Abram,
I suspect what it comes down to - I'm tossing this out off-the-cuff - is
that each new branch of maths involves new rules, new operations on numbers
and figures, and new ways of relating the numbers and figures to real
objects and sometimes new signs, period. And they aren't predictable or
derivable from previous ones. Set theory is ultimately a v. useful
convention, not an absolute necessity?
Perhaps this overlaps with our previous discussion, which could perhaps be
reduced to - is there a universal learning program - an AGI that can learn
any skill? That perhaps can be formalised as - is there a program that can
learn any program - a set of rules for learning any set of rules? I doubt
it. Especially if as we see with the relatively simple logic discussions on
this forum, people can't agree on which rules/conventions/systems to apply,
i.e. there are no definitive rules.
All this can perhaps be formalised neatly, near geometrically. (I'm still
groping you understand). If we think of a screen of pixels - can all the
visual games or branches of maths or art that can be expressed on that
screen - mazes/maze-running/2d geometry/ 3d geometry/Riemannian/ abstract
art/ chess/ go etc - be united under - or derived from - a common set of
metarules?
It should be fairly easy :) for an up-and-coming maths star like you to
prove the obvious - that it isn't possible. Kauffman was looking for
something like this. It's equivalent, it seems to me, to proving that you
cannot derive any stage of evolution of matter or life from the previous
one - that the world is fundamentally creative - that there are always new
ways and new rules to join up the dots.
Mike,
The answer here is a yes. Many new branches of mathematics have arisen
since the formalization of set theory, but most of them can be
interpreted as special branches of set theory. Moreover,
mathematicians often find this to be actually useful, not merely a
curiosity.
--Abram Demski
On Tue, Aug 26, 2008 at 12:32 PM, Mike Tintner <[EMAIL PROTECTED]>
wrote:
Valentina:In other words I'm looking for a way to mathematically define
how
the AGI will mathematically define its goals.
Holy Non-Existent Grail? Has any new branch of logic or mathematics ever
been logically or mathematically (axiomatically) derivable from any old
one? e.g. topology, Riemannian geometry, complexity theory, fractals,
free-form deformation etc etc
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