Le 25-juil.-07, à 17:31, Lennart Nilsson a écrit :
>
> Bruno Marchal <marchal.domain.name.hidden> wrote:
>
>> Concerning the math, do you know the book by Torkel Franzen on the
>> uses
>> and misuses of Godel theorems? Despite some big mistake I will talk
>> about, it is a quite excellent book which I would recommend
>
> I have read this book and would very much want to know what big
> mistake you
> are talking about.
Let me insist that I do love Torkel Franzen book, and recommend it
heartily to those who want a readable concise introduction to Godel's
theorem, and some warning on many possible abuses, indeed.
Having said this, here are the "big" mistakes I mentionned (just
recall that all good books have mistakes!):
First in the subtitle of the book: "An Incomplete Guide to its Use and
Abuse". But no "use" is really described. Franzen does not cite Post
positive use of incompleteness, nor does it mention Benacerraf,
Chihara, Reinhard, Webb ... or myself. Also, Church thesis is not
really explained despite proof of incompleteness based on
computability(*), nor are the provability logic mentionned.
(*)Compare with the first or second footnote in my Plotinus paper:
http://iridia.ulb.ac.be/~marchal/publications/CiE2007/SIENA.pdf
Second: after demolishing correctly some misuses of Godel in theology,
and assessing the role of faith in science, he concludes that Godel's
result are irrelevant in theology. Such a conclusion is just not valid.
Even if 100% or the current uses of Godel in theology are abuses, it
does not logically follow that Godel is irrelevant in that context.
Actually if you define the (pure) theology of the machine M by all the
true propositions *bearing on* the machine M (unprovable by machine M),
then it is a theorem that the (pure) theology is given by G* (minus G)
and their intensional variant. So pure theology, for a vast class of
machines, is given by the arithmetical interpretation of the difference
logics: G* \ G, X* \ X, Z* \ Z, G1* \ G1, X1* \ X1, etc.
Third: I disagree with his critics of Hawking use of Godel. Hawking has
been probably just a little not rigorous enough but it is easy to add
the arithmetical realist assumption (for example) to make his reasoning
rigorous. Yes, arithmetical incompleteness entails that we can build
machine such that no physical theory can ever describe completely their
long run behavior, which is part of the physical reality (assuming or
not the ontic (primitive) existence of a physical world, but being
realist (platonist) on number relations. So I think Torkel is a bit
unfair with Hawking analysis.
Fourth: I do think Franzen is also a bit unfair with Rudy Rucker too.
True, Rudy Rucker should have add in his seventh step (page 116 in
Torkel's book) that Godel knew the truth of the sentence only
conditionally from the assumption that the machine is correct, and of
course the machine knows that conditional truth too! Those who read
completely the "infinity and the mind" book by Rudy Rucker can
understand that he has seen this point, mainly trough his appendix on
man and robot.
Fifth: this is perhaps the most embarrassing mistake (or bad
pedagogical shortcut) in the misuse setting, especially with respect to
my own work (but I reassure: none of the many experts in logic who has
study my work has ever criticized it and there is nothing controversial
about it: I do have problem but only with literature-philosophers who
apparently never read my work).
Indeed page 125 in his book, Franzen asserts that a system cannot fully
even just postulate its consistency nor assert it, and that the
unprovability of consistency is really unassertibility of consistency.
This apparently goes against my dicto that consistency can *only* be
postulated, and certainly can be asserted, unlike self-soundness for
example. Here Torkel confuses the Rogerian sentence/machine with the
Godelian sentence/machine. The godelian sentence asserts its own
unprovability, and can be shown (by the lobian machine itself)
equivalent with a consistency statement. The Rogerian sentence/machine
asserts its own consistency, and can be shown (by the lobian machine
itself) equivalent with 0=1.
So a consistent machine can asserts its own consistency (on the
contrary a consistent machine cannot assert or even define its own
soundness). Three type of things can happen when a machine asserts its
own consistency. If the machine means it globally, the machine become a
Rogerian machine, and it will become inconsistent (prove the false). I
guess this is what Torkel was thinking about.
But the machine can means it locally, and just add its late consistency
as a new axiom. In that case the machine transforms itself in a *new*
machine and, not only remains sound (if it was before) but will be more
efficacious. It will speed itself up, shortening many proofs.
The machine can also assert it even more locally by defining a new
provability predicate, like the Z predicate Bp & Dp (Provable(p) and
Consistent(p)). In which case the machine discovers an intensional
variant of provability which obey some other provability logic, despite
its extensional equivalence with the Godel beweisbar predicate. This is
how the machine can discover the "person points of view" (hypostases).
All this can be proved by the machine:
Indeed G proves: (and thus the machine proves all this with p being any
sentence written in the language of the machine).
B(p <-> ~Bp) <-> B(p <-> Dt) (Godelian sentences)
B(p <-> B~p) <-> B(p <-> Bf) (Kreiselian sentences, or Smullyanian, I
would say)
B(p <-> ~B~p) <-> B(p <-> f) (Rogerian sentences)
B(p <-> Bp) <-> B(p <-> t) (Lobian sentences)
In french: the machine can prove that
a sentence provably equivalent to its own unprovability is provably
equivalent to consistency (Godel)
a sentence provably equivalent to its own refutability is provably
equivalent to inconsistency (Kreisel, Smullyan)
a sentence provably equivalent to its own consistency is provably
equivalent to a contradiction (Roger)
a sentence provably equivalent to its own provability is provably
equivalent to a tautology (Lob)
Note that you can substitute the word "sentence" by "machine", by
defining the machine dynamic by the modus ponens rule only (Hilbertian
"machines").
OK, this could *look* severe, so I insist that Franzen book is *quite*
excellent, and probably the best one optimizing the tradeoff
time/results. His appendices are also quite good and I do think that
book could help a lot in this list, for those interested in the
arithmetical UDA, alias the interview of the universal (and lobian)
machine.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Thu Jul 26 2007 - 10:13:00 PDT