May 18, 2012

Limiting case of intersection of manifolds? Mathematical truth?

Filed under: Letters from Ionia — Tags: , , — zankaon @ 7:28 pm

If two manifolds have a shared tangent i.e. apposed, contiguous; then might this be considered as limiting case for intersecting manifolds? That is, 3 elements might be considered sufficient to define continuum (inbetweeness) for a matching of respective elements of 2 such sets. Or limiting case of positive definite neighborhood of an element of a set? Hence such intersecting would change the cardinality of manifolds; hence also changing the topology? If 2 manifolds had different topology; then would such intersection give a mixed continua i.e. topological mixture?

Would Law of Inertia for manifolds be one lesser example of a more general theme, as in title?

One might further entertain the  theme of ‘apposed’ manifolds. For ‘tangents’ of respective surfaces (manifold, in a loose sense; but actually geometry); that is ‘adjacent’, but not coinciding tangents, one is not just speaking of 2 elements (i.e. points) of respective sets. Rather 3 elements of respective sets might seem required to define (match) ‘limit’ (justapproach‘ ?) a common continuum (open sets of co-mingling elements; but no matching, and hence no ascertaining of topological equivalence?) for 2 respective manifolds. Rather no intersection (i.e. no overlap) of such manifolds. So no binary tree intersection demonstration for manifolds as elements, in an actuality sense. Rather just a matching (mapping) of such 2 manifolds, to see if topologically equivalent.

However if no ‘limiting case’ for 2 approaching manifolds. That is, apposed, contiguous, and adjacent terminology, are just geometric descriptions, then geometrically, tangents of 2 respective manifolds  never coincide i.e. no mixing of elements of respective open sets. In contrast, if they did, then they would intersect. So manifolds would seem to intersect or are disjoint; with no limit concept description.Thus no eventual intersection i.e. no eventual overlap of disjoint manifolds. Would this seem consistent with the theme that these mathematical entities, called manifolds, behave as if they want to be left alone (unperturbed), both mathematically as well as in a physical modeling sense? Might this then constitute an example of so-called mathematical truth? The clustering of primes; another example of mathematical truth? Is mathematics then in part just an empirical discovery of mathematical truth? Also if manifolds can neither be created nor destroyed (for example, retrospective extrapolation of our manifold to pre-Big Bang stage?), then this would also seem consistent with not being able to construct a continuum (i.e. manifold) from null set, even though one can construct a number set. See above null set discussion. Also if manifolds can neither be created nor destroyed, then on finer scale than Planck scale, there would seem to be continuation i.e. no cut-off to manifold at Planck scale. TMM


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