*That* is what I was asking about when I asked which side you fell on.
Do you think such extensions are arbitrary, or do you think there is a
fact of the matter?
The extensions are clearly judged on whether or not they accurately reflect
the empirical world *as currently known* -- so they aren't arbitrary in that
sense.
On the other hand, there may not be just a single set of extensions that
accurately reflect the world so I guess that you could say that choosing
among sets of extensions that both accurately reflect the world is
(necessarily) an arbitrary process since there is no additional information
to go on (though there are certainly heuristics like Occam's razor -- but
they are more about getting a usable or "more likely" to hold up under
future observations or more likely to be easily modified to match future
observations theory . . . .).
The world is real. Our explanations and theories are constructed. For any
complete system, you can take the classical approach but incompleteness (of
current information which then causes undecidability) ever forces you into
constructivism to create an ever-expanding series of shells of stronger
systems to explain those systems contained by them.
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Monday, October 27, 2008 5:43 PM
Subject: Re: [agi] constructivist issues
Mark,
Sorry, I accidentally called you "Mike" in the previous email!
Anyway, you said:
"Also, you seem to be ascribing arbitrariness to constructivism which
is emphatically not the case."
I didn't mean to ascribe arbitrariness to constructivism-- what I
meant was that constructivists would (as I understand it) ascribe
arbitrariness to extensions of arithmetic. A constructivist sees the
fact of the matter as undefined for undecidable statements, so adding
axioms that make them decidable is necessarily an arbitrary process.
The classical view, on the other hand, sees it as an attempt to
increase the amount of true information contained in the axioms-- so
there is a right and wrong.
*That* is what I was asking about when I asked which side you fell on.
Do you think such extensions are arbitrary, or do you think there is a
fact of the matter?
--Abram
On Mon, Oct 27, 2008 at 3:33 PM, Mark Waser <[EMAIL PROTECTED]> wrote:
The number of possible descriptions is countable
I disagree.
if we were able to randomly pick a real number between 1 and 0, it would
be indescribable with probability 1.
If we were able to randomly pick a real number between 1 and 0, it would
be
indescribable with probability *approaching* 1.
Which side do you fall on?
I still say that the sides are parts of the same coin.
In other words, we're proving arithmetic consistent only by adding to
its
definition, which hardly counts. The classical viewpoint, of course, is
that
the stronger system is actually correct. Its additional axioms are not
arbitrary. So, the proof reflects the truth.
What is the stronger system other than an addition? And the viewpoint
that
the stronger system is actually correct -- is that an assumption? a truth?
what? (And how do you know?)
Also, you seem to be ascribing arbitrariness to constructivism which is
emphatically not the case.
----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Monday, October 27, 2008 2:53 PM
Subject: Re: [agi] constructivist issues
Mark,
The number of possible descriptions is countable, while the number of
possible real numbers is uncountable. So, there are infinitely many
more real numbers that are individually indescribable, then
describable; so much so that if we were able to randomly pick a real
number between 1 and 0, it would be indescribable with probability 1.
I am getting this from Chaitin's book "Meta Math!".
"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)."
Oh, I believe there is some confusion here because of my use of the
word "arithmetic". I don't mean grade-school
addition/subtraction/multiplication/division. What I mean is the
axiomatic theory of numbers, which Godel showed to be incomplete if it
is consistent. Godel also proved that one of the incompletenesses in
arithmetic was that it could not prove its own consistency. Stronger
logical systems can and have proven its consistency, but any
particular logical system cannot prove its own consistency. It seems
to me that the constructivist viewpoint says, "The so-called stronger
system merely defines truth in more cases; but, we could just as
easily take the opposite definitions." In other words, we're proving
arithmetic consistent only by adding to its definition, which hardly
counts. The classical viewpoint, of course, is that the stronger
system is actually correct. Its additional axioms are not arbitrary.
So, the proof reflects the truth.
Which side do you fall on?
--Abram
On Mon, Oct 27, 2008 at 1:03 PM, Mark Waser <[EMAIL PROTECTED]> wrote:
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
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