On 5/20/2012 12:24 AM, Russell Standish wrote:
On Sat, May 19, 2012 at 01:17:18PM -0400, Stephen P. King wrote:
  Dear Bruno,

     I finally found a good and accessible 
paper<http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&sqi=2&ved=0CEoQFjAA&url=http%3A%2F%2Fntrs.nasa.gov%2Farchive%2Fnasa%2Fcasi.ntrs.nasa.gov%2F20050243612_2005246604.pdf&ei=8NO3T9LmFu-d6AHAq_3uCg&usg=AFQjCNHMBmwAi1K7yJY3oRBnJrYRC2H9RA&sig2=yb-YNcKWR6LNPSVy8bQquA>
that discusses my bone of contention. To quote from it:

"A  theorem  proved by Markov  on  the  non-classifiability  of  the
4-manifolds  implies
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
four  dimensions."

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
Hi Russell,

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...

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.
Perelman solved the 3-d case of the Poincare conjecture AFAIK. I am pointing out something different, a bit more subtle.

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
space-time.

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.

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.

--
Onward!

Stephen

"Nature, to be commanded, must be obeyed."
~ Francis Bacon


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