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. It seems that this is the computer you 
have in mind to run the UD. That's OK and the argument works (assuming 
comp etc.), but in Platonia you have access to hypercomputers of the best 
and fastest kind. This does not invalidate CT - it still applies to the 
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.

Stathis Papaioannou
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