From: Ben Goertzel (ben@goertzel.org)
Date: Wed Jan 25 2006 - 19:05:21 MST
Marc,
I disagree with much of what you say in this thread, but I find that
your suggestion ties in interestingly with a prior thread on this
list, on Subjective Reality.
Firstly, I think it is probably a non sequitur to say that the
universe is a formal system. I find it more sensible to consider
formal systems as approximate and partial models of the universe.
So, in my view, the universe is neither consistent nor inconsistent,
any more than a brick is either consistent or inconsistent. There may
be mutually consistent or mutually inconsistent models of the
universe, or of a brick.
The question you have raised, in this perspective, is whether the
"best" (in some useful sense) way of understanding the universe
involves constructing multiple mutually logically inconsistent models
of the universe. You have suggested that this might be the case but
have (IMO) not presented any solid arguments in favor of your
contention.
An alternative philosophical perspective is that, though the universe
is not in itself a formal system, the "best" way of understanding it
involves constructing more and more comprehensive and sophisticated
consistent formal systems, each one capturing more aspects of the
universe than the previous. This is fairly close to being a
rephrasing of Charles S. Peirce's philosophy of science.
It seems nice to refer to these two perspectives as Inconsistent
versus Consistentist views of the universe. (Being clear however that
the inconsistency and consistency refer to models of the universe
rather than the universe itself.)
Potentially the Inconsistentist perspective ties in with a previous
thread on this list regarding the notion of Subjective Reality. It
could be that, properly formalized, the two models
A) The universe is fundamentally subjective, and the apparently
objective world is constructed out of a mind's experience
B) The universe is fundamentally objective and physical, and the
apparently subjective world is constructed out of physical structures
and dynamics
could be viewed as two
* individually logically consistent
* mutually logically inconsistent
* separately useful
models of the universe. If so, this would be a concrete argument in
favor of the Inconsistentist philosophical perspective.
-- Ben Goertzel
On 1/25/06, Marc Geddes <m_j_geddes@yahoo.com.au> wrote:
> 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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