Le 27-mars-05, à 07:58, Perry Bruce a écrit :
However as Bruno alludes to (I might be misinterpreting him): if there is a
definable correspondence between the physical universe and a formal system
then does not Gödel's theorem hold for the physical universe?
It will depend on the nature of the formal system you would associate to the universe.
The relation between physics and logics are as deep as the relation between knot theory and quantum statistics. The show is only beginning.
I guess most people know that quantum information cannot be duplicated, nor erased. This is a feature of what is called "weak linear logic" (which I am introducing slowly on the "everything-list" through the Shoenfinkel-Curry-Church theory of combinators for those who would be interested. links can be found at the end of the post: http://www.escribe.com/science/theory/m5952.html).
Girard "linear logic" can be seen as a "modal logic" extending that "weak linear logic".
One of the idea is that assumption cannot be used twice. In particular from
[(A & A) entails B]
You cannot derive that [ A entails B].
Girard motivates this through the following proposition (where it is supposed that a cigarette box cost one euro):
From [(I have one euro) and (I have one euro) then I can buy two cigarette boxes], I cannot infer that
[if I have one euro then I can buy two cigarette boxes].
That is (weak) linear logic is a logic with an implicit notion of resource. Godel's theorem does not apply to weak linear logic, but then such a logic is incomplete just because it is weak. Godel does apply to the non weak logic where a modality (written traditionnaly by an exclamation mark: "!" read "of course") where "of course A" that is !A is equivalent with A & A & A & A & A & A & A .... (infinite resource)
Such logic is useful to manage "parallel computation" and it admits many variant related to many sort of "tensorial product".
Actually a very interesting paper in a similar spirit, but more physics-oriented, is the following one by Christof Schmidhuber (2000):
Here is the abstract:
What are strings made of? The possibility is discussed that strings are purely mathematical objects, made of logical axioms. More precisely, proofs in simple logical calculi are represented by graphs that can be interpreted as the Feynman diagrams of certain large–N field theories. Each vertex represents an axiom. Strings arise, because these large–N theories are dual to string theories. These“logical quantum field theories” map theorems into the space of functions of two parameters: N and the coupling constant. Undecidable theorems might be related to non perturbative field theory effects.
My own approach is radically different because I extract the resource from general hypothesis about cognition, so the weak logic are directly derived from the Godel incompleteness.