My apologies. When Chalmers used the words "godelian argument" I thought he
was referring to Godel. Now I can see I misread it.
On Thu, Aug 23, 2012 at 9:09 PM, Jesse Mazer <laserma...@gmail.com> wrote:
> On Thu, Aug 23, 2012 at 8:41 PM, Richard Ruquist <yann...@gmail.com>wrote:
>> This is what Chalmers says in the 95 paper you link about the second
>> Penrose argument, the one in my paper:
>> " 3.5 As far as I can determine, this argument is free of the obvious
>> flaws that plague other Gödelian arguments, such as Lucas's argument and
>> Penrose's earlier arguments. If it is flawed, the flaws lie deeper. It is
>> true that the argument has a feeling of achieving its conclusion as if by
>> magic. One is tempted to say: "why couldn't F itself engage in just the
>> same reasoning?". But although there are various directions in which one
>> might try to attack the argument, no knockdown refutation immediately
>> presents itself. For this reason, the argument is quite challenging.
>> Compared to previous versions, this argument is much more worthy of
>> attention from supporters of AI. "
>> Chalmers finally concludes that the flaw for Godel, which Penrose also
>> assumed, is the assumption that we can know we are sound. So the other way
>> around, if Godel is correct, so is the Penrose second argument, which
>> Chalmers confirmed. However, Chalmers seems to be saying the Godel is
>> incorrect, hardly a basis for my paper.
> What do you mean "the flaw for Godel"? There is no doubt that Godel's
> mathematical proof is correct, and if you think Chalmers is suggesting any
> such doubt in his paper you are misreading him. The argument he's talking
> about is one specifically concerning human intelligence, which Godel's
> mathematical proof says nothing about (Godel did offer some brief comments
> about the implications of his mathematical proof for human intelligence,
> but they were very brief and somewhat ambiguous, see
> http://www.iep.utm.edu/lp-argue/#H4 ). And I already quoted his
> conclusions about the second argument, after the section you quote above:
> that although Chalmers agrees that Penrose's second argument does show that
> *either* our reasoning cannot be captured by a formal system *or* that we
> cannot be sure our reasoning is sound, Chalmers thinks Penrose is wrong to
> prefer the first option rather than the second.
>> Personally, when I am sound, I know I am sound. When I am unsound I
>> usually know that I am unsound. However, psychosis runs in my family, and
>> many times I have watched a relative lapse into psychosis without him
>> realizing it.
> Chalmers/Penrose aren't talking about "sound" in the ordinary colloquial
> sense of sanity or anything like that, they're talking about soundness in
> the sense of perfect mathematical certainty that there is absolutely no
> chance--not even a chance of 1 in 10^1000000000 or smaller, say--that they
> might have made an error in their judgement about the truth or falsity of
> some (potentially very complicated) proposition about arithmetic.
>> Besides I sent the paper to Chalmers and he had no problem with. But he
>> did wish me luck getting it published. He knew something I had not yet
> Did Chalmers offer any detailed commentary suggesting he had read through
> the whole thing carefully? If not it's possible he skimmed it and missed
> that sentence, or just read the abstract and decided it didn't interest
> him, but sent the note out of politeness.
>> On Thu, Aug 23, 2012 at 8:19 PM, Jesse Mazer <laserma...@gmail.com>wrote:
>>> A quibble with the beginning of Richard's paper. On the first page it
>>> 'It is beyond the scope of this paper and admittedly beyond my
>>> understanding to delve into Gödelian logic, which seems to be
>>> self-referential proof by contradiction, except to mention that Penrose in
>>> Shadows of the Mind(1994), as confirmed by David Chalmers(1995), arrived at
>>> a seemingly valid 7 step proof that human “reasoning powers cannot be
>>> captured by any formal system”.'
>>> If you actually read Chalmers' paper at
>>> definitely does *not* "confirm" Penrose's argument! He says in the paper
>>> that Penrose has two basic arguments for his conclusions about
>>> consciousness, and at the end of the section titled "the first argument" he
>>> concludes that the first one fails:
>>> "2.16 It is section 3.3 that carries the burden of this strand of
>>> Penrose's argument, but unfortunately it seems to be one of the least
>>> convincing sections in the book. By his assumption that the relevant class
>>> of computational systems are all straightforward axiom-and-rules system,
>>> Penrose is not taking AI seriously, and certainly is not doing enough to
>>> establish his conclusion that physics is uncomputable. I conclude that none
>>> of Penrose's argument up to this point put a dent in the natural AI
>>> position: that our reasoning powers may be captured by a sound formal
>>> system F, where we cannot determine that F is sound."
>>> Then when dealing with Penrose's "second argument", he says that Penrose
>>> draws the wrong conclusions; where Penrose concludes that our reasoning
>>> cannot be the product of any formal system, Chalmers concludes that the
>>> actual issue is that we cannot be 100% sure our reasoning is "sound" (which
>>> I understand to mean we can never be 100% sure that we have not made a
>>> false conclusion about whether all the propositions we have proved true or
>>> false actually have that truth-value in "true arithmetic"):
>>> "3.12 We can see, then, that the assumption that we know we are sound
>>> leads to a contradiction. One might try to pin the blame on one of the
>>> other assumptions, but all these seem quite straightforward. Indeed, these
>>> include the sort of implicit assumptions that Penrose appeals to in his
>>> arguments all the time. Indeed, one could make the case that all of
>>> premises (1)-(4) are implicitly appealed to in Penrose's main argument. For
>>> the purposes of the argument against Penrose, it does not really matter
>>> which we blame for the contradiction, but I think it is fairly clear that
>>> it is the assumption that the system knows that it is sound that causes
>>> most of the damage. It is this assumption, then, that should be withdrawn.
>>> "3.13 Penrose has therefore pointed to a false culprit. When the
>>> contradiction is reached, he pins the blame on the assumption that our
>>> reasoning powers are captured by a formal system F. But the argument above
>>> shows that this assumption is inessential in reaching the contradiction: A
>>> similar contradiction, via a not dissimilar sort of argument, can be
>>> reached even in the absence of that assumption. It follows that the
>>> responsibility for the contradiction lies elsewhere than in the assumption
>>> of computability. It is the assumption about knowledge of soundness that
>>> should be withdrawn.
>>> "3.14 Still, Penrose's argument has succeeded in clarifying some issues.
>>> In a sense, it shows where the deepest flaw in Gödelian arguments lies. One
>>> might have thought that the deepest flaw lay in the unjustified claim that
>>> one can see the soundness of certain formal systems that underlie our own
>>> reasoning. But in fact, if the above analysis is correct, the deepest flaw
>>> lies in the assumption that we know that we are sound. All Gödelian
>>> arguments appeal to this premise somewhere, but in fact the premise
>>> generates a contradiction. Perhaps we are sound, but we cannot know
>>> unassailably that we are sound."
>>> So it seems Chalmers would have no problem with the "natural AI"
>>> position he discussed earlier, that our reasoning could be adequately
>>> captured by a computer simulation that did not come to its top-level
>>> conclusions about mathematics via a strict axiom/proof method involving the
>>> mathematical questions themselves, but rather by some underlying fallible
>>> structure like a neural network. The bottom-level behavior of the simulated
>>> neurons themselves would be deducible given the initial state of the system
>>> using the axiom/proof method, but that doesn't mean the system as a whole
>>> might not make errors in mathematical calculations; see Douglas
>>> Hofstadter's discussion of this issue starting on p. 571 of "Godel Escher
>>> Bach", the section titled "Irrational and Rational Can Coexist on Different
>>> Levels", where he writes:
>>> "Another way to gain perspective on this is to remember that a brain,
>>> too, is a collection of faultlessly functioning element-neurons. Whenever a
>>> neuron's threshold is surpassed by the sum of the incoming signals,
>>> BANG!-it fires. It never happens that a neuron forgets its arithmetical
>>> knowledge-carelessly adding its inputs and getting a wrong answer. Even
>>> when a neuron dies, it continues to function correctly, in the sense that
>>> its components continue to obey the laws of mathematics and physics. Yet as
>>> we all know, neurons are perfectly capable of supporting high-level
>>> behavior that is wrong, on its own level, in the most amazing ways. Figure
>>> 109 is meant to illustrate such a class of levels: an incorrect belief held
>>> in the software of a mind, supported by the hardware of a faultlessly
>>> functioning brain."
>>> Figure 109 depicts the outline of a person's head with "2+2=5" appearing
>>> inside it, but the symbols in "2+2=5" are actually made up of large
>>> collections of smaller mathematical equations, like "7+7=14", which are all
>>> correct. A nice way of illustrating the idea, I think.
>>> I came up with my own thought-experiment to show where Penrose's
>>> argument goes wrong, based on the same conclusion that Chalmers reached: a
>>> community of "realistic" AIs whose simulated brains work similarly to real
>>> human brains would never be able to be 100% certain that they had not
>>> reached a false conclusion about arithmetic, and the very act of stating
>>> confidently in mathematical that they would never reach a wrong conclusion
>>> would ensure that they were endorsing a false proposition about arithmetic.
>>> See my discussion with LauLuna on the "Penrose and algorithms" thread here:
>>> On Thu, Aug 23, 2012 at 6:38 PM, Stephen P. King
>>>> Dear Richard,
>>>> Your paper <http://vixra.org/pdf/1101.0044v1.pdf> is very
>>>> interesting. It reminds me a lot of Stephen Wolfram's cellular automaton
>>>> theory. I only have one big problem with it. The 10d manifold would be a
>>>> single fixed structure that, while conceivably capable of running the
>>>> computations and/or implementing the Peano arithmetic, has a problem with
>>>> the role of time in it. You might have a solution to this problem that I
>>>> see that I did not deduce as I read your paper. How do you define time for
>>>> your model?
>>>> "Nature, to be commanded, must be obeyed."
>>>> ~ Francis Bacon
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