On Apr 25, 5:27 am, Bruno Marchal <marc....domain.name.hidden> wrote:
> Le 24-avr.-08, à 18:26, nichomachus a écrit :
>
>
>
>
>
>
>
> > On Apr 22, 11:28 pm, "Brian Tenneson" <tenn....domain.name.hidden> wrote:
> >> Perhaps Hilbert was right and Physics ought to have been axiomatized
> >> when he
> >> suggested it. ;) Then again, there might not have been a motivation
> >> to
> >> until recently with Tegmark's MUH paper and related material (like by
> >> David
> >> Wolpert of NASA).
>
> > The logical positivists were motivated to axiomatize in the predicate
> > calculus the laws of scientific theories in the early 20th century,
> > first because they believed that it would guarantee the cognitive
> > significance of theoretical terms in the theory (such as the
> > unphysical ether of maxwell's electromagnetism), and then later
> > because it had evolved into an attempt to specify the proper form of a
> > scientific theory. In practice this had too many problems and was
> > eventually abandoned. One of the consequences of this program was that
> > axiomatizing the laws of a theory in first order predicate calculus
> > with equality was that such a formulation of a theory always implied
> > various unintended interpretations. The amount of effort needed to
> > block these unintended interpretations was out of proportion with the
> > benefit received by axiomatization.
>
> It is a bit weird because it is just logically impossible to block
> those unintended interpretations. And This should not be a problem.
> The reason why physical theories are not axiomatize is more related to
> the fact that axiomatization does not per se solve or even address the
> kind of conceptual problem raised by physics.
Also to this point, that it is impossible to identify a theory with
any particular linguistic formulation of it. Theories are not
linguistic entities.
And since we’re on the subject: according to Max Tegmark, given the
apparent direction of inter-theoretic reduction, one may assume that
the foundational physics of our universe should be able to be
expressed in a completely “baggage-free” description that is without
reference to any human-specific concepts. This presumed most basic
law of the universe would be capable of being axiomatized without
unintended implications since the mathematical structure expressing
the most basic law would be isomorphic with the law itself to the
degree that it may appropriately be identified with it. The
mathematical laws which describe the phenomena of all of the emergent
levels or organization diverge from this ideal more and more the
further one proceeds from this unknown foundational theory.
> > Also, I
> > personally remain unconvinced that there is anything problematic about
> > the exitence of the universe of universes, or the ensemble of all
> > possible mathematical structures, thought it may not be well defined
> > at present. I don't believe that this is simply the union of all
> > axiomatic systems. If trying to define the Everything as a set implies
> > a contradiction, then fine -- it isn't a set, it's an ensemble, which
> > doesn't carry any of the connotations that are implied by the use of
> > "set" in the mathematical sense. Therefore each entity in the ensemble
> > is a unique collection of n axioms that has no necessary relationship
> > to any other axiom collection. What happens in an axiom system stays
> > in that axiom system, and can't bleed over to the next one on the
> > list. Some of these may be equivalent to each other.
>
> > A = The collection of all finite axiom systems
> > B = The collection of all consistent finite axiom systems
>
> I guess you mean "recursively enumerable" instead of finite. You would
> loose first order Peano Arithmetic (my favorite lobian machine :).
Really? It would seem that all recursively enumerable axiom systems
would exist in A.
> Note also that SAS occurs very quickly. SAS occur in theories which are
> much weaker than the SAS themselves (ex: SAS occur in Robinson
> Arithmetic, i.e. when you can define successor, addition and
> multiplication. SAS themselves need induction.
I don’t understand. Are you saying that Self Aware Substructures exist
in the Robinson Arithmetic?
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Received on Sat Apr 26 2008 - 00:55:09 PDT