On 5/24/2020 3:30 AM, Lawrence Crowell wrote:
On Saturday, May 23, 2020 at 6:48:06 PM UTC-5, Bruce wrote:
On Sun, May 24, 2020 at 9:37 AM Russell Standish
<[email protected] <javascript:>> wrote:
On Sat, May 23, 2020 at 12:05:08PM -0700, 'Brent Meeker' via
Everything List wrote:
>
>
> On 5/23/2020 4:42 AM, Bruno Marchal wrote:
> >
> > Well, those are theorem provable in very weak theories. It
is more a
> > question of grasping the proof than subscribing to a
philosophical idea.
> > That arithmetic executes all programs is a theorem similar
to Euclid’s
> > theorem that there is no biggest prima numbers. It is more
a fact, than
> > an idea which could be debated. I insist on this as I
realise this is
> > less known by the general scientists than 20 years ago. We
knew this
> > implicitly since Gödel 1931, and explicitly since Church,
Turing and
> > Kleene 1936.
>
> Recently you have said that your theory is consistent with
finitism, even
> ultrafinitism. But the idea that arithemtic exectues all
programs certainly
> requires infinities.
Only potential infinities, not actual infinities. For the UD
(a finite
object) to execute any given program, one only needs to wait a
finite
amount of time.
I thought the UD executing in arithmetic was timeless: so all the
infinity of possible programs have already been executed before
you even start thinking about it. So computationalism has actual
infinities built in.
Bruce
This depends on what one considers as the domain of computation,
whether that is some global content or what is accessible to a local
observer. In the first case it is infinite, at least countably
infinite. In the latter case it is unbounded above, but finite and
given by the finite area of an event horizon or boundary of space on a
holographic screen. Both constructions have some relevancy, for in the
infinite case we can imagine well enough there is some Cantor
diagonalization of quantum states, qubits or information the define a
horizon or limit. This would then enforce the locality of any possible
observer as bounded by a computational or epistemological horizon. So
the two perspective may have a sort of dualism.
That's still thinking of them being physically implemented. Bruno's UD
is simply a mathematical construct and hence exists Platonically
indpendent of spacetime.
Brent
LC
However, I would think that ultrafinitism would change COMP's
predictions, and in a sense be incompatibe with it. Some
programs will
not exist, because one would need to wait too long for them to be
executed by the UD. In fact, the choice of reference universal
machine
would be significant in ultrafinitism, IIUC.
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