On 03 Dec 2010, at 19:35, Brian Tenneson wrote:

If there is a "TOE," I would expect it to be pretty lengthy and
complicated.  The "TOE" would basically be a conjunction of all
answers to all questions.

But can this even be done in human terms?  Wouldn't there be
infinitely many questions (e.g., what is 1+1, what is 1+2, what is
1+3)?  That would mean that "TOE" is infinite in length.  So then the
question becomes can this infinite set of answers can be abridged into
a finite, yet equivalent, document?  And finally, can we find a
particular document with all the answers and be able to prove that is
the most succinct those answers could possibly be?  (IOW, can we find
the shortest document that contain all answers to all questions?)

Assuming Mechanism, arithmetical reality is (easily) shown to be a part of everything. Gödel's incompleteness makes impossible to formalize completely just arithmetic, and thus everything. In a non mechanist theory of mind, a sort of physicalist ultrafinitism cannot be excluded.




Well the subject of questions and answers related to all this is given
here:
http://arxiv.org/abs/0708.1362

Wolpert seems unaware of the work of Blum, Case and Smith in Inductive inference? Well, he seeems unaware of my work which justifies why we have to take into account the different person points of view. I will try to find some time to read more cautiously Wolpert (everyday people send me a lot of pdf asking me to read them, so this can take time ...). I think it would be more efficacious to study my work and to find the weakness of the papers yourself.




That which answers questions is called an inference device.

An inference device works from data to theory, or from input-outputs couples to programs. It is the "inverse" of proof or of computations, somehow.


 It
appears that there might be some interesting results concerning strong
inference devices.  It would vastly simplify things if something is
considered an answer only if that answer has a finite proof.  Then the
question would become "how many finite proofs are there?"  There are
infinitely many different finitely long proofs.

In reasonable theories, the number of proofs is aleph_zero.



That leads me to the conclusion that "TOE" is not expressible in a
finite document.

Right. I mean IF you want your TOE to be complete. But this does not exist with Mechanism.




However, if there are finitely many "categories of proofs" then the
document would just be a summary of the categories of proofs which
would make the "TOE" document finite.  Two proofs are in the same
category if their conclusions are equivalent and not in the same
category if their conclusions are not equivalent, meaning that they
are not merely restatements of one another.

Thus, there are finitely many categories of proof if and only if "TOE"
is a finite document.
There being finitely many categories of proof implies that "TOE" is a
finite document.
There being infinitely many categories implies that "TOE" is an
infinite document.


Yes. Any *complete* TOE is an infinite document (unless assuming some strong ultrafinitist physicalism).

Bruno

http://iridia.ulb.ac.be/~marchal/



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