On 12/19/2013 1:30 PM, Jesse Mazer wrote:
To me it seems like "thinking something is true" is much more of a fuzzy category that
"asserting something is true"
Maybe. But note that Bruno's MGA is couched in terms of a dream, just to avoid any
input/output. That seems like a suspicious move to me; one that may lead intuition astray.
Brent
(even assertions can be ambiguous when stated in natural language, but they can be made
non-fuzzy by requiring that each assertion be framed in terms of some formal language
and entered into a computer, as in my thought-experiment). Is there any exact point
where you cross between categories like "being completely unsure whether it's true" and
"having a strong hunch it's true" and "having an argument in mind that it's true but not
feeling completely sure there isn't a flaw in the reasoning" and "being as confident as
you can possibly be that it's true"? I never really feel *absolute* certainty that
anything I think is true, even basic arithmetical statements like 1+1=2, because I'm
aware of how I've sometimes made sloppy mistakes in thinking in the past, and because I
know intelligent people can seem to come to incorrect conclusions about basic ideas when
hypnotized, or when dreaming (like the logic of various characters in Alice in
Wonderland). I think of certain truth as being like an asymptote that an individual or
community of thinkers can continually get closer to but never quite reach.
If I consider the statement "Jesse Mazer will never think this statement is true", I may
imagine the perspective of someone else and see that from their perspective it must be
true if Jesse's thinking is trustworthy, but then I'll catch myself and see that this
imaginary perspective is really just a thought in Jesse's head--at that point, have I
had the thought that it's true? And at some point in considering it I can't really help
thinking some words along the lines of "oh, so then it *is* true" (it's hard to avoid
thinking something you know you are "forbidden" to think, like when someone tells you
"don't think of an elephant"), but is merely thinking the magic words enough to count as
having thought it's true, and therefore having made it false once and for all?
Jesse
On Thu, Dec 19, 2013 at 3:46 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
A nice exposition, Jesse. But it bothers me that it seems to rely on the
idea of
"output" and a kind of isolation like invoking a meta-level. What if
instead of
"Craig Weinberg will never in his lifetime assert that this statement is
true" we
considered "Craig Weinberg will never in his lifetime think that this
statement is
true"? Then it seems that one invokes a kind of paraconsistent logic in
which one
just refuses to draw any inferences from this sentence that one cannot
think either
true or false.
Brent
On 12/19/2013 8:08 AM, Jesse Mazer wrote:
The argument only works if you assume from the beginning that an A.I. is
unconscious or doesn't have the same sort of "mind" as a human (and
given your
views you probably do presuppose these things--but if the conclusion
*requires*
such presuppositions, then it's an exercise in circular reasoning). If
you are
instead willing to consider that an A.I. mind works basically like a
human mind
(including things like being able to make mistakes, and being able to
understand
things it doesn't "say out loud"), and are willing to "put yourself in
the
place" of an A.I. being faced with its own Godel statement, then you
can see
it's like a more formal equivalent of me asking you to evaluate the
statement
"Craig Weinberg will never in his lifetime assert that this statement is
true".
You can understand that if you *did* assert that it's true, that would
of course
make it false, but you can likewise understand that as long as you try
to
refrain from uttering any false statements including that one, it
*will* end up
being true.
Similarly, an A.I. who is capable of making erroneous statements, and of
understanding things distinct from its "output" to the world outside the
program, might well understand that its own Godel statement is
true--provided it
never outputs a formal judgment that the statement is true, which would
mean
it's false! So if the A.I. in fact avoided ever giving as output a
judgment
about that the statement is true, it need not be because it lacks an
understanding of what's going on, but rather just because it's caught
in a bind
similar to the one you're caught in with "Craig Weinberg will never in
his
lifetime assert that this statement is true".
To flesh this out a bit, imagine a community of human-like A.I.
mathematicians
(mind uploads, say), living in a self-contained simulated world with no
input
from the outside, who have the ability to reflect on various
arithmetical
propositions. Once there is a consensus in this community that a
proposition has
been proven true or false, they can go to a special terminal (call it
the
"output terminal") and enter it on the list of proven statements, which
will
constitute the simulation's "output" to those of us watching it run in
the real
world. Suppose also that the simulated world is constantly growing, and
that
they have an internal simulated supercomputer within their world to
help with
their mathematical investigations, and this supercomputer is constantly
growing
in memory too. So if we imagine a string encoding the *initial* state
of the
simulation along with the rules determining its evolution, although
this string
may be very large, after some time has passed the memory of the
simulated
supercomputer will be much larger than that, so it's feasible to have
this
string appear within the supercomputer's memory (and it's part of the
rules of
the simulation that the string automatically appears in the
supercomputer's
memory after some finite time T within the simulation, and all the A.I.
mathematicians knew that this was scheduled to happen).
Once the A.I. mathematicians have the program's initial conditions and
the rules
governing subsequent evolution, they can construct their own Godel
statement. Of
course they can never really be sure that the string they are given
correctly
describes the true initial conditions of their own simulated universe,
but let's
say they have a high degree of trust that it is--for example, they
might be mind
uploads of the humans who designed the original simulation, and they
remember
having designed it to ensure that the string that would appear in the
supercomputer's memory is the correct one. They could even use the
growing
supercomputer to run a simulation-within-the-simulation of their own
history,
starting from those initial conditions--the sub-simulation would always
lag
behind what they were experiencing, but they could continually verify
that the
events in the sub-simulation matched their historical records and
memories up to
some point in the past.
So, they have a high degree of confidence that the Godel statement
they've
constructed actually is the correct one for their own simulated
universe. They
can therefore interpret the conceptual meaning of the statement as
something
like "you guys living in the simulation will never enter into your
output
terminal a judgment that this statement is true". So they could
understand
perfectly well that if they ever *did* enter such a judgment into their
output
terminal, that would mean the statement was a false statement about
arithmetic.
But provided that they *don't* ever enter any such judgment into their
output
terminal, they can see it's a true statement about arithmetic (and can
discuss
this fact among themselves and reach a consensus about this fact, as
long as
they don't enter it as output to the terminal). If they are mathematical
platonists, they realize that this feeling of it being their choice
whether to
output the statement or not, with the statement's truth or falsity
depending on
that choice, is a sort of illusion--really the truth-value of the
statement is a
timeless fact about arithmetic. But presumably, in such a situation
they would
adopt a "compatibilist" view of free will as many real-world
philosophers have
done (http://plato.stanford.edu/entries/compatibilism/), a view which
sees no
conflict between the feeling of free will and the idea that our actions
are
ultimately completely determined by natural laws and initial conditions.
Jesse
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