Re: 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
>
>¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
>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.
Thanks for the lesson, Jon, but I disagree. The distinction between
discrete, continuous and differentiable is not arbitrary, and not a
mere matter of semiotics: it refers to the spaces themselves. (It is
also by the way not a matter of a choice of coordinate system.)
Depending on the kinds of space X, U or I might be, not all your
mappings will be invertible and not all of them will be composable;
you need to be more careful when using algebraic talk in an analytic
setting. And while one can of course use the discrete topology, it
doesnt provide any kind of bridge across that 'radical gulf'. For
example, consider a computer screen and give the set of pixels the
discrete topology; then the relation (in that topology) between two
adjacent pixels is the same as that between two pixels at opposite
corners of the screen, or indeed between *any* two pixels. The
discrete topology is like a car without wheels: it just abandons the
task of describing the spatial structure of the space in question.
There are some very real tensions between discrete and continuous
ways of understanding space, and they arise rather acutely when
trying to give a semantics for maps (real maps, the kind that people
use to find their way around; which, by the way, I recommend as a
much richer domain for semiotic explorations than Venn diagrams and
Boolean combinatorics, which have been kind of done to death). For
example, what counts as a 'symbol' on the surface of a road map? Is
every line representing a road a symbol? Every line *segment*? Are
there uncountably infinitely many symbols on the map, therefore? How
does one parse such a language? If the space of the map is dense, can
a map symbol have zero thickness? But if the map is discrete, how
does it represent a continuous geographical space? And so on; all
fascinating stuff. If you are interested, I can give you some hints:
for example, the map location of a symbol does not, in general,
denote the location of the thing symbolized.
Pat Hayes
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