SUO: Computable Manifolds & Discrete Topologies
pat hayes wrote:
>
> John, I think your requirement of differentiability is too strong.
> There are nondifferentiable surfaces even in geography, for example.
> Why do you need this? In fact, why do you even need to assume continuity?
> After all, computer screens have coordinate systems without continuity.
> If you relax these requirements I bet that one could apply dimensions
> even to mereology.
>
> Pat Hayes
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Pat,
The business about continuity and differentiability in the definition
of a manifold is really just an arbitrary "niceness" condition on the
ease of communication between two "observers" who are using different
"chart maps" from an object space to their personal coordinates, say,
"charts" i : X -> I and u : X -> U. Any application [u o i^-1](x) of
what is usually called the "transition map" [u o i^-1] : I -> U can be
interpreted in 2-person semiotic terms as "my name for what you call x",
or else in the 1-person way as "the new name for what I used to call x".
It is perfectly possible to consider other sorts of -- how do you say? --
"interoperability conditions" on these transition maps, say, ones that
might make eminent sense in a computational setting, like computability.
As far as the seemingly radical gulf between continuous and discrete,
one may always use the "discrete topology", which is defined to have
all sets be "open", and so all functions are "continuous", since the
topological definition of a "continuous function" is simply that the
inverse image (under the function in question) of an open set is open.
Jon Awbrey
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