Re: SUO: Re: Proposed SUO Content Outline
>John,
>
>It seems that your definition of coordinate system below could apply
>to anything
>that consists of a two sets and a function (okay, maybe a function that
>"is (locally at least) one-to-one, continuous, and differentiable")
>between them.
>
>As such, is this definition too general, in that it doesn't go far
>enough in distinguishing what is special about a coordinate system
>from other kinds of things that consist of two sets and a certain
>kind of function between them?
>
>At what point of generality should the thing we define as a coordinate system
>look different from other things?
Actually it occurs to me that we could insist that a coordinate be a
finite n-tuple of values each of which must be from some totally
ordered set. This would give a natural notion of dimension, and it
would allow 'spaces' to be things like discrete pixellated surfaces
as well as smooth manifolds. It would even allow things like the word
count in a text to be a kind of dimension; but it would not be
completely vacuous.
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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