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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