What I meant was, it seems like humans are
"logically complete" in some sense. In practice we are greatly limited
by memory and processing speed and so on; but I *don't* think we're
limited by lacking some important logical construct. It would be like
us discovering some alien species whose mathematicians were able to
understand each individual case of mathematical induction, but were
unable to comprehend the argument for accepting it as a general
principle, because they lack the abstraction. Something like that is
what I find implausible.
I like the phrase "logically complete".
The way that I like to think about it is that we have the necessary seed of
whatever intelligence/competence is that can be logically extended to cover
all circumstances.
We may not have the personal time or resources to do so but given infinite
time and resources there is no block on the path from what we have to
getting there.
Note, however, that it is my understanding that a number of people on this
list do not agree with this statement (feel free to chime in with you
reasons why folks).
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Wednesday, October 22, 2008 12:20 PM
Subject: Re: [agi] constructivist issues
Too many responses for me to comment on everything! So, sorry to those
I don't address...
Ben,
When I claim a mathematical entity exists, I'm saying loosely that
meaningful statements can be made using it. So, I think "meaning" is
more basic. I mentioned already what my current definition of meaning
is: a statement is meaningful if it is associated with a computable
rule of deduction that it can use to operate on other (meaningful)
statements. This is in contrast to positivist-style definitions of
meaning, that would instead require a computable test of truth and/or
falsehood.
So, a statement is meaningful if it has procedural deductive meaning.
We *understand* a statement if we are capable of carrying out the
corresponding deductive procedure. A statement is *true* if carrying
out that deductive procedure only produces more true statements. We
*believe* a statement if we not only understand it, but proceed to
apply its deductive procedure.
There is of course some basic level of meaningful statements, such as
sensory observations, so that this is a working recursive definition
of meaning and truth.
By this definition of meaning, any statement in the arithmetical
hierarchy is meaningful (because each statement can be represented by
computable consequences on other statements in the arithmetical
hierarchy). I think some hyperarithmetical truths are captured as
well. I am more doubtful about it capturing anything beyond the first
level of the analytic hierarchy, and general set-theoretic discourse
seems far beyond its reach. Regardless, the definition of meaning
makes a very large number of uncomputable truths nonetheless
meaningful.
Russel,
I think both Ben and I would approximately agree with everything you
said, but that doesn't change our disagreeing with each other :).
Mark,
Good call... I shouldn't be talking like I think it is terrifically
unlikely that some more-intelligent alien species would find humans
mathematically crude. What I meant was, it seems like humans are
"logically complete" in some sense. In practice we are greatly limited
by memory and processing speed and so on; but I *don't* think we're
limited by lacking some important logical construct. It would be like
us discovering some alien species whose mathematicians were able to
understand each individual case of mathematical induction, but were
unable to comprehend the argument for accepting it as a general
principle, because they lack the abstraction. Something like that is
what I find implausible.
--Abram
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agi
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