Le 31-août-07, à 16:54, Lennart Nilsson a écrit :

> Bruno says:
> "...the notion of computability is absolute."
> David Deutsch says: ....

OK, but on this point David, as he says himself, disagrees with 100% of 
the mathematicians.
OK, this *is* not an argument ....

> "We see around us a computable universe;

I am already not sure we can *see* a universe. But I am definitely sure 
we don't see a computable universe. This is only something which has to 
be inferred. Also, would Copenhagen QM be correct, we could as well say 
we see around us a non computable universe. You have to infer the MW 
for keeping Church thesis intact (this is not obvious to show).
Also, I recall you that platonists insists that seeing gives only 
appearances, and then comp implies (by UDA) that such appearances have 
to have non computationnal components.

> that is to say, of all
> possible mathematical objects and relationships, only an infinitesimal
> proportion
> are ever instantiated in the relationships of physical objects and 
> physical
> processes. (These are essentially the computable functions.)

David postulates here both comp and physical realism. This is simply 
not working by the UDA (and weak Occam).
Recall that: IF I am a Machine, then, whatever the "appearance of 
universe" can be it cannot be computable.

> Now it might
> seem that one approach to explaining that amazing fact, is to say "the
> reason
> why physical processes conform to this very small part of mathematics,
> 'computable mathematics,' is that physical processes really are 
> computations
> running on a computer external to what we think of as physical 
> reality."

Note that here comp really entails that physical processes cannot be 
computable. If I am a digitalizable machine then the universe cannot be 
a digitilizable machine. Don't confuse constructive physics or 
computational physics with computationnalist physics: which is the 
physics you have to derive from numbers once you take seriously enough 
the comp hyp. I know that all this is a bit counter-intuitive, that is 
why I like to ask people where in the UDA they stop to be convinced.

>  But
> that relies

relies wrongly

> on the assumption that the set of computable functions -- the
> Turing computable functions, or the set of quantum computable 
> operations
> -- is somehow inherently privileged within mathematics. 

This is Church thesis, and David agrees (and I think is even the first 
to show) that Quantum Computer does not violate Church thesis. The set 
of functions computable from N to N with a quantum computer is the same 
as the set of functions from N to N computable with Babbage machine or 
the one that can be described in <your favorite universal system (Java, 
Python, Lisp, Fortran, Game-of-life, etc.>.

> So that even a
> computer
> implemented in unknown physics (the supposed computer that we're
> all simulations on) would be expected to conform to those same notions 
> of
> computability, to use those same functions that mathematics designates 
> as
> computable.


> But in fact, the only thing that privileges the set of all
> computational
> operations that we see in nature, is that they are instantiated by
> the laws of physics.

I don't believe in this at all (and again, here I'm on the side of all 
mathematicians but OK don't take this as an argument).

>  It is only through our knowledge of the physical world
> that we know of the difference between computable and not computable.

Not at all, except in the weak sense that you have to live to begin 
with for  being interested in question like that.. Wait until I explain 
Church Thesis, there are deep purely arithmetical reasons to believe in 
the necessity of the computable and the many uncomputables.

> So
> it's only through our laws of physics that the nature of computation 
> can be
> understood. It can never be vice versa."

I don't believe in this. It has not been proved, and actually this 
cannot be maintained if comp is true.

> http://www.qubit.org/people/david/Articles/PPQT.pdf
> If it is only through our knowledge of the physical world
> that we know of the difference between computable and not computable, 
> and I
> don´t see any flaw in David´s argument that leads up to that 
> statement, then
> the notion of computability definitely is not absolute.

Well, thank you for providing me still more motivations to explain why 
the concept of computability is the most absolute epistemological 
notion ever discovered by the mathematician, and why Church thesis, 
although a very strong statement (philosophically) is very well 
grounded both with the facts and conceptually.

You could have chosen a better moment because next week I have exams 
and will not be in my office, but the week after I will try to explain 
this. It is necessary to get the UDA, and even more for the AUDA (the 
lobian interview).



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