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The Emeror's New Mind

#1 User is offline   helene_t 

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Posted 2006-June-22, 07:04

I just read The Emeror's New Mind by Roger Penrose.

I think his presentations of general relativity and quantum mechanics are excelent (even to the point that I could understand parts of it), but his main point is still troubling me. It goes something like this:

1) There is more to the mind (and, in particular, to consciouness) than a Turing Machine.
2) Quantum mechanics is incomplete in that it cannot explain why a particle's wave function must collapse before we can observe the particle.
3) A new physical paradigma that can predict when wave functions collapse will be the key to a theory of beyond-Turing information procesing and, in particular, to consciousness.

His main case for 1) is that Goedel's theorem gives us an example of a proposition which cannot be proved by a Turing machine yet with human insight it can be "seen" to be true. I have a vague feeling that Goedel's theorem is just yet another "This statement is false" paradox, but of course, there must be more to it since logicians consider it important. However, since it's trivial to write a Turing machine that can reproduce Goedel's own proof of his theorem, this interpretation seems to be a contradiction. Obviously, I did not understand this correctly. But even so, it puzzles me that such a simple thing can give rise to controversy. After all, therre are also some clever minds who keep believing in strong AI and thus they must somehow interpret Goedl's theorem differently.

2) seems so obvious the way he explains it that it's hard to see how anyone could disagree. But apparently, Penrose's view is highly controversial. What are the alternatives?

As for 3), it seems very odd to me to speculate about a relation between physics and psychology. The hard problem seems to be to define what consciousness is. Before we have done that, I think it's premature to speculate about how to explain it in terms of neuropfysiology, let alone in terms of particle physics. We don't even know what kind of consequences Penrose's yet-to-be-discovered theory will have for chemical bonds, let alone for neurophysiological processes!

Any thoughts?
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#2 User is offline   david_c 

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Posted 2006-June-22, 07:31

helene_t, on Jun 22 2006, 02:04 PM, said:

As for 3), it seems very odd to me to speculate about a relation between physics and psychology.

All I can say is that when I read the book this seemed very "right" to me. I can't really explain why, just that it goes along with my intuition.
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#3 User is offline   han 

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Posted 2006-June-22, 08:24

I think that those humans who think that their intuition can tell them something more about (for example) the axiom of choice are misguided. So if this is really Penrose's arguement for (1) then I think that it is rubbish. I don't know enough about (2) or (3), nor have I read the book.
Please note: I am interested in boring, bog standard, 2/1.

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#4 User is offline   mike777 

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Posted 2006-June-22, 09:12

I recently reread this book. It is the counterargument to Kurzweil, see my other posts.


What does it mean to be self aware? Even if computers cannot become self aware can they become intelligent and selfreplicate? In any case Robots have been involved in a reported 77 human deaths.

Kurzweil points out the blurring of man and machine. We know we can put machine parts into humans and we know we can put Human DNA into machines.

Kurzweil argues that a speed of 10^16 computers come close to matching the speed of the human mind. Along with similiar growth factors in memory, access speed etc. he expects to reach this around 2020. With increases in brain mapping he expects to see software developments around 2030 and speeds reaching the combined speed of all human brains around 2040 10^30 and by 2050 speeds in the range of 10^43. This is when he argues the singularity occurs. The point when machines become self aware and the future becomes unknowable.

In the Turing tests, tests whether we can tell the difference whether we are speaking to a machine or a human, he argues with the blending of man and machine that definitions become critical.

Even tv shows have lightly touched on this subject. See Battlestar Galattica where machines become self aware, battle humans and they are having sex and babies with each other blending the dividing line between man and machine.
Think of the Borg where human babies are born on baby farms but with machine implants.
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#5 User is offline   barmar 

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Posted 2006-June-22, 09:53

I've never read this book, although I've known for a long time that I probably should. But I've read numerous other books that respond to his points, so I think I have an idea of the gist of his argument. And I just don't believe it.

It seems like his argument is basically: We don't understand how the mind works, and we can't seem to reproduce it with a deterministic process. The collapse of the wave function is a non-deterministic process, I guess it must be the basis of how the mind works. The problem is that quantum mechanics operates at too microscopic a level of the universe to have a direct correspondence to the macroscopic behavior of a collection of neurons.

What I think is actually going on is that the brain is a very complex, dynamic system. Dynamic systems give the appearance of non-determinism, simply because there are too many states for us to track. I think this is basically what Chaos Theory is about.

Perhaps quantum mechanics is also non-deterministic because it's also a dynamic system, maybe involving strings, branes, or whatever the subatomic theory is of the day. But these are still very different levels of the universe, and it seems unlikely that there is any direct correlation between them.

#6 User is offline   helene_t 

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Posted 2006-June-22, 10:08

That's excatly the way I think about it, Barmar. You put it very well.
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#7 User is offline   Sigi_BC84 

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Posted 2006-June-22, 11:13

helene_t, on Jun 22 2006, 03:04 PM, said:

His main case for 1) is that Goedel's theorem gives us an example of a proposition which cannot be proved by a Turing machine yet with human insight it can be "seen" to be true. I have a vague feeling that Goedel's theorem is just yet another "This statement is false" paradox, but of course, there must be more to it since logicians consider it important. However, since it's trivial to write a Turing machine that can reproduce Goedel's own proof of his theorem, this interpretation seems to be a contradiction. Obviously, I did not understand this correctly. But even so, it puzzles me that such a simple thing can give rise to controversy. After all, therre are also some clever minds who keep believing in strong AI and thus they must somehow interpret Goedl's theorem differently.

There are some misconceptions about Gödel's Incompleteness Theorem here.

First of all, it is not considered merely "important" but one of the most profound theorems dealing with logic and provability. At the beginning of the century, one of the goals of mathematics was to find an algorithm that could be used to prove all of mathematics -- so to speak the universal maths problem solver. Gödel showed with his proof that this is, in principle, unachievable.

Gödel's Theorem is not paradoxical in any way. It makes a statement about formal systems (possibly inconsistent ones), but that does not make it paradoxical or inconsistent itself.

From the Wikipedia article (http://en.wikipedia.org/wiki/Goedel's_incompleteness_theorem):

Gödel's Incompleteness Theorem: For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable in the theory. That is, any consistent theory of a certain expressive strength is incomplete.

This means that whatever (useful) formal system you choose to research mathematics, there will always be thruths hidden from you while within your chosen system because you cannot prove them (i.e. show their truth value in a consistent way).

Example: it could have been possible that Fermat's Last Theorem is true but unprovable. It could be that P=NP (or P!=NP) is true but unprovable.

The proof of the incompleteness theorem is not even hard (the hard part lies in getting the necessary formalisms into place).

--Sigi
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#8 User is offline   kfgauss 

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Posted 2006-June-22, 11:51

I read parts of this book a long time ago and recall thinking that the philosophy part of it was entirely rubbish. (I'm no philosopher, of course, though.)

If I recall correctly, his big philosophical claim is that because of Gödel's incompleteness theorem (yes, at the heart of it is a "this statement is false"-type paradox), there will always be true statements that a given robot can not realize are true, but that humans can.

Two (rather different) responses to this:

1) A robot, though deterministic, needn't be a machine for rigorously proving theorems. Humans believe things that they cannot prove (often contradictory things), and it doesn't seem hard to create a (deterministic) robot with this capability. (It does seem hard [as an engineering/computing problem] to create a robot with this ability that will "believe" the correct sorts of things most of the time, but this isn't my problem, as long as I've gotten around his philosophical point.)

2) Ignoring the idea in point (1) that humans believe things they cannot prove, note that he starts with one given robot and notes that humans will be able to realize the truth of some statement that it cannot. He does not, however, complete the argument by arguing/showing that there is no such statement for a given human. (And indeed, though I'm now being somewhat facetious, I'm certain there are [very complicated] statements that are impossible for me to be able to realize are true that robots will.) [There is a slight sticky point here, which is that if he is indeed claiming that all humans will be able to realize the truth of the statement the robot cannot, then surely the same can't be true for any human. However, I don't believe the statement with the proviso "all" for various reasons.]

Andy
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#9 User is offline   whereagles 

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Posted 2006-June-22, 12:22

There are two books by neurologist António Damásio on which he tries to explain what counciousness, emotions and feelings are. These are:

"The Feeling of What Happens: Body and Emotion in the Making of Consciousness"
"Looking for Spinoza: Joy, Sorrow, and the Feeling Brain"

I read the latter one. Basically his idea is that the brain is a place where the state of the body gets mapped into. The mapped information is then processed so as to take actions towards the survival of the body.

I find Damásio's ideas very convincing, simple and common-sense. So I really have to think Penrose is going a bit too far with his quantum-psychology hypothesis...
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#10 User is offline   mike777 

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Posted 2006-June-22, 12:30

Again what is a brain and what is a body, I assume you can have a brain or a body that is not human but fit a reasonable definition being called a brain or a body. If they self replicate they can certainly be alive, bacteria for instance.

Then it seems to just come down to a definition of self aware.

I think the issue of being able to fool a bunch of random humans in a blind test will be an easy one to pass. It would not surprise me if some humans are called robots in a blind test. You do not need to know the correct answers or even a logical nonanswer to be considered self aware.

It will not be that difficult to confuse our brain with 5 sensory inputs with a virtual reality. So what is reality?

If something has a brain and a body and can self replicate and we interact with it and cannot be sure if it is a human or not what will it matter?

If you can love "it", communicate with "it", have sex with "it", marry "it", and have babies in a "womb" the distinction of is it alive will matter to few in time. I assume "it" will be stronger and smarter than many of us, whatever stronger or smarter means. But this may comeback to the melding of man, machine and the chemical chart.

I guess that is why they call it the soul of the machine, if "it" prays is it self aware?
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#11 User is offline   joshs 

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Posted 2006-June-22, 13:16

I read this a number of years ago, but don't remember much about it other than I wasn't impressed with his argument. Personally, Godel's incompleteness theorem ("Any sufficiently complex (so that it at least can handle the natural numbers) formal system must contain a true proposition that can be stated within the formal system but can't be proved within the formal system." - OK, this is a loose statement of the theorem.) only says formal systems all have limitations- e.g. not set of axioms can lead to proofs of everything- there will always be true but unprovable statements. This is no more a statement that there are limitations on what a computer can know, than it is a statement about what humans can know, and neither has any bearing on whether someone or something is intellegent. Just because I can't deduce all true statements, doesn't make me not intellegent...

On the other hand I have always been impressed with Richard Hofstader's book Godel, Escher, Bach. Hofstader had an interesting concept that "intellegence" comes from a complex system that has multiple levels of organization (each which symbolic processing power), with with enough feedback between the levels to change the "rules" by which the levels evolve (in the dynamical system sense). He claims that "brains", computers, ant colonies, and many other complex systems all have the basic architecture needed for intelligence to "emerge". Anyway, its a brilliant book and a very subtle argument.
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#12 User is offline   david_c 

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Posted 2006-June-22, 13:40

barmar, on Jun 22 2006, 04:53 PM, said:

It seems like his argument is basically: We don't understand how the mind works, and we can't seem to reproduce it with a deterministic process.  The collapse of the wave function is a non-deterministic process, I guess it must be the basis of how the mind works.

I don't see any problem with this. Certainly it is nothing more than a guess; but making educated guesses is an important part of the scientific process.

Quote

The problem is that quantum mechanics operates at too microscopic a level of the universe to have a direct correspondence to the macroscopic behavior of a collection of neurons.

What makes you think that? You certainly can't prove it. Why can't there be some mechanism in the brain for magnifying quantum-mechanical effects? Or alternatively, maybe it could be possible for a large collection of particles to act in a quantum-mechanical way? I believe there has been some research into this possibility, though I don't know what came of it.

Quote

What I think is actually going on is that the brain is a very complex, dynamic system.  Dynamic systems give the appearance of non-determinism, simply because there are too many states for us to track.  I think this is basically what Chaos Theory is about.

Maybe there is some element of dynamical systems involved, but that would still leave an awful lot to be explained, to say the least.
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#13 User is offline   kfgauss 

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Posted 2006-June-22, 13:42

joshs, on Jun 22 2006, 07:16 PM, said:

On the other hand I have always been impressed with Richard Hofstader's book Godel, Escher, Bach. [snip]

You mean Douglas Hofstadter. Richard Hofstadter appears to have been a historian.

I also quite liked that book. It paints a pretty picture of things, though one doesn't really have any way of knowing whether it's an accurate picture. Lots of other good stuff in there too.

Andy
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#14 User is offline   joshs 

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Posted 2006-June-22, 13:51

Yes of course, Douglas. I read Richard's books in high school history class :)
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#15 User is offline   helene_t 

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Posted 2006-June-23, 00:40

Hannie, on Jun 22 2006, 04:24 PM, said:

I think that those humans who think that their intuition can tell them something more about (for example) the axiom of choice are misguided. So if this is really Penrose's arguement for (1) then I think that it is rubbish.

No, then I did not summarize it clearly enough. It goes like this:

Consider the countable set of all proofs, natural number, statements, predicates and propositions that can be written in a particular language. Let's enumerate that set. Consider the predicate:

q(n): the system contains no proof for the statement p_n(n), i.e. the n'th predicate with n as an argument.

Since we just described the predicate q without using any tricky stuff, we can assume that q belongs to our enumerated set. Let's call it's index k, i.e. p_k = q.

Now consider the statement q(k). True or false?

The answer is that it is true but that the system can contain no proof for it.

This is contrary to how Sigi phrased it. Sigi says that there are truths that are hidden for us. But this is not hidden for us, it's just hidden for the system.

Penrose claims that a Turing machine cannot convince itself that q(k) is true since it's system contains no proof for it. Yet with human mathematical insight (something external to the system), it can be proved.

This surprises me since Goedel's theorem, as I understand it, is what Sigi explains. Besides, Penrose's book could easily be assumed to belong to the enumerated set. This would turn q(k) into a paradox so it only proves that there is some logical fault in what I just wrote. Maybe there is a logical fault in Penrose's book or maybe I just misunderstood it. Anyway, I do not think that it's particulary relevant to the issue of consciusness. (See posts from kfgauss ans joshs, I agree with both of them).
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#16 User is offline   barmar 

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Posted 2006-June-23, 07:56

kfgauss, on Jun 22 2006, 01:51 PM, said:

1) A robot, though deterministic, needn't be a machine for rigorously proving theorems. Humans believe things that they cannot prove (often contradictory things), and it doesn't seem hard to create a (deterministic) robot with this capability.

In fact, computersoften don't bother proving things, and "believe" false things quite often, if you deduce what they believe from their behavior. For instance, buffer overflow bugs are due to a computer program "believing" that the input it will receive fits into the fixed-size memory area it has set aside for it.

#17 User is online   P_Marlowe 

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Posted 2006-June-23, 10:12

helene_t, on Jun 23 2006, 01:40 AM, said:

Hannie, on Jun 22 2006, 04:24 PM, said:

I think that those humans who think that their intuition can tell them something more about (for example) the axiom of choice are misguided. So if this is really Penrose's arguement for (1) then I think that it is rubbish.

No, then I did not summarize it clearly enough. It goes like this:

Consider the countable set of all proofs, natural number, statements, predicates and propositions that can be written in a particular language. Let's enumerate that set. Consider the predicate:

q(n): the system contains no proof for the statement p_n(n), i.e. the n'th predicate with n as an argument.

Since we just described the predicate q without using any tricky stuff, we can assume that q belongs to our enumerated set. Let's call it's index k, i.e. p_k = q.

Now consider the statement q(k). True or false?

The answer is that it is true but that the system can contain no proof for it.

This is contrary to how Sigi phrased it. Sigi says that there are truths that are hidden for us. But this is not hidden for us, it's just hidden for the system.

Penrose claims that a Turing machine cannot convince itself that q(k) is true since it's system contains no proof for it. Yet with human mathematical insight (something external to the system), it can be proved.

This surprises me since Goedel's theorem, as I understand it, is what Sigi explains. Besides, Penrose's book could easily be assumed to belong to the enumerated set. This would turn q(k) into a paradox so it only proves that there is some logical fault in what I just wrote. Maybe there is a logical fault in Penrose's book or maybe I just misunderstood it. Anyway, I do not think that it's particulary relevant to the issue of consciusness. (See posts from kfgauss ans joshs, I agree with both of them).

Hi Helen,

if I understand your example correctly,
your example is just a case of the Russell
paradoxon, well known to set theory:

You get contradictions, if you allow, that
the set is an element of itself.
And you certainly know, how this gets prevented:
this is forbidden :P .

I did not long enough think about your
sentences, to much Math, ... and my days
as a math studend are gone since several
years and I am lazy, so you may excuse me,
if I my impression was wrong.

With kind regards
Marlowe
With kind regards
Uwe Gebhardt (P_Marlowe)
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#18 User is offline   han 

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Posted 2006-June-23, 10:12

Thanks for brushing up my logic, it clearly needed it (and probably still needs more!).

Point (1) still seems wrong though. It doesn't allow the Turing machine to step outside of the language, but does allow us humans to do this to see that the statement is indeed true. Perhaps I can phrase this better if I think about it some more (or I come to the conclusion that I'm again missing the point).
Please note: I am interested in boring, bog standard, 2/1.

- hrothgar
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#19 User is offline   helene_t 

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Posted 2006-June-23, 10:32

Marlowe: this indeed resembles Russel's paradox - it also resembles the proof for the proposition that the real numbers are not countable.

While Russel's paradox is based on the assumption that there is a set defined as the set of all sets that don't belong to itself, I cannot see a similar (dubious) assumption underlying the above.
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#20 User is offline   kfgauss 

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Posted 2006-June-23, 11:10

helene_t, on Jun 23 2006, 06:40 AM, said:

Hannie, on Jun 22 2006, 04:24 PM, said:

I think that those humans who think that their intuition can tell them something more about (for example) the axiom of choice are misguided. So if this is really Penrose's arguement for (1) then I think that it is rubbish.

No, then I did not summarize it clearly enough. It goes like this:

Consider the countable set of all proofs, natural number, statements, predicates and propositions that can be written in a particular language. Let's enumerate that set. Consider the predicate:

q(n): the system contains no proof for the statement p_n(n), i.e. the n'th predicate with n as an argument.

Since we just described the predicate q without using any tricky stuff, we can assume that q belongs to our enumerated set. Let's call it's index k, i.e. p_k = q.

Now consider the statement q(k). True or false?

The answer is that it is true but that the system can contain no proof for it.

This is contrary to how Sigi phrased it. Sigi says that there are truths that are hidden for us. But this is not hidden for us, it's just hidden for the system.

Penrose claims that a Turing machine cannot convince itself that q(k) is true since it's system contains no proof for it. Yet with human mathematical insight (something external to the system), it can be proved.

This surprises me since Goedel's theorem, as I understand it, is what Sigi explains. Besides, Penrose's book could easily be assumed to belong to the enumerated set. This would turn q(k) into a paradox so it only proves that there is some logical fault in what I just wrote. Maybe there is a logical fault in Penrose's book or maybe I just misunderstood it. Anyway, I do not think that it's particulary relevant to the issue of consciusness. (See posts from kfgauss ans joshs, I agree with both of them).

I'm not really an expert on this, but here goes:

You ask two separate questions. I'll try to paraphrase (tell me if I don't state them correctly).

1) How is it that we can prove q(k) with our human mathematical insight, yet Sigi claims certain truths will be hidden from us due to the formal system we choose to work with?

I believe there's another point of confusion running around here, which is that mathematicians sometimes work with axioms and sometimes work within models. When working with axioms, there is no truth beyond provability, but there are undecidable statements. In this case, we may have some intuition that we'd like something undecidable to be true. Then one might add an appropriate axiom (that one intuitively likes) so that one can prove it (e.g. the Axiom of Choice and, somewhat less frequently, the Continuum Hypothesis are occasionally used).

When working within a model, however, we've essentially chosen a mathematical universe and any statement we can make will be true or false (in this model). If this model is big enough (contains a copy of the natural numbers etc) then the incompleteness theorem will apply. Going to a bigger model, one will be able to prove some things one previously couldn't, but now there will be more statements, etc. A nice example from the Wolfram page on Gödel's Incompleteness Theorem is that apparently the consistency of arithmetic can be proved if one uses transfinite induction, but if one wants to prove the consistency of arithmetic + transfinite induction, one would need to expand further.

(I'm not a logician, so some terminology and/or claim may be slightly off. Don't ask me what the difference between "model" and "formal system" is, or if there indeed is one.)

2) Couldn't you just add (the ideas in) Penrose's book to a formal system, so that it would be able to prove its own q(k) because it would have the idea that such q(k) are true?

I'm not really sure what you mean by add (the ideas in) Penrose's book to the formal system. Probably the answer is that this just doesn't make sense and/or can't be done.

Andy
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