Le 09-mai-07, à 18:50, Brent Meeker a écrit :

> Bruno Marchal wrote:
>> Le 09-mai-07, à 09:08, [EMAIL PROTECTED] a écrit :
>>> Of course reality doesn't change.  The question of map versus
>>> territory is *not* an all or nothing
>>> question.  *sometimes* the map equals the territory.  Most of the 
>>> time
>>> it does not.
>> This is an important point where I agree with Marc. With or without
>> comp the necessity of distinguishing the map and the territory cannot
>> be uniform, there are "meaning"-fixed-point, like when a map is
>> embedded continuously in the territory (assuming some topology in the
>> map and in the territory, this follows by a fixed point theorem by
>> Brouwer, which today admits many interesting computational
>> interpretations.
>> Bruno
> I don't think you can define a topology on "meaning" that will allow 
> the fixed point theorem to apply.

A whole and rather important subfield of computer science is entirely 
based on that. It is know as denotational semantics. Most important 
contributions by Dana Scott. You can look on the web for "Scott 
domains".  Or search with the keyword "fixed point semantics topology".
Sometimes the topology is made implicit through the use of partial 
order and  a notion of continuous function in between. Indeed the 
topological space used in this setting are not Hausdorff space, and are 
rather different from those used in geometry or analysis.

A nice book is the one by Steve Vickers. It has provide me with 
supplementary reason to single out the Sigma_1 sentences as 
particularly important in the comp frame. (Topology via Logic, 
Cambridge University Press, 1989: 

Here are some links:




I could say more in case I come back on the combinators. Cf:
because there are strong relation between fixed point semantics, the 
paradoxical combinators, and the use of the (foirst and second) 
recursion theorem in theoretical computer science. But ok all that 
could become very technical of course.

... I thought you knew about fixed point semantics, perhaps I miss your 
point ...



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