On 24 Aug 2012, at 03:15, Richard Ruquist wrote:
My apologies. When Chalmers used the words "godelian argument" I
thought he was referring to Godel. Now I can see I misread it.
OK. be careful. That is why we have always to separate clearly a "pure
theory" from its application in some domain. But even such an
application can be made as clean cut that the pure theory, if we
follow the definition and translation rules. In cognitive science,
being fuzzy on this can lead to infinite circular boring vocabulary
discussion.
The "last application", to reality, is alway conjectural, in science
we never know as such, or we never know for sure.
Bruno
On Thu, Aug 23, 2012 at 9:09 PM, Jesse Mazer <[email protected]>
wrote:
On Thu, Aug 23, 2012 at 8:41 PM, Richard Ruquist <[email protected]>
wrote:
Jesse,
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 learned.
Richard
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.
Jesse
On Thu, Aug 23, 2012 at 8:19 PM, Jesse Mazer <[email protected]>
wrote:
A quibble with the beginning of Richard's paper. On the first page
it says:
'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 http://web.archive.org/web/20090204164739/http://psyche.cs.monash.edu.au/v2/psyche-2-09-chalmers.html
he 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: http://groups.google.com/group/everything-list/browse_thread/thread/c92723e0ef1a480c/429e70be57d2940b?#429e70be57d2940b
Jesse
On Thu, Aug 23, 2012 at 6:38 PM, Stephen P. King <[email protected]
> wrote:
Dear Richard,
Your paper 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?
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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