From: Ben Goertzel (ben@goertzel.org)
Date: Sun Apr 27 2003 - 16:57:34 MDT
> > > It is literally the set of all formal systems,
> > > because it is the set
> > > of all consistent mathematical systems.
> >
> > I consider formal systems to be a subset of the set of
> > consistent mathematical systems but you have equated
> > them.
>
> What other sort of mathematics is there outside of formal systems?
Hmmm... most real mathematics is in fact NOT fully formalized, but I'll
never tire of advertising the Mizar project (www.mizar.org) which has gone a
remarkably long way toward formalizing real math like calculus, topology,
abstract algebra, etc. Like Russell, Hilbert and the whole crew, I tend to
believe that all math is indeed formalizable in principle, even though most
of it has not been fully formalized in practice. Admittedly this is not
*proved* but I'm not sure I've heard anyone dispute it before! I'm curious,
what is an example of a consistent mathematical system that you believe is
not fully formalizable?
> > Surely the most primitive entities are also the most numerous.
>
> Nope. Not at all, for the same reason that the Chaitin-Kolmogrov
> information of the set of all integers is tiny compared to that of
> almost all individual integers.
Hmmm... the "more primitive entities are more numerous" idea definitely
needs some refinement...
As you point out, it is wrong if "primitive" means algorithmic information
content, but there could be some other definition of "primitive" that made
it correct...
-- Ben G
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