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 -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.