On 5/20/2012 12:24 AM, Russell Standish wrote:
On Sat, May 19, 2012 at 01:17:18PM -0400, Stephen P. King wrote:
I finally found a good and accessible
that discusses my bone of contention. To quote from it:
"A theorem proved by Markov on the non-classifiability of the
that, given some comprehensive specification for the topology
of a manifold (such as
its triangulation, a la Regge calculus, or instructions for
constructing it via cutting
and gluing simpler spaces) _there exists no general algorithm
to decide whether the
manifold is homeomorphic to some other manifold _ [l]. The
impossibility of classifying
the 4-manifolds is a well-known topological result, the proof of
which, however, may
not be well known in the physics community. It is
potentially a result of profound
physical implications, as the universe certainly appears to
be a manifold of at least
Funnily enough, I remember from the dim-distant undergraduate days,
that the classifiability of 3 and 4-manifolds were open problems. 1&
2-manifolds had known classifications (2-manifolds are classified by
the number of "holes" (aka genus), for instance). Manifolds of
dimension higher than 4 are known to be unclassifiable. So a result
that 4-manifolds are unclassifiable would be a significant topological
result. What's suspicious is the claim that this was proved in
1960. Also suspicious in light of the Wikipedia entry claiming the
problem is still open: http://en.wikipedia.org/wiki/4-manifold
Could you be a bit more exact? The paper that I linked and quoted
was considering classification in terms of general algorithms. This is a
rather narrow case, no? I am not discussing the Poincare conjecture...
Perelman solved the 3-d case of the Poincare conjecture AFAIK. I am
pointing out something different, a bit more subtle.
Conversely, as for the 3-manifold problem, this looks it might have been
solved by Perelman's work that also solved the more famous Poincare
conjecture in 2003. If there's anybody about that more knowledgeable on these
matters, please comment.
Yes, the possibility that dovetailing via general algorithm is not
possible for 4-manifolds. This is important because if our perceived
physical world has a structure that cannot be defined by a general
algorithm then some other explanation is necessary. Bruno is trying to
convince us that our experiences of a physical world is nothing more
than the shared dreams of numbers. I believe that this is false, numbers
cannot form a primitive ontological basis from which our experiences of
our universe and its physics obtains.
I remember there was something peculiar about 4-dimensional space that
wasn't true of any other dimension - unfortunately, the sands of time
have erased the details from my memory. But I remember people were
speculating that it was a possible reason for why we lived in 4D
It is my opinion that we "live" in a 4D space-time because of this
non-computable feature. It cannot be specified in advance, thus we
actually have to go through the process of computing finite
approximations to the general problem of 4-manifold classification. This
problem and the one of QM (of finding boolean Satisfiable lattices of
Abelian von Neuman subalgebras or equivalent) are both places where
physics is not reducible to a pre-existing string of numbers.
My discussion of Leibniz' Monadology and its flawed idea of
pre-established harmony was an attempt to show how this problem has
shown up in philosophy many years ago and we are only now finding
solutions to it.
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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