On 18 Mar 2011, at 19:34, meekerdb wrote:
On 3/18/2011 8:12 AM, Bruno Marchal wrote:
Hi John,
In computer science there is something interesting which can be
seen as a critics or as a vindication of what you are saying. That
thing is the Church thesis, also called Church-Turing thesis, (CT)
and which has been proposed independently by Babbage (I have
evidence for that), Emil Post (the first if we forget Babbage),
Kleene, Turing, Markov, but not by Church (actually).
The thesis has many versions. One version is that ALL computable
functions can be defined in term of lambda expressions, or in term
of Turing machines, or in term of Markov algorithm, or in term of
Post production system, etc. All those versions are provably
equivalent.
Such a thesis *seems* to be in opposition with your idea that
complete knowledge is impossible. But it is not.
The contrary happens. Indeed the thesis concerns only completeness
with respect to computability, and then, as I have already explain
on this list, it entails the incompleteness of any effective
knowability concerning just the world of what machines can do.
By "machines" here I assume you mean digital machines/computers.
Yes.
I think this doesn't apply to machines described by real numbers.
But of course we think it is unlikely that real number machines
exist and that the reals are just a convenient fiction for dealing
with arbitrarily fine divisions of rationals.
I don't know. I have not yet find a machine genuinely based on real
numbers, but that might make sense. DM assumes I am digitalizable, so
that if I am a machine based on the reals, they are not relevant for
my identity preservation.
But there can be a cardinality between the integers and the reals.
You mean that the continuum hypothesis is undecidable in ZF? OK.
I wonder what this implies about computability?
I am not sure there is any.
Set theories have too much metaphysical baggage which I find more
distracting than genuine in the comp frame. But they might be needed
to solve complex problem in the future, like the distribution of the
primes might need complex analysis. It is hard to tell in advance. The
incompleteness phenomenon can be used to predict that the comp theory
will split into a huge numbers of variants. It will not stop the war
on religious matter! Some will perhaps accept the continuum
hypothesis, and others will reject it, leading to different
consequences of the comp assumption, but today, the continuum
hypothesis is nothing but an example of a an idea which is not used
in math or in physics and computer science.
Bruno
Brent
Church thesis makes it impossible to find *any* complete theory
about the behavior of machines. I explain this in the first
footnote of the Plotinus' paper. I can explain if someone ask more.
It is proved by a typical use of the (Cantor) diagonalization
procedure.
It vindicates what you say, really. We can sum up this by
Completeness with respect of computability provably entails a
strong form of incompleteness for our means of knowability and
provability about machines' possible behavior.
This can be proved rigorously in few lines. It is stronger and
easier than Gödel's incompleteness, and it entails Gödel's
incompleteness once we can show that the propositions on the
computable function can be translated into arithmetical
propositions (the lengthy tedious part of Gödel's proof).
Not only Church thesis makes it possible to think about
'everything', but it makes us able to prove our (machine's)
limitation about the knowledge about that everything. In any case,
this makes us modest, because either CT is wrong and we are
incomplete for computability, or CT is true and we are incomplete
about our knowledge about computability, machines, and numbers.
Best,
Bruno
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to [email protected]
.
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
