From: Simon Gordon (sim_dizzy@yahoo.com)
Date: Sun Apr 27 2003 - 18:39:54 MDT
--- Ben Goertzel <ben@goertzel.org> wrote: > > With
regards to informal systems, they are
> conceivable
> > so sure they exist. Formal systems are but a tiny
> > infinitesimal subset of the set of all
> mathematical
> > systems, according to my intuition ;)
> >
> I don't understand how you're defining "mathematical
> systems" ...
>
system def= a coherent structure of interelating
elements.
"mathematical system" is as opposed to a physical
system, i.e. a system which exists as an independent
abstraction. Mathematical systems do not have to
conform to any formal rule structure, they just have
to "live" in the same abstract mathematical plain as
the other systems e.g. algebra, topology, systems of
set theory. You have to remember that there are a vast
many branches of mathematics that havent even been
conceived yet, clearly the SIAIs have got their work
cut out, we have only scratched the tip of the iceberg
mathematically. The sort of mathematics we have got at
the moment is presumably very basic and simple in
comparison, but its hard to imagine that ALL the
mathematical discoveries that SIs will ever make will
fall into the "formalizable" category. At some point
we are bound to hit some really weird stuff which
cannot even be axiomatized!
Simon.
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