Bruno Marchal writes:
> Le 21-oct.-06, à 06:02, Stathis Papaioannou a écrit :
> >
> >
> > Bruno Marchal writes:
> >
> >>>> The UD is both massively parallel
> >>>> and massively sequential. Recall the UD generates all programs and
> >>>> executes them all together, but one step at a time. The "D" is for
> >>>> dovetailing which is a technic for emulating parallelism 
> >>>> sequentially.
> >>>
> >>> Given that no actual physical hardware is needed to run it, why did
> >>> you choose the UD to generate all the computations rather than just
> >>> saying they are all run in parallel. There is enough room in Platonia
> >>> for infinite parallel virtual machines, isn't there?
> >>
> >>
> >> This is an interesting and key question. It is also a rather difficult
> >> one. To answer it we have to dig deeper on the importance and
> >> miraculous aspect of Church thesis, which makes existing a universal
> >> dovetailer, and which makes precise what a computational states is, 
> >> and
> >> why we have to postulate Arithmetical Realism, and why we have to be
> >> cautious with any form of larger mathematical platonism (but such
> >> platonism is not prohibited per se).
> >> Now with comp, and Church thesis in particular, it can be shown that
> >> the computational states can be said to exist (in the same sense than
> >> numbers) and it can be defined thoroughly by the UD. If you introduced
> >> infinite machines (and I agree that it is defensible that some of such
> >> machine exists in Platonia) , either you will lose Church thesis, or
> >> you will lose the "YES DOCTOR", at least in the form I usually gave 
> >> it.
> >> Your move here can be done, nevertheless, without changing the
> >> mathematical structure of the hypostases, but this asks for a non
> >> trivial generalization of comp, and of Church thesis in particular. I
> >> would not do that unless it is needed to get the physics (and then 
> >> this
> >> would be a refutation of comp, or more precisely here: of Sigma_1
> >> comp).
> >
> > The Chuch thesis concerns what can in theory be computed by a physical
> > computer with unlimited resources.
> Church thesis just assert that a universal turing machine can compute 
> all computable functions from N to N.
> It relate a mathematical object with a human cognitive notion. It does 
> not invoke physical machine at all.

In a sense that is true, but a TM is still a model of what could possibly be 
in a physical universe such as ours. Of course the model is still valid 
of the existence of a physical machine or indeed a physical universe, but if 
abandon the idea of a physical universe there is no need to constrain yourself 
models based on one. So I suppose the two questions I have (which you partly 
answer below) are, having arrived at step 8 of the UDA could you go back and 
say that the UD is not really necessary but all the required computations exist 
eternally without any generating mechanism or program (after all, you make this 
assumption for the UD itself), or alternatively, could you have started with 
8 and eliminate the need for the UD in the argument at all?

> > It seems that this is the computer you
> > have in mind to run the UD.
> Only for providing a decor for a story. This assumption is eliminated 
> when we arrive (step eight of UDA-8) at the conclusion that universal 
> digital machine cannot distinguish any "reality" from an arithmetical 
> one.
> > That's OK and the argument works (assuming
> > comp etc.), but in Platonia you have access to hypercomputers of the 
> > best
> > and fastest kind.
> Fastness is relative in Platonia. Universal machine can always been 
> sped up on almost all their inputs (There is a theorem by Blum and 
> Marquez to that effect). Then indeed there are the "angels" and 
> hierachies of "non-comp" machine. A vast category of "angels" can be 
> shown to have the same hypostases (so we cannot tested by empirical 
> means if we are such angels). Then they are entities very closed to the 
> "one", having stronger hypostases, i.e. you need to add axioms to G and 
> G* (or V, V* with explicit comp) to get them.

Of course I was joking when I said "best and fastest". In Platonia there is 
no actual time and everything is as fast and as perfect as you want it.

> > This does not invalidate CT - it still applies to the physical
> > world, such as it is - but it does make it unnecessary when the 
> > resources of
> > Platonia are available. Also, I don't see how introducing infinite 
> > machines
> > invalidates "yes doctor", since if anything it adds to the choice of 
> > "hardware"
> > when considering your replacement brain.
> You are right here, I agree. But this makes the thought experiment 
> longer to describe. Mathematically Sigma1-comp, which is the standard 
> traditionnal comp, is easier to handle than their "hyper" or "super" 
> generalization. G, G* and all the hypostases remains correct and 
> complete for much weaker form of comp. That's correct.

Stathis Papaioannou
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