Hi Barry,
Le 18-déc.-07, à 18:52, Barry Brent a écrit :
>
> Bruno--
>
> Ahh, my amateur status is nakedly exposed. I'm going to expose my
> confusion even further now.
That is the courageous attitude of the authentic scientists.
I like "amateur" because they have less prejudices, they have inner
motivations, and rarely follow authoritative arguments.
>
> Never heard of a universal language. I thought I was familiar with
> Church's thesis, but apparently no.
As I said a while ago, coming back from an international meeting on
computability (in Siena, where my Plotinus' paper has been accepted), I
got the feeling that few people really grasp Church Thesis, including
some
"experts".
My "Brussels thesis" has been criticized for being too much pedagogical
on Church thesis, but each time people try to debunk my work, soon
enough I realize they have a problem with Godel or Church, never (yet)
with
my contribution.
The worst is that most people *feel* at ease with CT but apparently are
not.
I take as an honor to explain this too you, I *do* appreciate your work
in
Number Theory, as far as I understand it. Possible links could emerge.
(You make me discover also the nice paper on "prime percolation" by
Vardi:
I love percolation. Not just because I am an amateur of good coffe, but
because
exact percolation problem have led to the Temperley Lieb
algebra/category; which
makes links between knot theory, combinators/lambda-calculus, quantum
computations, and eventually number theory, if not the number 24
itself).
> I thought it was the claim that
> two or three or four concepts (including recursive function and
> computable function) were extensionally equivalent. I have heard of
> the lambda calculus, but I don't know what it is, or what its
> connection is with Church's thesis. I have a rough guess, based on
> what you're saying. I'm surprised. I imagine that the claim of
> existence of a universal language must be made in the context of a
> theory of languages? Never heard of that theory.
This happens because the expression "theory of languages" is used in
the context of "non universal languages", like in the Chomsky hierarchy
of
languages for example. Universal languages and machines appears in
what is called "computability theory" or "recursion theory".
>
> Well, if I imagine such a theory, it must involve both syntax and
> semantics, yes? Semantics connects a language to a world, right?
> (The experts are cringing, I'm sure....) Can one language encompass
> all possible worlds? Can't we imagine worlds, the structures of
> which are so dissonant, that their languages could never be
> consistently subsumed under some single larger ("universal")
> language? (More cringing, no doubt....) Or, is it that when we
> restrict the worlds in question to some suitable realm--say, numbers--
> all these things work out? (Cringes redoubled!) I can imagine other
> ways out. Maybe we're concerned with just one world, suitably
> described. Maybe structural inconsistencies of possible worlds are
> no more an impediment to being expressible in one language than
> logical inconsistencies? (But how do we know?)
We have to distinguish logic and computability. In logic we will have
language in which sentences are to be interpreted in some world/model.
But in computability we can go very far by just interpreting them in
some
procedural way. The expressions in computer language are really basic
instruction like in the coffee-bar machine. Eventually we can describe
them
all in term of NAND gates, delay and electrical current.
A computing language is then universal if all computable functions
(from N
to N, or from finite things coded in N to finite things coded in N) can
be
computed by following a finite set of instructions in the language.
>
> What about cardinality? From your remarks, I imagine that the number
> of elementary symbols in any language, including the universal
> language, is supposed to be finite,
Yes.
> so that the set of algorithms is
> countable?
Yes. And so are the computable functions.
> If there are lots of worlds and languages, I wonder how
> people make that work.
Because all the languages or machines which have been
invented for computing (computable) functions from N to N, have
been shown equivalent, and that the closure of the set of computable
functions by those machines, for the (transcendental) diagonalization
procedure, give a powerful argument that those language/machine
are universal.
Careful: they are universal with respect to the class of computable
functions. They are not universal with respect to the propositions they
can express or prove. A universal language/machine is not a theory.
Most universal language don't even have a way to assert propositions,
just some sort of commands (cf the coffee-bar instructions as example).
> Is such a language going to be adequate for
> expressing propositions about all possible worlds? How do we know?
In computing, well ... like in Number theory, there is only one "world",
the world of natural numbers. But universal language just describes
functions from N to N. Propositions will occur when we will introduce
basic beliefs in the machine. Such beliefs will NEVER be universal.
Universality is a computability notion. By Godel, no theories can be
universal with respect of provability.
>
> My poor excuse is, I've only been on this list a little while, and I
> made my (sketchy) acquaintance with these ideas a very long time
> ago. Sorry if I'm way off topic.
I think you are quite in the topic. To explain a bit of my
work (and related TOE things) I have to provide some
help for people to distinguish clearly many things which are quite
different
but which are also deeply related. It is the relation in between those
notions which are important. They are often confused.
Those notions are (forgetting the most important one "truth"):
- Computability (the only one which has a possible notion of
universality)
- Provability (which *can* be universal with respect to computability,
but which is never universal with respect to provability of
propositions).
This will be illustrated with the notion of "lobian machine".
- Knowability (which can be proved equivalent extensionally with
provability, but will appears to have quite different intensional
logics)
- Observability (the same remark applies).
- Sensibility
Etc.
More later ... I will come back on the "key post" asap.
> Tell me to go look it up
> somewhere, or stop wasting time, if you want to...
Actually I'm afraid your are just motivating me for trying to be
clearer and
simpler. You are the one helping here, but feel free to organize your
computability-time as
you have to.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Dec 19 2007 - 08:57:17 PST