Re: Turing completeness of the functional paradigm?

Babylonian wrote:
> Tom wrote:
> > Babylonian wrote:
> >
> > > Sequentiality, as denoted by the successor relation, is a ordinarily
> > > thought of as a property of time, not space.
> >
> > But some guy once said that the universe is a timeless manifold, his
> > name was ... no, I will not recall. E.. something.
>
> It seems pretty ad-hoc for a Platonist (of any variety) to invoke
> modern physics to justify assumptions. As your first link states, the
> world of forms is nonmaterial and presumably inacessible to physical
> sensation and measurement. Moreover, general relativity does not state
> that time does not exist. Do you claim that sequentiality, as denoted
> by the successor relation, is a property of space and not time?
>
> >
> > > With great trepidation I
> > > ask - why is this so serious?
> >
> > Look where it gets you:
> >
> > I show how Plato's theory, as it applies to mathematical objects, is
> > essentially a primitve version of modern recursion theory, which has
> > all the essential elements of the ancient theory.
> > http://home.ican.net/~arandall/Plato/
> >
> > We believe that metaphysics should strive for no less than mathematics.
> > http://mally.stanford.edu/computation.pdf
> >
> > Thank you for writing.
> > Tom
>
> I do not comprehend much of that material, but I see that the
> metaphysical content appears to be based on the Church-Turing thesis,
> which for many philosophers amounts to a convenient back-door way of
> seeming to justify determinism. A quote from your first link:
>
>
> > 1. All computable things correspond to a lambda-form. (Mathematical
> > Version)
> > 2. All physical things correspond to a lambda-form. (Physical Version)
> > 3. All thinkable things correspond to a lambda-form. (Cognitive
> > Version)
> > 4. All things correspond to a lambda-form. (Ontological Version)
>
>
> I may regret this, but here is my answer to the four theses:
>
>
> (1)
> The Church-Turing thesis doesn't figure large in my own thinking, since
> a constructivist cannot identify a class of all constructible functions
> in the sense of having a proof that f is computable if and only if it
> can be written in a certain language. If he had such a (constructive)
> proof, diagonalization gives him a technique for creating a new
> function outside that class. (As the other side of the coin, you do not
> know if a Turing machine computes a function unless you know a related
> halt function of that machine. What then, is the class of
> Turing-computable functions?) So, I just never saw 'real information'
> in the Church-Turing thesis; the two classes of functions that it
> states equivalence for, are 'nonconstructivities'.
>
> (2)
> You assume determinism, don't you? I warn you that nondeterminism is
> the observed fact. Quite often, I see someone who bases some
> philosophical position on a claim that some facts are hidden, unseen,
> unobserved. Sometimes, as in the case of Bohmian mechanics, I am forced
> to admit that the 'hidden fact' is different from the 'observed fact',
> but when I consider what a 'fact' is, I claim the 'observed fact' is
> true and the 'hidden fact' is sophistry. I assign neither 'true' nor
> 'false' to 'sophistry'. I took the time to read (part of) Randall's
> essay on quantum superposition, and I see that he expects a physical
> theory to explain 'objective reality' when in fact it only needs to
> relate observations. It is a very common illusion.
>
> (3)
> f(x)=1 if I agree with (3)
> f(x)=0 if I disagree with (3)
> Is there a lambda-form for f that someone else can evaluate? Or is this
> one of those unsolvable problems so that neither you nor I can ever
> know the value of f?
> (Note that the continuity principle does not hold for finite choice
> sequences.)
>
> (4)
> The only thing truly forced upon me as 'real', is awareness. Even
> 'mind' is merely a linguistic contrivance for packaging
> generalizations. It is normal to 'trust' our awareness, but I guess I
> just don't share your feeling that metaphysics can justify or explain
> this.

My intention was to say that all notions non-definable within Peano's
system are a mistake. In pure logic there is even no time.

> Well, I guess I just made quite enough enemies so I'll just shut up now
> and suffer the endless ridicule.

Oh, I am sorry, every time someone uses the word "ridicule" I take it
personally. That will cease when I have grasped what a human grasp
should.

Thank You for Your kind help.
Tom

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