Re: Turing completeness of the functional paradigm?



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.



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

.