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 <[email protected]>
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.
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".
If that is the case, when is it determined (for us) that a certain
program terminates? Is it when the first being anywhere in any
universe tests it, when someone in our universe tests it, when
someone in our past light cone tests it, when you test it yourself
or read about someone who did? Would it ever be possible for two
beings in two different universes to find different results
regarding the same program? If not, then what enforces this
agreement?
But if it leads to paradoxes or absurdities we should just modify
our invention keeping the good part and avoiding the paradoxes if
we can. Peano's arithmetic will still be there in Platonia and
sqrt(2) will be irrational there. But the diagonal of a unit
square may depend on how we measure it or what it's made of.
Does this instrumentalist approach prevents one from having a
theory of reality?
Who said it's instrumentalist? Just because it considers a finite
model of reality? When Bruno proposes to base things on arithmetic
and leave analysis and set theory alone, does that make him an
instrumentalist?
Of course not. As the comp hypothesis use a non instrumentalist
interpretation of arithmetic. It makes only comp being a finitism (not
an ultrafinitism). There is no axiom of infinity at the ontological
level. Infinity is a "correct" illusion from inside, and mainly due to
the FPI, and the fact that for all x, s(x) > x.
Bruno
Brent
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