On 30 Oct 2012, at 18:39, Stephen P. King wrote:
On 10/30/2012 12:51 PM, Bruno Marchal wrote:
On 30 Oct 2012, at 17:04, meekerdb wrote:
On 10/30/2012 4:30 AM, Bruno Marchal wrote:
My argument is that concepts of truth and provability of
theorems apply only to the concepts of numbers and their
constructions, not to numbers themselves.
Truth applies to proposition, or sentences representing them for
some machine/numbers. If not, comp does not even makes sense.
So your are agreeing? "Two" has no truth value, but "Two equals
one plus one." does.
Yes I agree. It seems I insisted on this a lot.
But in this context, it seems that Stephen was using this to assert
that the truth of, say "Two equals one plus one." depend on some
numbers or subject having to discover it, or prove it.
Bruno
http://iridia.ulb.ac.be/~marchal/
Dear Bruno,
My point is that a number is not a capable of being an
ontological primitive
Then I can stop reading as you need to assume the numbers (or anything
Turing equivalent) to get them.
*and* having some particular set of values and meanings.
I just assume
x + 0 = x
x + s(y) = s(x + y)
x *0 = 0
x*s(y) = x*y + x
And hope you understand.
A statement, such as 2 = 1+1 or two equals one plus one, are said
truthfully to have the same meaning because there are multiple and
separable entities that can have the agreement on the truth value.
In the absence of the ability to judge a statement independently of
any particular entity capable of "understanding" the statement,
there is no meaning to the concept that the statement is true or
false. To insist that a statement has a meaning and is true (or
false) in an ontological condition where no entities capable of
judging the meaning, begs the question of meaningfulness!
You are taking for granted some things that your arguments
disallow.
Do you agree that during the five seconds just after the Big Bang
(assuming that theory) there might not have been any possible
observers. But then the Big Bang has no more sense.
I think Brent is right, and Quentin. You confuse 1+1=2 with human
expression for pointing on that proposition. You obviously needs human
to understand those " "1+1=2" ", but the content of "1+1=2" has simply
no relation at all with the human, or with a physical universe.
I asked you some time ago if you agree with the use of the excluded
middle in arithmetic. It asserts that for any arithmetical proposition
P, even highly non computably verifiable, you can accept as new
arithmetical truth the proposition asserting that P v ~P. Which
intuitive meaning that the proposition is unambiguously either true,
or false, despite you have no idea if it is P or ~P which is the true
one. To accept this means that you accept that such truth are
independent of the means to prove or verify them.
Even intuitionist (who are sort of mathematical solipsist) accept, for
P arithmetical, the proposition
~ ~ (P v ~P), which makes them already realist in the sense used in
comp.
Bruno
http://iridia.ulb.ac.be/~marchal/
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