On 1/22/2014 1:38 AM, Bruno Marchal wrote:
On 22 Jan 2014, at 01:02, meekerdb wrote:
On 1/21/2014 3:30 PM, Jason Resch wrote:
On Tue, Jan 21, 2014 at 3:30 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 1/21/2014 8:13 AM, Jason Resch wrote:
Why would you want to do that? It seems like an unnecessary extra axiom that
doesn't have any purpose or utility.
It prevents the paradoxes of undeciability, Cantor diagonalization, and it
corresponds more directly with how we actually use arithmetic.
I'm not sure it helps. What you may gain from avoiding paradoxes makes many
of
our accepted proofs false. E.g. Euclids proof of infinite primes. Or Euler's
identity. Most of math would be ruined. A circle's circumference would not
even
be pi*diameter.
Would this biggest number be different for different beings in different
universes? What is it contingent on?
You're taking an Platonic view that there really is an arithmetic and
whether
there's a biggest number is an empirical question.
Ah! I just said that is was not. Somehow you deny the reality of math.
Which math? Finite arithmetic, Peano arithmetic, set theory, homotopy theory,...? Or in
short, yes.
I'm saying it's an invention. We invented an system in which you can
always add 1
because that was convenient; you don't have to think about whether you can
or not.
So to use this same line of reasoning, would you say there is no definite (a priori)
fact of the matter of whether or not a given program terminates, unless we actually
build a machine executing that program and observe it terminate?
That's kind of mixing categories since 'program' (to you) means something in Platonia
and there you don't need a machine to run it. In the physical world there is no
question, all programs running on a machine terminate, for one reason or another.
Non-terminating programs are the result of over idealization.
What makes you sure that the idea that all programs terminates is not also an
idealisation (about a finite universal reality)?
Also, if all programs terminate, there is no more real numbers. I guess you will say
that there are idealisation. You seem to "know" that there is a concrete reality, but
the comp approach to the mind-body problem asks to, temporarily perhaps, doubt such
"certainty".
Of course I'm not *certain*, all theories are defeasible outside of Platonia. But it
seems like a well supported theory; at least as certain as "you can always add one more".
Brent
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