I, being of the classical persuasion, believe that arithmetic is either
consistent or inconsistent. You, to the extent that you are a
constructivist, should say that the matter is undecidable and therefore
undefined.
I believe that arithmetic is a formal and complete system. I'm not a
constructivist where formal and complete systems are concerned (since there
is nothing more to construct).
On the other hand, if you want to try to get into the "meaning" of
arithmetic . . . .
= = = = = = =
since the infinity of real numbers is larger than the infinity of
possible names/descriptions.
Huh? The constructivist in me points out that via compound constructions
the infinity of possible names/descriptions is exponentially larger than the
infinity of real numbers. You can reference *any* real number to the extent
that you can define it. And yes, that is both a trick statement AND also
the crux of the matter at the same time -- you can't name pi as a sequence
of numbers but you certainly can define it by a description of what it is
and what it does and any description can also be said to be a name (or a
"true name" if you will :-).
If the Gödelian truths are unreachable because they are undefined, then
there is something *wrong* with the classical insistence that they are
true or false but we just don't know which.
They are undefined unless they are part of a formal and complete system. If
they are part of a formal and complete system, then they are defined but may
be indeterminable. There is nothing *wrong* with the classical insistence
as long as it is applied to a limited domain (i.e. that of closed systems)
which is what you are doing.
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Monday, October 27, 2008 12:29 PM
Subject: Re: [agi] constructivist issues
Mark,
An example of people who would argue with the meaningfulness of
classical mathematics: there are some people who contest the concept
of real numbers. The cite things like that the vast majority of real
numbers cannot even be named or referenced in any way as individuals,
since the infinity of real numbers is larger than the infinity of
possible names/descriptions.
"OK. But I'm not sure where this is going . . . . I agree with all
that you're saying but can't see where/how it's supposed to address/go
back into my domain model ;-)"
Well, you already agreed that classical mathematics is meaningful.
But, you also asserted that you are a constructivist where meaning is
concerned, and therefore collapse Godel's and Tarski's theorems. I do
not think you can consistently assert both! If the Godelian truths are
unreachable because they are undefined, then there is something
*wrong* with the classical insistence that they are true or false but
we just don't know which.
To take a concrete example: One of these truths that suffers from
Godelian incompleteness is the consistency of arithmetic. I, being of
the classical persuasion, believe that arithmetic is either consistent
or inconsistent. You, to the extent that you are a constructivist,
should say that the matter is undecidable and therefore undefined.
--Abram
On Mon, Oct 27, 2008 at 12:04 PM, Mark Waser <[EMAIL PROTECTED]> wrote:
Hi,
It's interesting (and useful) that you didn't use the word meaning until
your last paragraph.
I'm not sure what you mean when you say that meaning is constructed,
yet truth is absolute. Could you clarify?
Hmmm. What if I say that meaning is your domain model and that truth is
whether that domain model (or rather, a given preposition phrased in the
semantics of the domain model) accurately represents the empirical world?
= = = = = = = =
I'm a classicalist in the sense that I think classical mathematics needs
to be accounted for in a theory of meaning.
Would *anyone* argue with this? Is there anyone (with a clue ;-) who
isn't
a classicist in this sense?
I am also a classicalist in the sense that I think that the
mathematically true is a proper subset of the mathematically provable,
so
that Gödelian truths are not undefined, just unprovable.
OK. But that is talking about a formal (and complete -- though still
infinite) system.
I might be called a constructivist in the sense that I think there needs
to be a tight, well-defined connection between syntax and semantics...
Agreed but you seem to be overlooking the question of "Syntax and
semantics
of what?"
The semantics of an AGI's internal logic needs to follow from its
manipulation rules.
Absolutely.
But, partly because I accept the
implementability of super-recursive algorithms, I think there is a
chance to allow at least *some* classical mathematics into the
picture. And, since I believe in the computational nature of the mind,
I think that and classical mathematics that *can't* fit into the
picture is literally nonsense! So, since I don't feel like much of
math is nonsense, I won't be satisfied until I've fit most of it in.
OK. But I'm not sure where this is going . . . . I agree with all that
you're saying but can't see where/how it's supposed to address/go back
into
my domain model ;-)
----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Monday, October 27, 2008 11:05 AM
Subject: Re: [agi] constructivist issues
Mark,
I'm a classicalist in the sense that I think classical mathematics
needs to be accounted for in a theory of meaning. (Ben seems to think
that a constructivist can do this by equating classical mathematics
with axiom-systems-of-classical-mathematics, but I am unconvinced.) I
am also a classicalist in the sense that I think that the
mathematically true is a proper subset of the mathematically provable,
so that Godelian truths are not undefined, just unprovable.
I might be called a constructivist in the sense that I think there
needs to be a tight, well-defined connection between syntax and
semantics... The semantics of an AGI's internal logic needs to follow
from its manipulation rules. But, partly because I accept the
implementability of super-recursive algorithms, I think there is a
chance to allow at least *some* classical mathematics into the
picture. And, since I believe in the computational nature of the mind,
I think that and classical mathematics that *can't* fit into the
picture is literally nonsense! So, since I don't feel like much of
math is nonsense, I won't be satisfied until I've fit most of it in.
I'm not sure what you mean when you say that meaning is constructed,
yet truth is absolute. Could you clarify?
--Abram
On Mon, Oct 27, 2008 at 10:27 AM, Mark Waser <[EMAIL PROTECTED]> wrote:
Hmmm. I think that some of our miscommunication might have been due to
the
fact that you seem to be talking about two things while I think that I'm
talking about third . . . .
I believe that *meaning* is constructed.
I believe that truth is absolute (within a given context) and is a proper
subset of meaning.
I believe that proof is constructed and is a proper subset of truth (and
therefore a proper subset of meaning as well).
So, fundamentally, I *am* a constructivist as far as meaning is concerned
and take Gödel's theorem to say that meaning is not completely defined or
definable.
Since I'm being a constructionist about meaning, it would seem that your
statement that
A constructivist would be justified in asserting the equivalence of
Gödel's incompleteness theorem and Tarski's undefinability theorem,
would mean that I was "correct" (or, at least, not wrong) in using
Gödel's
theorem but probably not as clear as I could have been if I'd used Tarski
since an additional condition/assumption (constructivism) was required.
So, interchanging the two theorems is fully justifiable in some
intellectual circles! Just don't do it when non-constructivists are
around :).
I guess the question is . . . . How many people *aren't* constructivists
when it comes to meaning? Actually, I get the impression that this
mailing
list is seriously split . . . .
Where do you fall on the constructivism of meaning?
----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Sunday, October 26, 2008 10:00 PM
Subject: Re: [agi] constructivist issues
Mark,
After some thought...
A constructivist would be justified in asserting the equivalence of
Godel's incompleteness theorem and Tarski's undefinability theorem,
based on the idea that truth is constructable truth. Where classical
logicians take Godels theorem to prove that provability cannot equal
truth, constructivists can take it to show that provability is not
completely defined or definable (and neither is truth, since they are
the same).
So, interchanging the two theorems is fully justifiable in some
intellectual circles! Just don't do it when non-constructivists are
around :).
--Abram
On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser <[EMAIL PROTECTED]>
wrote:
OK. A good explanation and I stand corrected and more educated. Thank
you.
----- Original Message ----- From: "Abram Demski"
<[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Saturday, October 25, 2008 6:06 PM
Subject: Re: [agi] constructivist issues
Mark,
Yes.
I wouldn't normally be so picky, but Godel's theorem *really* gets
misused.
Using Godel's theorem to say made it sound (to me) as if you have a
very fundamental confusion. You were using a theorem about the
incompleteness of proof to talk about the incompleteness of truth, so
it sounded like you thought "logically true" and "logically provable"
were equivalent, which is of course the *opposite* of what Godel
proved.
Intuitively, Godel's theorem says "If a logic can talk about number
theory, it can't have a complete system of proof." Tarski's says, "If
a logic can talk about number theory, it can't talk about its own
notion of truth." Both theorems rely on the Diagonal Lemma, which
states "If a logic can talk about number theory, it can talk about its
own proof method." So, Tarski's theorem immediately implies Godel's
theorem: if a logic can talk about its own notion of proof, but not
its own notion of truth, then the two can't be equivalent!
So, since Godel's theorem follows so closely from Tarski's (even
though Tarski's came later), it is better to invoke Tarski's by
default if you aren't sure which one applies.
--Abram
On Sat, Oct 25, 2008 at 4:22 PM, Mark Waser <[EMAIL PROTECTED]>
wrote:
So you're saying that if I switch to using Tarski's theory (which I
believe
is fundamentally just a very slightly different aspect of the same
critical
concept -- but unfortunately much less well-known and therefore less
powerful as an explanation) that you'll agree with me?
That seems akin to picayune arguments over phrasing when trying to
simply
reach general broad agreement . . . . (or am I misinterpreting?)
----- Original Message ----- From: "Abram Demski"
<[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Friday, October 24, 2008 5:29 PM
Subject: Re: [agi] constructivist issues
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