From: Marc Geddes (m_j_geddes@yahoo.com.au)
Date: Wed Jan 25 2006 - 18:23:48 MST
Daniel >>>
>I do not believe there is such a thing as a partially
consistent mathematical system. A system is either
consistent or inconsistent, and if it is inconsistent,
it doesn't work.
*A* particular formal system has to be consistent
(because in an inconsistent system you can prove
anything), but my suggestion was that a full
description of reality may require *several*
over-lapping formal systems. Each system *in itself*
would be consistent, but the different systems would
not be fully consistent *with each other*. An analogy
here would be a 3-D movie. To get the 3-D effect two
different versions of a scene are shot - each version
is shifted slightly in space (one version for each
eye). Each version of the scene is consistent in
itself (left eye version or right eye version), but
the two versions are not fully consistent with each
other. (Consider the two versions to be analogous to
several formal systems).
>What he has said is that there is no a priori reason
to say you are wrong. There is also no reason to say
you are right. So why should we believe you?
Here are my reasons:
If there's only one unitary mathematics which is fully
consistent describing reality, then there appears to
be no way to explain the existence of mathematics
itself. The trouble is that in a formal system which
is fully consistent and complete (and at least complex
enough to include arithmetic) there are true
statements that can be phrased in the language of the
system which cannot be proved true within the system -
this of course follows from the Godel theorems. So if
the formal system running reality was fully consistent
and complete then one would have no choice to conclude
that there must exist an endless number of true axioms
(since for any number finite number of axioms there
would be some true math statements which escaped the
net - Godel). So with a single formal system it's an
endless tower of turtles all the way down. But the
explanation as to *why* this system of mathematics
existed could never be explained. It would just have
to be accepted as a brute fact.
Now this to me seems to run contrary to the scientific
method, which by its very nature assumes that there
are no unexplained supernatural brute facts of
reality. And mathematics itself is a part of reality.
So a full explanation of reality should require
science to explain mathematics as well. And as I just
pointed out, with a single formal system running
reality, this couldn't be done - there'd be an endless
tower of turtles all the way down.
Now if, however, we adopt my superficially
'unappealing' suggestion that reality as a whole is
inconsistent, the situation changes. There would then
be several different over-lapping formal systems
needed to fully describe reality. *And each formal
system could be used to provide an explanation of the
others* Thus mathematics could 'explain itself' and
there need be no endless unexplained tower of Godelian
turtles. So you see, my suggestion is not so
unappealing after all.
"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder’s eye on the last day”
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