Excellent, Bruno, Thanks!
Barry
On Dec 19, 2007, at 7:57 AM, Bruno Marchal wrote:
>
> 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/
>
>
> >
Dr. Barry Brent
barrybrent.domain.name.hidden
http://home.earthlink.net/~barryb0/
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Received on Wed Dec 19 2007 - 15:10:58 PST