LauLuna wrote:
>
>On 29 jun, 19:10, "Jesse Mazer" <laserma....domain.name.hidden> wrote:
> > LauLuna wrote:
> >
> > >On 29 jun, 02:13, "Jesse Mazer" <laserma....domain.name.hidden> wrote:
> > > > LauLuna wrote:
> >
> > > > >For any Turing machine there is an equivalent axiomatic system;
> > > > >whether we could construct it or not, is of no significance here.
> >
> > > > But for a simulation of a mathematician's brain, the axioms wouldn't
>be
> > > > statements about arithmetic which we could inspect and judge whether
> > >they
> > > > were true or false individually, they'd just be statements about the
> > >initial
> > > > state and behavior of the simulated brain. So again, there'd be no
>way
> > >to
> > > > inspect the system and feel perfectly confident the system would
>never
> > > > output a false statement about arithmetic, unlike in the case of the
> > > > axiomatic systems used by mathematicians to prove theorems.
> >
> > >Yes, but this is not the point. For any Turing machine performing
> > >mathematical skills there is also an equivalent mathematical axiomatic
> > >system; if we are sound Turing machines, then we could never know that
> > >mathematical system sound, in spite that its axioms are the same we
> > >use.
> >
> > I agree, a simulation of a mathematician's brain (or of a giant
>simulated
> > community of mathematicians) cannot be a *knowably* sound system,
>because we
> > can't do the trick of examining each axiom and seeing they are
>individually
> > correct statements about arithmetic as with the normal axiomatic systems
> > used by mathematicians. But that doesn't mean it's unsound either--it
>may in
> > fact never produce a false statement about arithmetic, it's just that we
> > can't be sure in advance, the only way to find out is to run it forever
>and
> > check.
>
>Yes, but how can there be a logical impossibility for us to
>acknowledge as sound the same principles and rules we are using?
The axioms in a simulation of a brain would have nothing to do with the
high-level conceptual "principles and rules" we use when thinking about
mathematics, they would be axioms concerning the most basic physical laws
and microscopic initial conditions of the simulated brain and its simulated
environment, like the details of which brain cells are connected by which
synapses or how one cell will respond to a particular electrochemical signal
from another cell. Just because I think my high-level reasoning is quite
reliable in general, that's no reason for me to believe a detailed
simulation of my brain would be "sound" in the sense that I'm 100% certain
that this precise arrangement of nerve cells in this particular simulated
environment, when allowed to evolve indefinitely according to some
well-defined deterministic rules, would *never* make a mistake in reasoning
and output an incorrect statement about arithmetic (or even that it would
never choose to intentionally output a statement it believed to be false
just to be contrary).
> >
> > But Penrose was not just arguing that human mathematical ability can't
>be
> > based on a knowably sound algorithm, he was arguing that it must be
> > *non-algorithmic*.
>
>No, he argues in Shadows of the Mind exactly what I say. He goes on
>arguing why a sound algorithm representing human intelligence is
>unlikely to be not knowably sound.
He does argue that as a first step, but then he goes on to conclude what I
said he did, that human intelligence cannot be algorithmic. For example, on
p. 40 he makes quite clear that his arguments throughout the rest of the
book are intended to show that there must be something non-computational in
human mental processes:
"I shall primarily be concerned, in Part I of this book, with the issue of
what it is possible to achieve by use of the mental quality of
'understanding.' Though I do not attempt to define what this word means, I
hope that its meaning will indeed be clear enough that the reader will be
persuaded that this quality--whatever it is--must indeed be an essentail
part of that mental activity needed for an acceptance of the arguments of
2.5. I propose to show that the appresiation of these arguments must involve
something non-computational."
Later, on p. 54:
"Why do I claim that this 'awareness', whatever it is, must be something
non-computational, so that no robot, controlled by a computer, based merely
on the standard logical ideas of a Turing machine (or equivalent)--whether
top-down or bottom-up--can achieve or even simulate it? It is here that the
Godelian argument plays its crucial role."
His whole Godelian argument is based on the idea that for any computational
theorem-proving machine, by examining its construction we can use this
"understanding" to find a mathematical statement which *we* know must be
true, but which the machine can never output--that we understand something
it doesn't. But I think my argument shows that if you were really to build a
simulated mathematician or community of mathematicians in a computer, the
Godel statement for this system would only be true *if* they never made a
mistake in reasoning or chose to output a false statement to be perverse,
and that therefore there is no way for us on the outside to have any more
confidence about whether they will ever output this statement than they do
(and thus neither of us can know whether the statement is actually a true or
false theorem of arithmetic).
It's true that on p. 76, Penrose does restrict his conclusions about "The
Godelian Case" to the following statement (which he denotes 'G'):
"Human mathematicians are not using a knowably sound algorithm in order to
ascertain mathematical truth."
I have no objection to this proposition on its own, but then in Chapter 3,
"The case for non-computability in mathematical thought" he does go on to
argue (as the chapter title suggest) that this proposition G justifies the
claim that human reasoning must be non-computable. In discussing objections
to this argument, he dismisses the possibility that G might be correct but
that humans are using an unknowable algorithm, or an unsound algorithm, but
as far as I can see he never discusses the possibility I have been
suggesting, that an algorithm that faithfully simulated the reasoning of a
human mathematician (or community of mathematicians) might be both knowable
(in the sense that the beings in the simulation are free to examine their
own algorithm) and sound (meaning that if the simulation is run forever,
they never output a false statement about arithmetic), but just not knowably
sound (meaning that neither they nor us can find a *proof* that will tell us
in advance that the simulation will never output a false statement, the only
way to check is to run it forever and see).
>
> >
> >
> > >And the impossibility has to be a logical impossibility, not merely a
> > >technical or physical one since it depends on Gödel's theorem. That's
> > >a bit odd, isn't it?
> >
> > No, I don't see anything very odd about the idea that human mathematical
> > abilities can't be a knowably sound algorithm--it is no more odd than
>the
> > idea that there are some cellular automata where there is no shortcut to
> > knowing whether they'll reach a certain state or not other than actually
> > simulating them, as Wolfram suggests in "A New Kind of Science".
>
>The point is that the axioms are exactly our axioms!
Again, the "axioms" would be detailed statements about the initial
conditions and behavior of the most basic elements of the simulation--the
initial position and velocity of each simulated molecule along with rules
for the molecules' behavior, perhaps--not the sort of high-level conceptual
axioms we use in our minds when thinking about mathematics. If we can't even
predict whether some very simple cellular automata will ever reach a given
state, I don't see why it should be surprising that we can't predict whether
some very complex physical simulation of an immortal brain and its
environment will ever reach a given state (the state in which it decides to
output the system's Godel statement, whether because of incorrect reasoning
or just out of contrariness).
>
> >In fact I'd
> > say it fits nicely with our feeling of "free will", that there should be
>no
> > way to be sure in advance that we won't break some rules we have been
>told
> > to obey, apart from actually "running" us and seeing what we actually
>end up
> > doing.
>
>I don't see how to reconcile free will with computationalism either.
I am only talking about the feeling of free will which is perfectly
compatible with ultimate determinism (see
http://en.wikipedia.org/wiki/Compatibilism ), not the philosophical idea of
"libertarian free will" (see
http://en.wikipedia.org/wiki/Libertarianism_(metaphysics) ) which requires
determinism to be false. If we had some unerring procedure for predicting
whether other people or even ourselves would make a certain decision in the
future, it's hard to see how we could still have the same subjective sense
of making choices whose outcomes aren't certain until we actually make them.
Jesse
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Received on Thu Jul 05 2007 - 16:14:14 PDT