypothetical perfect artificial copy of the human cognitive system would also be "intelligent."

Interesting Ian
We must accept no such thing, I certainly don't accept it.

But suppose you have a machine that takes a university exam paper and generates answers (say for Mathematics or English literature) and a human marks it, believing the paper to be the result of a human and gives the paper a high distinction.

How could you not describe that process as intelligent? You could certainly say that it was not conscious, or that that it was not a mind. But it would clearly be intelligent.

[Interesting Ian]

Originally posted by Robin
Silly question really. The human mind is a machine so one machine already thinks as a human mind does.

And I imagine that no justification will be given for this outrageous assertion. :rolleyes:

Wheel out your conscious robot, then I too will believe that we are merely machines. Yer going to have to do that since all philosophical arguments which attempt to establiush we are machines are somewhat lacking.

[Interesting Ian]

Originally posted by Robin
But suppose you have a machine that takes a university exam paper and generates answers (say for Mathematics or English literature) and a human marks it, believing the paper to be the result of a human and gives the paper a high distinction.

How could you not describe that process as intelligent? You could certainly say that it was not conscious, or that that it was not a mind. But it would clearly be intelligent.

OK, that's fine, I merely deny it is conscious.

[Robin]

Originally posted by Interesting Ian
And I imagine that no justification will be given for this outrageous assertion. :rolleyes:

Wheel out your conscious robot, then I too will believe that we are merely machines. Yer going to have to do that since all philosophical arguments which attempt to establiush we are machines are somewhat lacking.
Ah, finally the reaction I was trying to provoke! Unfortunately I don't have time to answer properly at the moment, I will do so in a couple of days.

It is interesting to note that the participants in the debate that Penrose is responding to (for example Searle, Dennet, McCarthy) all stipulate this as a premise.

[Robin]

Interesting Ian,

Just so I don't waste my time, if you don't agree that the human mind is a machine, do you agree that the human brain is a machine?

[Interesting Ian]

Originally posted by Robin
Interesting Ian,

Just so I don't waste my time, if you don't agree that the human mind is a machine, do you agree that the human brain is a machine?

Yes, although I think the processes within it can be influenced by the mind.

[phildonnia]

Man, that took a while. I've been laying Ian-bait all over this thread.

Originally posted by Interesting Ian
We human beings can recognise truths that do not appear to have been derived from an algorithmic process.

Such as?

Be prepared to discuss
1) how we know they are truths.
2) how we know they can not derived from an algorithmic process.

[Correa Neto]

Originally posted by Interesting Ian
OK, that's fine, I merely deny it is conscious.

Why?

Here there have been lots of discussions using hypothetical machines. Please allow me to propose one more.

Suppose at last a sci-fi robot or computer is created (wheeled out, using your sentence). Let it be Robin´s machine, C3P0, one of Asimov´s robots, whatever. This hipothetical machine can gather information on its environment and its own components, it can proccess these informations, learn (it even manage to successfully finish a course at an university), propose solutions to problems based on all the iformation it gathered, is aware of its own skills, of the data it has stored in its databanks and can perform tasks by itself, without being comanded do do so, it avoids being damaged, tries to repair itself and gather energy to continue its operations.

You are having a dialogue with it at an internet forum, at a chat, or at the telephone, without knowing its not a human being.

Would you be able to recognize it as artificial and therefore labell it as non counsient? If so, based on what aspects? What types of criteria do you propose to use in order to diferentiate a counsient entity from one that just behaves like as a counsient being?

Or you are assuming that it will not be really counsient because its a machine, a non-organic entity? Or because its not a human being?

[Interesting Ian]

Originally posted by Correa Neto
Why?

Here there have been lots of discussions using hypothetical machines. Please allow me to propose one more.

Suppose at last a sci-fi robot or computer is created (wheeled out, using your sentence). Let it be Robin´s machine, C3P0, one of Asimov´s robots, whatever. This hipothetical machine can gather information on its environment and its own components, it can proccess these informations, learn (it even manage to successfully finish a course at an university), propose solutions to problems based on all the iformation it gathered, is aware of its own skills, of the data it has stored in its databanks and can perform tasks by itself, without being comanded do do so, it avoids being damaged, tries to repair itself and gather energy to continue its operations.

You are having a dialogue with it at an internet forum, at a chat, or at the telephone, without knowing its not a human being.

Would you be able to recognize it as artificial and therefore labell it as non counsient? If so, based on what aspects? What types of criteria do you propose to use in order to diferentiate a counsient entity from one that just behaves like as a counsient being?

Or you are assuming that it will not be really counsient because its a machine, a non-organic entity? Or because its not a human being?

We would need to feel it is conscious. I propose that even an android that behaved exactly like a human being would not arouse such a feeling. This is because I suspect we know that other people are conscious partially through anomalous cognition.

[jay gw]

The debate that Penrose appears to be entering is whether a machine that appears in every respect to think, reason, use common sense, solve problems like a human can truly be said to be a 'mind'.

Plenty of humans can't do that - who said artificial intelligence has to have "common sense"? What is common sense?

solve problems like a human

How do I solve problems, exactly? Please explain.

Yes, it is necessary to map the brain before you can understand it. The map provides the blueprint. But then someone wanting to duplicate/build it from nothing would have to understand how all the parts interact. That is probably the thing that keeps AI researchers stumped.

The brain seems to be divided into components, then assembled into a whole. Very simplistic yes I know, but the verbal/art/math/conscience abilities are not randomly distributed. They're in the same location, grouped together.

One of the big problems that AI faces is that circuits made of metal and plastic cannot, physically can't, duplicate the circuits in the brain. I don't know how anyone will get around this.

At some point, it will be possible to build mechanical material that can simulate cells.

Maybe it's wrong, but it seems that if every circuit in the brain were recreated AND grouped as the brain is, and energy introduced, something very interesting would happen.

People discuss "consciousness" and "intuition" as if they're something metaphysical. I think they are natural outcomes from the circuits interacting with senses and the environment.

[Robin]

Originally posted by Interesting Ian
OK, that's fine, I merely deny it is conscious.

And in fact I probably agree with you. I think that it is a valuable distinction to make - we are debating what John Searle calls the "Strong AI" hypothesis, that an artificially intelligent machine is necessarily a mind.

[Interesting Ian]

Originally posted by phildonnia
Man, that took a while. I've been laying Ian-bait all over this thread.



Such as?

Be prepared to discuss
1) how we know they are truths.
2) how we know they can not derived from an algorithmic process.

I admit I know very little about this subject, but did not Goedel show that human beings can know things that cannot be known by computation?

Anyway, we can't specify how these truths are known - perhaps not surprising as these truths are simply intuited. That is to say they are simply seen to be true by an inner understanding.

[phildonnia]

Originally posted by Interesting Ian
We would need to feel it is conscious. I propose that even an android that behaved exactly like a human being would not arouse such a feeling.

I think we can all agree that if there's some extra-physical component to consciousness, then a physical system can't match up. That is, of course, a big "if", and I've certainly never seen any evidence for it.

We don't even need Goedel's theorem for that, except to point to the existence of "truth" outside of a physical system.

This is because I suspect we know that other people are conscious partially through anomalous cognition.

Thus, in your opinion, the Turing test will always fail, since we can always detect through "anomalous cognition" whether we're talking to a machine or not. I'll just mention an often observed phenomenon called the "Eliza Effect" where people are fooled by the most superficial imitations of human behavior into supposing the existence of intelligence. So if we have a consciousness-detector, it's really not used very well.

[Jeff Corey]

The Eliza program has evolved in some weird ways. Rather than just just parroting Rogerian non-directive therapy, variants have gone beyond that.
They have even done Bruce Sterling stuff with spiderbots impersonating people on websites.
I figger that's what Ian is.

[Robin]

Originally posted by Jeff Corey
The Eliza program has evolved in some weird ways. Rather than just just parroting Rogerian non-directive therapy, variants have gone beyond that.
They have even done Bruce Sperling stuff with spiderbots impersonating people on websites.
I figger that's what Ian is.

I have my suspicions in this respect about some JREF contributors, but not Interesting Ian.

[Robin]

Originally posted by Interesting Ian
I admit I know very little about this subject, but did not Goedel show that human beings can know things that cannot be known by computation?


No.

[Jeff Corey]

Originally posted by Robin
I have my suspicions in this respect about some JREF contributors, but not Interesting Ian.
Nor De-Bunk nor Pillory.
Show me the evidences for the schoolarship?

[Interesting Ian]

Originally posted by phildonnia


Thus, in your opinion, the Turing test will always fail, since we can always detect through "anomalous cognition" whether we're talking to a machine or not. I'll just mention an often observed phenomenon called the "Eliza Effect" where people are fooled by the most superficial imitations of human behavior into supposing the existence of intelligence. So if we have a consciousness-detector, it's really not used very well. [/B]

No, not the turing test. I said an android. For example you would need to gaze into someones eyes when they express emotions. Merely reading a paragraph of text is insufficient to establish whether consciousness is there!

[Interesting Ian]

Originally posted by Interesting Ian
I admit I know very little about this subject, but did not Goedel show that human beings can know things that cannot be known by computation?

Robin
No.

I refer you to this article here. (http://users.ox.ac.uk/~jrlucas/Godel/mmg.html)


Gödel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met.1 This I attempt to do.

Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot {44} be proved-in-the-system, but which we can see to be true.

[Robin]

Originally posted by Interesting Ian
I refer you to this article here. (http://users.ox.ac.uk/~jrlucas/Godel/mmg.html)

If you are quoting John Lucas then you should have said "...did not John Lucas show that human beings can know things that cannot be known by computation?"

Because Lucas referred to Godel in his article does not mean that the views reflect those of Godel.

[Interesting Ian]

Originally posted by Robin
If you are quoting John Lucas then you should have said "...did not John Lucas show that human beings can know things that cannot be known by computation?"

Because Lucas referred to Godel in his article does not mean that the views reflect those of Godel.

I wasn't quoting Lucas in my original statement. I was merely drawing upon my general knowledge.

[Robin]

Originally posted by Interesting Ian
I wasn't quoting Lucas in my original statement. I was merely drawing upon my general knowledge.
You should read the Lucas article in the context of the definition he is using:
In fact we should say briefly that any system which was not floored by the Gödel question was eo ipso not a Turing machine, i.e., not a machine within the meaning of the act.
So he is talking about Turing machines so even if he is right he is not precluding the idea that some physical device might be able to do all that a mind can.

(Anybody who is about to jump in with the "Church-Turing thesis" please note that they did not claim a Turing machine can do anything that any physical system can)

[Interesting Ian]

Originally posted by Robin
Originally posted by Interesting Ian
I wasn't quoting Lucas in my original statement. I was merely drawing upon my general knowledge.

Robin
You should read the Lucas article in the context of the definition he is using:

quote:In fact we should say briefly that any system which was not floored by the Gödel question was eo ipso not a Turing machine, i.e., not a machine within the meaning of the act.

Robin
So he is talking about Turing machines so even if he is right he is not precluding the idea that some physical device might be able to do all that a mind can.



But any machine will function according to physical laws. Cannot all physical laws be expressed algorthimically? After all, physical laws are characterized by their mathematical form. Thus there are no non-turing machines.

[Robin]

Originally posted by Interesting Ian
But any machine will function according to physical laws. Cannot all physical laws be expressed algorthimically? After all, physical laws are characterized by their mathematical form. Thus there are no non-turing machines.
No non-turing machines? Certainly Lucas appears to be suggesting the possibility of one in the second to last paragraph of the article you quoted.

Turing never suggested that there were no non-Turing machines. There are things that happen in real machines like randomness, parallelism that have no place in a Turing machine.

If all physical laws could be expressed algorithmically that would appear to negate the very premise of Lucas's article.

[Robin]

See for example the following entry in the Stanford Encyclopedia of Philosphy:

A myth seems to have arisen concerning Turing's paper of 1936, namely that he there gave a treatment of the limits of mechanism and established a fundamental result to the effect that the universal Turing machine can simulate the behaviour of any machine. The myth has passed into the philosophy of mind, generally to pernicious effect.

[Kopji]

I don't see Gödel's theorems addressing the possibility of artificial intelligence one way or another.

We could eventually create human-like artificial intelligence but not fully understand how or why it works.

[T'ai Chi]

Originally posted by phildonnia

As far as we can tell, the physical world does work like + and *.


What does that mean for the world to work like + and * ?

I'm sitting here with a copy of Godel's actual proof in front of me, and I can't manage to see how a theorem for systems of numbers and some operations, which concluded there exists propositions in it that are neither provable or disprovable, translates to the 'real world'.

It is an interesting analogy to be sure, but I just can't see how it relates to the real, as you say, physical, world.

[Kopji]

A practical real world application of Gödel might be computer virus detection (at least polymorphic ones). Since viruses break the normal rules, detecting them also requires, to some extent, getting outside the system 'rules'.

[hammegk]

A myth seems to have arisen concerning Turing's paper of 1936, namely that he there gave a treatment of the limits of mechanism and established a fundamental result to the effect that the universal Turing machine can simulate the behaviour of any machine.

Is homo sap a machine?

What other machine has been proposed that cannot be 'simulated' by a (theoretical) Turing machine, real-time considerations aside?

[Interesting Ian]

Originally posted by Robin
No non-turing machines? Certainly Lucas appears to be suggesting the possibility of one in the second to last paragraph of the article you quoted.

Turing never suggested that there were no non-Turing machines. There are things that happen in real machines like randomness, parallelism that have no place in a Turing machine.

If all physical laws could be expressed algorithmically that would appear to negate the very premise of Lucas's article.

Randomness? You mean intrinsic randomness as described by QM? I don't know what you mean by parallelism.

But sure, a non-turing machine, meaning something that operates by physical processes, some of which cannot be expressed algorithmically, would be immune to such criticisms. But something like randomness could be built into a computer couldn't it?

[drkitten]

Originally posted by Interesting Ian
I admit I know very little about this subject, but did not Goedel show that human beings can know things that cannot be known by computation?


Um, no. He didn't. Not in the slightest. I don't think the word "human" appears at all in any of his work on logic.

[drkitten]

Originally posted by Interesting Ian
But any machine will function according to physical laws. Cannot all physical laws be expressed algorthimically?

No. Again. Re-read your QM, and specifically the Bell Inequalities.

[Interesting Ian]

Originally posted by Kopji
I don't see Gödel's theorems addressing the possibility of artificial intelligence one way or another.

We could eventually create human-like artificial intelligence but not fully understand how or why it works.

If you're declaring it would be conscious, I say that we need to understand what consciousness is first before trying to create it.

[drkitten]

Originally posted by Interesting Ian
But something like randomness could be built into a computer couldn't it?

Into a computer, yes, but not into a Turing machine. You'd need an oracle for randomness, a device that is physically easy to build, but not part of the TM formalism.

And once you've attached such a device to a TM, the resulting assembly is no longer a TM.

[Interesting Ian]

Originally posted by new drkitten
Um, no. He didn't. Not in the slightest. I don't think the word
"human" appears at all in any of his work on logic.

It's supposed to be a direct implication of his proof. You need to justify your assertion "not in the slightest" (this is not to say you're wrong, I simply do not know enough about the subject).


No. Again. Re-read your QM, and specifically the Bell Inequalities.


Well, I would need to read it in the first place. Well look, if algorithms cannot describe QM despite the fact that QM is described by mathematics like all other physical laws, and presumably quantum processes occur in the brain, why do most materialists say that a brain is simply a computer??

[drkitten]

Originally posted by Interesting Ian
It's supposed to be a direct implication of his proof. You need to justify your assertion "not in the slightest" (this is not to say you're wrong, I simply do not know enough about the subject).


Well, his proof applies to axiomatic systems of logic.

Humans aren't axiomatic systems of logic.

Godel's theorem doesn't apply to ham sandwiches, or pieces of chalk, either, for approximately the same reason.




Well, I would need to read it in the first place. Well look, if algorithms cannot describe QM despite the fact that QM is described by mathematics like all other physical laws,

Algorithms and mathematics are not synonymous terms; lots of QM cannot be expressed in algorithmic terms because of the apparent randomness and interactivity involved. It's possible to build an interactive machine with a random oracle, but such a machine is neither axiomatic (unless you postulate the underlying hidden variables that Bell disproved) or nor a Turing machine, by the definition of TM.

[Interesting Ian]

Originally posted by new drkitten
Well, his proof applies to axiomatic systems of logic.

Humans aren't axiomatic systems of logic.

Godel's theorem doesn't apply to ham sandwiches, or pieces of chalk, either, for approximately the same reason.





Algorithms and mathematics are not synonymous terms; lots of QM cannot be expressed in algorithmic terms because of the apparent randomness and interactivity involved. It's possible to build an interactive machine with a random oracle, but such a machine is neither axiomatic (unless you postulate the underlying hidden variables that Bell disproved) or nor a Turing machine, by the definition of TM.

I think the idea might be that QM is irrelevant to the brain because the effects of QM would be too miniscule to have any impact on the functioning of the brain? Thus the brain can be described algorithmically, and hence is essentially a computer.

[drkitten]

Originally posted by Interesting Ian
I think the idea might be that QM is irrelevant to the brain because the effects of QM would be too miniscule to have any impact on the functioning of the brain? Thus the brain can be described algorithmically, and hence is essentially a computer.

Well, that might be the idea, but it's at best unproven (and directly contradicts the theories of Penrose in the opening post, as a start). Notice, however, if you make this argument, you are still faced with the problem that the computer, being interactive, is not an axiomatic system of logic, and Godel's theorem doesn't apply to it. As a general rule, if you want to know whether or not Godel's theorem applies to a given system, simply enumerate the (formal) axioms under which it operates. If you can't do that, then Godel is irrelevant.

There's a very interesting debate about whether or not intelligence can be obtained by a computer.

None of it happens anywhere the word "Godel."

In the context of the artificial intelligence debate, the only meaning that the word "Godel" has is that one of the debaters has essentially zero understanding of any of the fields of logic, psychology, or computer science. This applies to J.R. Lucas, it applies to Roger Penrose, and applies specifically to a number of participants in this thread that forum rules prevent me from naming directly.

[Interesting Ian]

Originally posted by new drkitten


Humans aren't axiomatic systems of logic.




What does this actually mean? Humans operate according to physical laws. These laws are precisely encapsulated by mathematics. So you're saying mathematics is not an axiomatic system of logic??

[AWPrime]

Originally posted by Robin
I have my suspicions in this respect about some JREF contributors

1inChrist?

- posted 24 hours
- posts contain no intelligence

[AWPrime]

Originally posted by AWPrime
Why does the poll question differ from the title question?

Poll question: Will machines ever be able to think as humans do?

Title question: Did Godel disprove the idea of artificial intelligence?


A NO to the title question doesn't mean a NO to the poll question. Is this a attempt to manipulate the poll?

[Bodhi Dharma Zen]

The brain IS NOT a computer.

We dont know what it is "AI"

Godel is complex to understand.

[Interesting Ian]

Originally posted by Bodhi Dharma Zen
The brain IS NOT a computer.


There seem to be quite a lot of people maintaining this. First of all, does anyone disagree?

Secondly, if the brain is not a computer, then this would imply computers cannot produce consciousness, yes?

But a brain can, and presumably therefore people think this is because of QM events in the brain? I presume that ones interaction with the environment does not produce consciousness per se since a computer could collect information from the environment?

I'm just wondering what's so special about QM events that they can magically produce consciousness?

[drkitten]

Originally posted by Interesting Ian
What does this actually mean? Humans operate according to physical laws. These laws are precisely encapsulated by mathematics. So you're saying mathematics is not an axiomatic system of logic??

Fallacy of equivocation.

Humans "operate according to" physical laws. This does not mean that they "are" physical laws.

Physical laws are "described by" mathematics. This does not mean that they "are" mathematics.

Mathematics "includes" axiomatic systems of logic. This does not mean that it "is" an axiomatic system of logic.

Godel's theorem applies to things that "are" axiomatic systems of logic.

[drkitten]

Originally posted by Interesting Ian
There seem to be quite a lot of people maintaining this. First of all, does anyone disagree?

Of course the brain isn't a computer. It's not silicon-based, has no transistors, does not require a battery or external electrical power, et cetera.

[QUOTE]
Secondly, if the brain is not a computer, then this would imply computers cannot produce consciousness, yes?


Why on earth would it imply this? Leopards are spotted. But a giraffe is not a leopard, and so this implies a giraffe cannot be spotted?


But a brain can, and presumably therefore people think this is because of QM events in the brain?


This has indeed been proposed, mostly by people (such as Penrose) with litle knowledge of either the brain or of quantum mechanics. There's very little reason to give intellectual weight to this "presumption," as there's no evidence to support it other than as an abstract and ill-founded supposition.

Other people think that a brain can produce consciousness for a wide variety of other reasons, for which you are handwaved generally to "the literature." It's fair to say that the number of people who actually believe that QM causes consciousness is an extreme minority, mostly spearheaded by Roger Penrose. It is neither a well-regarded nor widely-accepted theory, and the regard and acceptance decrease the higher up the food chain you get.




I presume that ones interaction with the environment does not produce consciousness per se

That has also indeed been proposed (demonstrably, since you just proposed it, but it's also been independently proposed by people such as Searle). It's been the subject of much debate and many scholars (Chalmers, the Churchlands, Hofstadter) reject it out of hand.

since a computer could collect information from the environment?


Irrelevant unless you assume that a computer cannot be conscious, which is at best both unproven and controversial.


I'm just wondering what's so special about QM events that they can magically produce consciousness?

You're not the only one. In context, QM is usually dragged up to support the ill-founded Godelian argument you posed above. But since the argument itself is clearly false, it doesn't matter what mechanism is proposed.

[Interesting Ian]

Originally posted by new drkitten
Fallacy of equivocation.

Humans "operate according to" physical laws. This does not mean that they "are" physical laws.

Physical laws are "described by" mathematics. This does not mean that they "are" mathematics.

Mathematics "includes" axiomatic systems of logic. This does not mean that it "is" an axiomatic system of logic.

Godel's theorem applies to things that "are" axiomatic systems of logic.

Let's get to the nitty gritty. I imagine you would also assert that an android, whose behaviour is indistinguishable from a human being's, would also not be an axiomatic system of logic?

I'll wait for your reply before proceeding.

[drkitten]

Originally posted by Interesting Ian
Let's get to the nitty gritty. I imagine you would also assert that an android, whose behaviour is indistinguishable from a human being's, would also not be an axiomatic system of logic?


Trivially, yes. For example, an android would have weight and occupy space -- an axiomatic system of logic, being an abstraction, does neither.

You're conflating fundamentally different levels of reality here.

[hammegk]

Originally posted by new drkitten
Into a computer, yes, but not into a Turing machine. You'd need an oracle for randomness, a device that is physically easy to build, but not part of the TM formalism.

And once you've attached such a device to a TM, the resulting assembly is no longer a TM.
Interesting contention. Why is the TM itself unable to correctly incorporate a random-generator algorithm, or a multitude of them?


You're not the only one. In context, QM is usually dragged up to support the ill-founded Godelian argument you posed above. But since the argument itself is clearly false, it doesn't matter what mechanism is proposed.
Clearly false?

Are you avering that calcium ions involved in brain function move so rapidly that qm effects are irrelevant? (An effect which of course could be digitally simulated.)

[Interesting Ian]

Originally posted by new drkitten
Originally posted by Interesting Ian
There seem to be quite a lot of people maintaining this. First of all, does anyone disagree?


Dr
Of course the brain isn't a computer. It's not silicon-based, has no transistors, does not require a battery or external electrical power, et cetera.



Let's try to be sensible about this. I mean that people are not maintaining that a brain is effectively a computer.


Secondly, if the brain is not a computer, then this would imply computers cannot produce consciousness, yes?

Dr
Why on earth would it imply this?



Because brains produce consciousness which a priori is an astonishing fact. So it would therefore be extremely surprising if some other physical processes, of quite a characteristically different nature, also produced consciousness. You seem to be leaning towards the view that all things are conscious, even rocks etc.


But a brain can, and presumably therefore people think this is because of QM events in the brain?


Dr
This has indeed been proposed, mostly by people (such as Penrose) with litle knowledge of either the brain or of quantum mechanics. There's very little reason to give intellectual weight to this "presumption," as there's no evidence to support it other than as an abstract and ill-founded supposition.



So hang on a sec. You're denying that the execution of algoritms produce consciousness, and you're denying that throwing QM events into the mix produces consciousness, so therefore what is it that does produce consciousness?? :eek:




Other people think that a brain can produce consciousness for a wide variety of other reasons, for which you are handwaved generally to "the literature." It's fair to say that the number of people who actually believe that QM causes consciousness is an extreme minority, mostly spearheaded by Roger Penrose. It is neither a well-regarded nor widely-accepted theory, and the regard and acceptance decrease the higher up the food chain you get.



I fail to see what else there could be apart from the execution of algorithms or QM events which produce consciousness. Please enlighten me.





II
I presume that ones interaction with the environment does not produce consciousness per se


Dr
That has also indeed been proposed (demonstrably, since you just proposed it, but it's also been independently proposed by people such as Searle). It's been the subject of much debate and many scholars (Chalmers, the Churchlands, Hofstadter) reject it out of hand.



Sorry? They reject what?





II
since a computer could collect information from the environment?


Dr
Irrelevant unless you assume that a computer cannot be conscious, which is at best both unproven and controversial.



Huh? You implied that was your position since a brain is not (effectively) a computer. Do you hold that absolutely all processes/things are conscious then? This would tend to be the consequence of your position.



II
I'm just wondering what's so special about QM events that they can magically produce consciousness?

Dr
You're not the only one. In context, QM is usually dragged up to support the ill-founded Godelian argument you posed above. But since the argument itself is clearly false, it doesn't matter what mechanism is proposed.


I think their position is that the pure execution of algorithms is not capable of generating some truths, but which we can clearly see are true. Therefore something else is required. What else could there be apart from QM? Nothing physical it would seem. I say a non-physical substantial self.

[drkitten]

Originally posted by Interesting Ian
I'll wait for your reply before proceeding.

I've decided to take pity on you and try to frame the question properly that I believe you want to ask. As I said, you are conflating fundamentally different levels of reality when you ask whether or not human beings are axiomatic systems of logic (or something like that), because human beings are concrete entities, not abstractions like logic. You can't even meaningfully ask whether or not the behavior of human beings is an axiomatic system of logic, because behavior, being a series of events, occurs at specific space/time locations.

The question that I think you want to ask is : "Can human behavior [or more specifically cognitive behavior] be modelled by an axiomatic system of logic?"

I think we will agree that this is a very deep question; in a nutshell, it's a major component of the Strong AI hypothesis. A definitive "no" answer would send many current AI researchers scurrying for other fields of practice and rather deplete the ranks of current graduate students. On the other hand, it would also specifically not exclude all the currently taken approaches to AI that are being studied; there are a number of approaches that are not axiomatic or logic based, and even if the above answer were a definitive "no," these other approaches might work.

Now, let's look at what Godel's theorem really says. For a sufficiently complex axiomatic system of logic (systematically complex basically means can do arithmetic correctly), then there is either an undecidable sentence (a sentence that can neither be proven true or proven false), or or a sentence that can be proven both true and false. Hence any such system is either "incomplete" or "inconsistent."

Let us take as a working assumption that the answer to the question above is "yes," that we can describe human cognitive behavior with an axiomatic system. In that case, we either have the case that

1) the system can't do arithmetic correctly
2) there are some statements that the system will never be able to validate as either true or false.
3) there are some statements that the system will believe to be both true and false.

If you make two further assumptions, first that the second case above holds, and second that human beings will be able to determine if the statement in question is true or false, then you have a "proof" that human cognitive behavior is capable of doing something that no axiomatic system can do. In other words, no axiomatic system can perfectly model human cognition. That's the Lucas/Penrose argument in a very quick summary.

It's also total tosh, specifically because of the two "further assumptions" detailed above. Let's look at the three cases further, but apply them to human beings above.

1) the system can't do arithmetic correctly
2) there are some statements that the system will never be able to validate as either true or false.
3) there are some statements that the system will believe to be both true and false.

Case 1 -- is there anyone here who claims to be able to do arithmetic "correctly"? As in, never makes a mistake, and can handle problems of unlimited size? I doubt it. Humans have rather serious cognitive limitations in terms of, for example, attention span, short-term memory, and reliability/reproducibility of cognitive tasks. If humans really could do arithmetic "correctly," everyone in the world would have gotten perfect scores on their standardized tests. I think it's fairly clear that an axiomatic system that perfectly models human cognitive behavior wouldn't necessarily do arithmetic "correctly."

Similarly, case 3 -- there's lots of work done, including Nobel-prize work, about the various long-standing cognitive biases that cause people to simultaneously believe contradictory things. Type "Linda is a feminist bank teller" into your web browser if you don't believe me. Humans simply aren't perfectly consistent reasoners, and so there's no reason to believe their models should be either.

So what we're really talking about if you want to make the Lucas/Penrose argument about AI go through is not a statement about human-like intelligence, but about God-like inhuman intelligence, the sort that has access to truths and abilities beyond the merely human. But in this case, if we already assume that such truths exist, then we've already assumed that case 2 above applies, not only to the formal system, but to human cognition itself. We've assumed that there are statements whose truth is inaccessible to humans. In this case, it should be no surprise that a detailed model of us finds that exact same truth inaccessible. Now, a different axiomatic system of logic may be able to establish the truth of the proposition, but only via a chain of logic so long and so convoluted that we will never understand or be able to follow the proof. In fact, we already have a candidate for such a statement in the four-color theorem, which has been "proved," but by a computer program of such complexity that no human could duplicate or even understand the proof in its entirety. This theorem may well be our equivalent of the Godel sentence, because the fundamental limitations on how we perform our thoughts limit our ability to understand that proof.

So, basically, Godel's theorem says nothing at all about whether or not there are truths inaccessible to humans, except in the very general statement that if you assume (without justification) that the theorem applies, then one of three cases holds --- and a good case, consistent with our current knowledge of psychology, can be made for any of the three cases. Alternatively, you can assume that Godel's theorem does not apply to human cognition, in which case all bets are off for both the humans and for the computers that model them. In no case does quantum mechanics amount to more than an intellectual detour, and usually an attempt to blind the readership with science that they don't understand.

In that regard, it's fairly good. Most people who understand mathematics well enough to know Godel know little psych or QM. Most physicists don't know psych. Et cetera. Unfortunately, the set of people who know all three fields do not include either Lucas, Penrose, or a number of unnamed participants on this thread.

[drkitten]

Originally posted by hammegk
Interesting contention. Why is the TM itself unable to correctly incorporate a random-generator algorithm, or a multitude of them?


For approximately the same reason you can't have three wheels on a unicycle. If you build a unicycle, and put two more wheels on it, it's no longer a unicycle.

The notion of a "random-generator algorithm" is inherently contradictory and does not exist. What laymen call "random number generators" are more properly called "pseudo-random number generators," as the number they generate aren't really random, but look random enough for most purposes if you don't examine them really closely. For people for whom the quality of random numbers really matters (e.g. cryptographers), there's a tremendous amount of research effort establishing and improving the statistical quality of algorithmically generated sequences, exactly because Turing machines, and by extension algorithms in general, cannot generate them.



Clearly false?

Are you avering that calcium ions involved in brain function move so rapidly that qm effects are irrelevant? (An effect which of course could be digitally simulated.)

No. I am averring that the argument that Godel's theorem prohibits axiomitization of human-like intelligence is clearly false. If you do not assume that human-like intelligence is non-axiomatizable, there is no reason to invoke quantum mechanics to explain why.

(I also note in passing that digital similation is insufficient, due to the butterfly effect.)

[drkitten]

Originally posted by Interesting Ian
Let's try to be sensible about this. I mean that people are not maintaining that a brain is effectively a computer.


Assuming that by "computer" you mean algorithmic computational device, then, um, you're flat-out wrong. Lots of people maintain this -- Dennett, McCarthy, the Churchlands, Minsky, Hofstadter, Fodor, Berwick, Pinker, Marcus, Plunkett, &c.

The question is not whether the position is popular, but whether
it is true.




Because brains produce consciousness which a priori is an astonishing fact. So it would therefore be extremely surprising if some other physical processes, of quite a characteristically different nature, also produced consciousness.


Ah, yes. We call this "argument from incredulity" and fail students for using it where I come from. Lots of surprising things happen. Is this one of them? Do you have any evidence that it isn't?

I'm not affirming, or denying, anything. I am specifically rejecting arguments that cannot be supported either rationally (through reasoned, non-fallacious argumentation) or empirically (through generally available and accepted "scientific" data).



I think their position is that the pure execution of algorithms is not capable of generating some truths, but which we can clearly see are true.


This is indeed their position. It is at best unproven, since they have given no explanation about how we can clearly see they are true. If they use that as an assumption, then they're clearly begging the question. (As such, it's not rationally supported.) It also contradicts widely available and accepted data -- for example, I can easily show you an algorithm that is capable of generating every truth (but also every falsehood). Therefore, the statement that "the pure execution of algorithms is not capable of generating some truths" is not empirically supported, either, and they're committing a simple error in logical reasoning. Finally, their assessment of human capabilities contradicts widely available psychological data.

In other words, bollocks to them.

[hammegk]

Originally posted by new drkitten
For approximately the same reason you can't have three wheels on a unicycle. If you build a unicycle, and put two more wheels on it, it's no longer a unicycle.
If you say so.


The notion of a "random-generator algorithm" is inherently contradictory and does not exist. .... there's a tremendous amount of research effort establishing and improving the statistical quality of algorithmically generated sequences, exactly because Turing machines, and by extension algorithms in general, cannot generate them.
So true, in (practical) fact. We could also debate what is or even could be actually "random" in any finite system.

Yet what is wrong with my position that any "random number" no matter how generated can be simulated by a Turing Machine?


If you do not assume that human-like intelligence is non-axiomatizable, there is no reason to invoke quantum mechanics to explain why.
Agreed, although I haven't invoked anything. My understanding is that qm effects do govern calcium ion propagation in the brain.

And if you do assume human intelligence is an axiom, hello Turing machine.


(I also note in passing that digital similation is insufficient, due to the butterfly effect.)
Not currently forwards computable, but amenable to simulation in any case.

[Interesting Ian]

Originally posted by new drkitten
Trivially, yes. For example, an android would have weight and occupy space -- an axiomatic system of logic, being an abstraction, does neither.

You're conflating fundamentally different levels of reality here.

Indeed, and I am doing so in a forlorn endeavour to ease communication. The android is conscious and is merely the execution of algorithms. This very strongly suggests that the brain is merely the execution of algorithms too.

Now let's have a straight answer from you instead of going off into an irrelevant tangency like you are prone to do.

Do you agree that the brain is merely a algorithmic machine or not?

[MESchlum]

One point I got out of reading GEB (and other things) is that the undecidable postulates in your system are not true or false in an absolute sense.

Example: geometry. Euclid's postulate about parallel lines is non provable using the others. Assume it's true, you get Euclidian geometry. Assume it's false, you get non-Euclidian geometry.

Both geometries work, in their specific contexts. The undecidable postulate can be true, or false, and you will get a different system in either case - but both systems are valid.

The assumption that it is possible to "know" whether an undecidable postulate is correct or not is false. One version can seem more obvious (Euclidian geometry), but this does not invalidate the other.

This makes me ignore claims stating that humans (or whatever) can go beyond "mere" algorithms because they "know" if an undecidable proposition is true or not. If you have a better argument, I'm happy to hear it - but this one, as stated, does not hold water (to me).

[Interesting Ian]

Originally posted by new drkitten


The question that I think you want to ask is : "Can human behavior [or more specifically cognitive behavior] be modelled by an axiomatic system of logic?"



Before even reading the rest of your post, I just want to address this. I wasn't trying to ask this question, because surely it obviously can be?? Even *I*, who believes that we are souls, thinks that our behaviour can be so modelled.

Why?

Because if enough knowledge were available regarding someone's psyche, and therefore we can in principle predict exactly how that person will react under appropriate circumstances, then surely this can be modelled using the appropriate algorithms?? In practice an appropriately programmed android's behaviour would be indistinguishable from a human being's (but maybe not absolutely indistinguishable).

No, I was trying to ask the question:

"Can consciousness be modelled by an axiomatic system of logic?" Or maybe, "is consciousness nothing but the execution of algorithms"?

Now to read the rest of your post. Thanks for taking the time to respond in detail :)

[69dodge]

Originally posted by new drkitten
for example, I can easily show you an algorithm that is capable of generating every truth (but also every falsehood).Yes, exactly.

People are getting their quantifiers mixed up. Goedel didn't show that some truth exists which no algorithm can produce. He showed that each algorithm has some truth, possibly specific to it, which it cannot produce. (Unless the algorithm also produces some falsehoods, that is. But then it's a "bad" algorithm, so we don't much care about it.) But another algorithm always exists which can produce that truth.

Of course, the other algorithm has its own limitations.

Which a third algorithm doesn't have.

And so on, ad infinitum.

So when a person "intuitively sees" the truth of a Goedel sentence, it is not the case that he knows something that no algorithm can produce; it is merely the case that he knows something that one particular algorithm can't produce. A Goedel sentence is only a Goedel sentence relative to a particular axiomatic system. It's not provable within that system, but other systems exist in which it is provable.

Unless people are infinitely smart, which clearly they aren't, Goedel's theorem provides no reason to suppose that there's more to their reasoning abilities than algorithms.

[Interesting Ian]

Originally posted by new drkitten
I've decided to take pity on you and try to frame the question properly that I believe you want to ask. As I said, you are conflating fundamentally different levels of reality when you ask whether or not human beings are axiomatic systems of logic (or something like that), because human beings are concrete entities, not abstractions like logic. You can't even meaningfully ask whether or not the behavior of human beings is an axiomatic system of logic, because behavior, being a series of events, occurs at specific space/time locations.

The question that I think you want to ask is : "Can human behavior [or more specifically cognitive behavior] be modelled by an axiomatic system of logic?"

I think we will agree that this is a very deep question;



Not at all; surely it can be?

[Dr Adequate]

Originally posted by Interesting Ian
I admit I know very little about this subject, but did not Goedel show that human beings can know things that cannot be known by computation?
No.

[Interesting Ian]

Originally posted by new drkitten

The question that I think you want to ask is : "Can human behavior [or more specifically cognitive behavior] be modelled by an axiomatic system of logic?"

I think we will agree that this is a very deep question; in a nutshell, it's a major component of the Strong AI hypothesis. A definitive "no" answer would send many current AI researchers scurrying for other fields of practice and rather deplete the ranks of current graduate students. On the other hand, it would also specifically not exclude all the currently taken approaches to AI that are being studied; there are a number of approaches that are not axiomatic or logic based, and even if the above answer were a definitive "no," these other approaches might work.

Now, let's look at what Godel's theorem really says. For a sufficiently complex axiomatic system of logic (systematically complex basically means can do arithmetic correctly), then there is either an undecidable sentence (a sentence that can neither be proven true or proven false), or or a sentence that can be proven both true and false. Hence any such system is either "incomplete" or "inconsistent."

Let us take as a working assumption that the answer to the question above is "yes," that we can describe human cognitive behavior with an axiomatic system. In that case, we either have the case that


1) the system can't do arithmetic correctly
2) there are some statements that the system will never be able to validate as either true or false.
3) there are some statements that the system will believe to be both true and false.

If you make two further assumptions, first that the second case above holds, and second that human beings will be able to determine if the statement in question is true or false, then you have a "proof" that human cognitive behavior is capable of doing something that no axiomatic system can do. In other words, no axiomatic system can perfectly model human cognition. That's the Lucas/Penrose argument in a very quick summary.

It's also total tosh, specifically because of the two "further assumptions" detailed above. Let's look at the three cases further, but apply them to human beings above.

1) the system can't do arithmetic correctly
2) there are some statements that the system will never be able to validate as either true or false.
3) there are some statements that the system will believe to be both true and false.

Case 1 -- is there anyone here who claims to be able to do arithmetic "correctly"? As in, never makes a mistake, and can handle problems of unlimited size? I doubt it. Humans have rather serious cognitive limitations in terms of, for example, attention span, short-term memory, and reliability/reproducibility of cognitive tasks. If humans really could do arithmetic "correctly," everyone in the world would have gotten perfect scores on their standardized tests. I think it's fairly clear that an axiomatic system that perfectly models human cognitive behavior wouldn't necessarily do arithmetic "correctly."



Well of course it wouldn't. But the point is that it couldn't whether or not it models human cognitive behaviour. Also the fact that in practise no-one can do arithmetic perfectly is not relevant. I think the argument might be (although I don't know, only having as much knowledge as a average man in the street regarding Goedel's theorem) that in principle there is no reason we couldn't. But in principle there is a reason why a computer couldn't. Therefore we are not algorithmic machines.

[Interesting Ian]

Originally posted by new drkitten


Similarly, case 3 -- there's lots of work done, including Nobel-prize work, about the various long-standing cognitive biases that cause people to simultaneously believe contradictory things. Type "Linda is a feminist bank teller" into your web browser if you don't believe me. Humans simply aren't perfectly consistent reasoners, and so there's no reason to believe their models should be either.



None of this matters. There is no logical reason why we are inconsistent; only psychological reasons . .or reasons of intellectual deficiency etc.

Although I know nothing about mathematics, or Goedel's theorem, from what little knowledge I do know, I'm pretty sure you're simply not getting the arguments employed to show that we are not simply algorithmic machines.

I agree though I can't really argue for it. I might look into this Goedel stuff some more

(PS how do you put those 2 dots above the "o"?)

[Interesting Ian]

Originally posted by new drkitten
Originally posted by hammegk
Interesting contention. Why is the TM itself unable to correctly incorporate a random-generator algorithm, or a multitude of them?


Dr
For approximately the same reason you can't have three wheels on a unicycle. If you build a unicycle, and put two more wheels on it, it's no longer a unicycle.

The notion of a "random-generator algorithm" is inherently contradictory and does not exist. What laymen call "random number generators" are more properly called "pseudo-random number generators," as the number they generate aren't really random, but look random enough for most purposes if you don't examine them really closely. For people for whom the quality of random numbers really matters (e.g. cryptographers), there's a tremendous amount of research effort establishing and improving the statistical quality of algorithmically generated sequences, exactly because Turing machines, and by extension algorithms in general, cannot generate them.


Not all "random" number generators are pseudo-random number generators. Indeed, pseudo-random number generators are not random at all, on the converse, they are determined!




No. I am averring that the argument that Godel's theorem prohibits axiomitization of human-like intelligence is clearly false. If you do not assume that human-like intelligence is non-axiomatizable, there is no reason to invoke quantum mechanics to explain why.

(I also note in passing that digital similation is insufficient, due to the butterfly effect.) [/B]

Why does everyone keep talking about intelligence rather than consciousness?? Of course computers can be "intelligent", but that's not interesting. I want to know if they could be conscious!

[Interesting Ian]

Originally posted by new drkitten
Originally posted by Interesting Ian
Let's try to be sensible about this. I mean that people are not maintaining that a brain is effectively a computer.


Dr
Assuming that by "computer" you mean algorithmic computational device, then, um, you're flat-out wrong. Lots of people maintain this -- Dennett, McCarthy, the Churchlands, Minsky, Hofstadter, Fodor, Berwick, Pinker, Marcus, Plunkett, &c.

The question is not whether the position is popular, but whether
it is true.


I meant people on this thread like yourself. But apparently I was getting the wrong impression in your case. And yup, I wasn't talking about quantum computers ;)





II
Because brains produce consciousness which a priori is an astonishing fact. So it would therefore be extremely surprising if some other physical processes, of quite a characteristically different nature, also produced consciousness.

Dr

Ah, yes. We call this "argument from incredulity" and fail students for using it where I come from.



I'm worried that they fail students for what they erroneously regard as an "argument from incredulity".
My argument is not. Materialism has just one problem, and it's a huge one. Namely how do brain processes produce consciousness? Now you want to compound that by saying that brain processes are not algorithmic but produce consciousness, but also algorithmic processes produce consciousness too. Now, if you're maintaining that not everything is conscious (eg an obvious algorthmic process such as a boulder rolling down a hill)), then basically you're making your metaphysic that much more complex than is apparently warranted. This is because you're saying some algorithmic processes produce consciousness, but others don't.

BTW, these students you fail? I guess they're not allowed to dispute their failure by show that it is in fact you who is in error?? :rolleyes: Typical of teachers/lecturers. Just justifies my often repeated contention that formal education is a waste of time. For a kick off, too many teachers/lecturers are intellectual deficient.




II
I think their position is that the pure execution of algorithms is not capable of generating some truths, but which we can clearly see are true.


Dr
This is indeed their position. It is at best unproven, since they have given no explanation about how we can clearly see they are true.



Yes, this is because an explanation is not possible. It's a sudden understanding. Maybe we're making temporary contact with the Platonic world of forms?



If they use that as an assumption, then they're clearly begging the question. (As such, it's not rationally supported.)



Well, are they not pointing to certain limitations as to the truths the execution of algorithms can reach? Are we like wise logically limited? Only psychologically limited it would seem, which is a completely different thing.


It also contradicts widely available and accepted data -- for example, I can easily show you an algorithm that is capable of generating every truth (but also every falsehood). Therefore, the statement that "the pure execution of algorithms is not capable of generating some truths" is not empirically supported, either, and they're committing a simple error in logical reasoning.


I'm sorry, but I do not understand this error.



Finally, their assessment of human capabilities contradicts widely available psychological data.

In other words, bollocks to them.



I haven't read their arguments, so I do not know what they are. But I'm guessing they're saying that there is no logical reason why we cannot get to know all truths, although they would certainly acknowledge the psychological reasons.

[69dodge]

Originally posted by MESchlum
The assumption that it is possible to "know" whether an undecidable postulate is correct or not is false.How did Goedel show that the sentence he constructed was undecidable, in the first place? He did it by constructing a sentence that said, basically, "I'm not provable". Such a sentence can't be provable, if the system is to be consistent. (If it were provable and yet it says, "I'm not provable", that would mean it's false. But only in an inconsistent system could a false sentence be provable.)

Well, if says it's not provable, and in fact it's not provable, that means it's true, right?

[drkitten]

Originally posted by hammegk

So true, in (practical) fact. We could also debate what is or even could be actually "random" in any finite system.

Yet what is wrong with my position that any "random number" no matter how generated can be simulated by a Turing Machine?


The same thing that is wrong with a position that any number of wheels can be attached to a unicycle.

Basically, what's wrong is the meanings of the words you are using.

[Interesting Ian]

Originally posted by 69dodge
Yes, exactly.

People are getting their quantifiers mixed up. Goedel didn't show that some truth exists which no algorithm can produce. He showed that each algorithm has some truth, possibly specific to it, which it cannot produce. (Unless the algorithm also produces some falsehoods, that is. But then it's a "bad" algorithm, so we don't much care about it.) But another algorithm always exists which can produce that truth.

Of course, the other algorithm has its own limitations.

Which a third algorithm doesn't have.

And so on, ad infinitum.

So when a person "intuitively sees" the truth of a Goedel sentence, it is not the case that he knows something that no algorithm can produce; it is merely the case that he knows something that one particular algorithm can't produce. A Goedel sentence is only a Goedel sentence relative to a particular axiomatic system. It's not provable within that system, but other systems exist in which it is provable.

Unless people are infinitely smart, which clearly they aren't, Goedel's theorem provides no reason to suppose that there's more to their reasoning abilities than algorithms.

They don't need to be infinitely smart. They just need to be not logically limited unlike any algorithm is.

[drkitten]

Originally posted by Interesting Ian
I think the argument might be (although I don't know, only having as much knowledge as a average man in the street regarding Goedel's theorem) that in principle there is no reason we couldn't.


Well, no. In principle, human beings are limited by their psychological abilities as much as their physical ones, which is why concepts like "short term memory" and "digit span" exist.

Talking about "in principle there is no reason why we couldn't have an infinite digit span is exactly as realistic as a statement that "in principle, there's no reason we couldn't jump 300m in the air without aid." You're talking about an utterly unrealistic idealization of the human capacity here.


But in principle there is a reason why a computer couldn't. Therefore we are not algorithmic machines.

The fundamental reason that axiomatic systems cannot do everything is because they're finite. The fundamental reason that humans cannot do anything is because they're finite, too. Not only is there a reason, in principle, but in extremely broad terms it's the same darn reason.

[drkitten]

Originally posted by Interesting Ian

Although I know nothing about mathematics, or Goedel's theorem, from what little knowledge I do know, I'm pretty sure you're simply not getting the arguments employed to show that we are not simply algorithmic machines.

Should this go down as another classic Ian quote? "Although I know nothing about the subject, I'm pretty sure you're not getting it."

You're right. If you assume, first of all, that human capacity is not limited by their physical or psychological makeup, and second of all, that human beings are capable of performing acts that are logically impossible, then there's no reason to conclude that they are in any way limited in the same way that systems limited by logic, physics, or psychology would be. If men were gods, they would have different capacities.

I just consider that an unrealistic starting assumption. In what way do you consider a human with an infinitely long lifespan, an infinitely long attention span, and total recall of every detail s/he has ever experienced to be a legitimate standard of human capacity? But even if we assume that such a creation existed, there's still nothing in evidence that such a creature would be able to unerringly sort truth from falsehood, or would be able to determine every truth in the universe. In fact, Tarski's work suggests quite the opposite.

[MESchlum]

Originally posted by 69dodge
How did Goedel show that the sentence he constructed was undecidable, in the first place? He did it by constructing a sentence that said, basically, "I'm not provable". Such a sentence can't be provable, if the system is to be consistent. (If it were provable and yet it says, "I'm not provable", that would mean it's false. But only in an inconsistent system could a false sentence be provable.)

Well, if says it's not provable, and in fact it's not provable, that means it's true, right?

Well...

If:

*a statement that is not false is true
* the system is consistent

Then you've just proven (within the system) that the statement is true. So it must be false, since it's been proven. Of course, if a statement that is not false can also be not true, you have no problem (besides, perhaps, consistency?)

By my interpretation of the setup (which, I will grant, is a bit rusty), we can build a new system, containing the statement, and use that.

"I'm not provable in system X"

Is not provable in X, and can be assumed to be true (in system Y which contains X and "the sentence is provable in Y") or not (system Z). You can then build up a sentence that says "I'm not provable in Y", or course.

[Interesting Ian]

Originally posted by new drkitten
Originally posted by Interesting Ian
I think the argument might be (although I don't know, only having as much knowledge as a average man in the street regarding Goedel's theorem) that in principle there is no reason we couldn't.


Dr
Well, no. In principle, human beings are limited by their psychological abilities as much as their physical ones, which is why concepts like "short term memory" and "digit span" exist.

Talking about "in principle there is no reason why we couldn't have an infinite digit span is exactly as realistic as a statement that "in principle, there's no reason we couldn't jump 300m in the air without aid." You're talking about an utterly unrealistic idealization of the human capacity here.



Yes. But it doesn't matter. It still doesn't alter the case that we are not logically limited. Any algorithm (but not of course an infinte number of them) is logically limited. You need to argue that we are likewise logically limited, not merely psychologically limited or even nomologically limited.



II
But in principle there is a reason why a computer couldn't. Therefore we are not algorithmic machines.


Dr
The fundamental reason that axiomatic systems cannot do everything is because they're finite. The fundamental reason that humans cannot do anything is because they're finite, too. Not only is there a reason, in principle, but in extremely broad terms it's the same darn reason.

No, this is an error. Although the reason might be because of finite capacities in both instances, that doesn't entail that that which limits in both cases is identical.

You gat a fail Dr Kitten!

Love to be in one of your classes and say that! :D

[drkitten]

Originally posted by Interesting Ian
Yes. But it doesn't matter. It still doesn't alter the case that we are not logically limited. Any algorithm (but not of course an infinte number of them) is logically limited. You need to argue that we are likewise logically limited, not merely psychologically limited or even nomologically limited.


I will be happy to do so when you provide working definitions of the distinctions between logical limitations, psychological limitations, and nomological limitations. Because from where I sit, a psychological limitation such as short term memory can be proven to be isomorphic and equivalent to a logical limitation.

(Actually, that particular isomorphism is rather fundamental to a lot of the field called "theory of computation" and appears in a lot of contexts. For example, any finite approximation to the set of all palindromes is finitely computable (and can be captured by a regular expression), but the set itself is not. The fundamental difference here can be expressed as a "psychological" limitation on the capacity of the processing unit, or alternatively as a formal, logical limitation on the sets themselves. So you see, the difference that you attempt to draw is one that most practitioners in the field would disbelieve. Not merely ignore, but actively claim to be nonexistent.)

[Bodhi Dharma Zen]

Originally posted by new drkitten
Fallacy of equivocation.

Humans "operate according to" physical laws. This does not mean that they "are" physical laws.

Physical laws are "described by" mathematics. This does not mean that they "are" mathematics.

Mathematics "includes" axiomatic systems of logic. This does not mean that it "is" an axiomatic system of logic.

Godel's theorem applies to things that "are" axiomatic systems of logic.

Well exposed. A great percentage of the discussions on this forum would be absurd if we all knew how to argue.

[Interesting Ian]

Originally posted by Bodhi Dharma Zen
Originally posted by new drkitten
Fallacy of equivocation.

Humans "operate according to" physical laws. This does not mean that they "are" physical laws.

Physical laws are "described by" mathematics. This does not mean that they "are" mathematics.

Mathematics "includes" axiomatic systems of logic. This does not mean that it "is" an axiomatic system of logic.

Godel's theorem applies to things that "are" axiomatic systems of logic.


Bodhi Dharma Zen
Well exposed. A great percentage of the discussions on this forum would be absurd if we all knew how to argue.

I don't think it is well exposed at all. He is being deliberately pedantic in order to ignore the essence of my points. In other words it was a complete irrelevancy.

[Interesting Ian]

Originally posted by new drkitten
I will be happy to do so when you provide working definitions of the distinctions between logical limitations, psychological limitations, and nomological limitations. Because from where I sit, a psychological limitation such as short term memory can be proven to be isomorphic and equivalent to a logical limitation.



Really, what can I say to a person who cannot understand, on one hand, the distinction between a psychological limitation/nomological limitation, and on the other hand a logical limitation?

A logical limitation means that you cannot achieve the logically impossible -- it is inherently inconsistent. For example, it is logically impossible for an object to be both simultaneously a sphere and a cube. A psychological/nomological limitation simply means that, as a matter of fact, we cannot do something -- although if circumstances have been different we could have. For example, in another logically possible universe, we could have been like gods and have stupendous powers. There is no logical inconsistency in this. However, for any algorithm, even with the element of randomness introduced within it, it will always be logically impossible, in any logically possible universe , for it to prove some thing that happens to be true. But it seems that we are not likewise logically limited.

[hammegk]

Originally posted by new drkitten
The same thing that is wrong with a position that any number of wheels can be attached to a unicycle.
Strange sentence, huh?


Basically, what's wrong is the meanings of the words you are using.
Basically, what's wrong is the meanings of the words you are using.


Basically, what's wrong is the meanings of the words you are using.


Basically, what's wrong is the meanings of the words you are using.


BTW, what is "consciousness"? Do you contend Strong AI is "conscious"?

[MESchlum]

Originally posted by Interesting Ian
Really, what can I say to a person who cannot understand, on one hand, the distinction between a psychological limitation/nomological limitation, and on the other hand a logical limitation?

A logical limitation means that you cannot achieve the logically impossible -- it is inherently inconsistent. For example, it is logically impossible for an object to be both simultaneously a sphere and a cube.

[Raise hand] Me! Me! Me!

Consider a diameter on a sphere. It's a circle, right? Now look at four points placed at regular intervals on the diameter. The diameter is therefore (relative to the sphere) a quadrilateral with all sides equal. What's more, the angle at each point is the same, so it's a square.

Hence, with the proper geometry (non-Euclidian, by the way, see other posts) you can have a square that is simultaneously a circle.

Go up a dimension or two, and I'm not sure if it still works, but it should...

A psychological/nomological limitation simply means that, as a matter of fact, we cannot do something -- although if circumstances have been different we could have. For example, in another logically possible universe, we could have been like gods and have stupendous powers. There is no logical inconsistency in this.

Okay. So a "logical impossibility" is something that contradicts the basic axioms used (1+1=3 if you have stated earlier that 1+1=2), while the other kind is something else?

Listing all the axioms and seeing how they interact will get you into a lot of trouble, but if that's what you want...

By the way, if my logically stupendous power is to be immovable and yours is to move anything you like, what happens?

However, for any algorithm, even with the element of randomness introduced within it, it will always be logically impossible, in any logically possible universe , for it to prove some thing that happens to be true. But it seems that we are not likewise logically limited.

I'm confused.

There are statements that, within a given system cannot be proved. You can use a larger system, that contains the old one, and postulates the statement is true. Or you can postulate it's false (Euclidian vs. non-Euclidian geometry, both are valid).

So this "we" you mention is limiting itself to a single answer to every issue where two are feasible, and valuable? Give me a computer that includes the curvature of space (and so can deal with Euclidian and non-Euclidian geometry) instead.

Now if you're arguing about understanding a system, and its limitations, I'll grant that we've got a head start over computers. We can often "see", for simple systems, where the inconsistencies lie (parallel postulate, division by zero, etc.).

Though a modern computer armed with geometry (minus the parallel postulate) would not intuit the postulate, it should (via Godel's technique) be able to find statements that are non provable. Then, by assuming one such statement is true, it would derive a new geometry A. And by assuming it's false, it would derive a new geometry B.

[Interesting Ian]

Here is a appropriate quote from that article I referenced earlier which answers some of the objections that people have raised.


This is the answer to one objection put forward by Turing.3 He argues that the limitation to the powers of a machine do not amount to anything much. Although each individual machine is incapable of getting the right answer to some questions, after all each individual human being is fallible also: and in any case "our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing simultaneously over all machines." But this is not the point. We are not discussing whether machines or minds are superior, but whether they are the same. In some respect machines are undoubtedly superior to human minds; and the question on which they are stumped is admittedly, a rather niggling, even (118) trivial, question. But it is enough, enough to show that the machine is not the same as a mind. True, the machine can do many things that a human mind cannot do: but if there is of necessity something that the machine cannot do, though the mind can, then, however trivial the matter is, we cannot equate the two, and cannot hope ever to have a mechanical model that will adequately represent the mind. Nor does it signify that it is only an individual machine we have triumphed over: for the triumph is not over only an individual machine, but over any individual that anybody cares to specify---in Latin {50} quivis or quilibet, not quidam---and a mechanical model of a mind must be an individual machine. Although it is true that any particular "triumph" of a mind over a machine could be "trumped" by another machine able to produce the answer the first machine could not produce, so that "there is no question of triumphing simultaneously over all machines", yet this is irrelevant. What is at issue is not the unequal contest between one mind and all machines, but whether there could be any, single, machine that could do all a mind can do. For the mechanist thesis to hold water, it must be possible, in principle, to produce a model, a single model, which can do everythi