Re: SUO: Re: Computable Manifolds & Discrete Topologies
> > 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?
>
>Rhetorical questions? Or do you really want to find out?
Well, I've spend several years (and a fair bit of the taxpayer's
money) trying to, so I guess the answer must be yes.
>Do I dare suggest a source?
Please do. I have read everything that I could find on the topic from
as many sources as I could find, but if you have any more, please
point me to them. If you have any actual thoughts on these matters
yourself, I would be delighted to discuss them with you. However,
your next paragraph:
> Or do you really prefer to
>reinvent for yourself the wheels of this car that you
>do not see spinning all the while? Where I come from,
>being a symbol is not an absolutely essential category
>but a relatively interpretive category, so none of the
>questions that you asked above would be considered as
>making any sense beyond the context of a sign relation
>that someone or another has in mind for them. That is
>to say, there is no fact of the matter, independent of
>every circumstance, there are only facets of how it is
>in fact minded, when it comes to saying whether a thing
>is a symbol or not.
... sounds to me like an elaborately expressed, though in fact
rather facile, excuse for avoiding doing the real work. (If you were
serious about what you say, then shouldn't you be studying
psychology, rather than thinking about Venn diagrams?)
To side-step your excuse, let me rephrase the question about how many
symbols there *are*, into how many symbols could a map be intepreted
to have? The point being that any kind of Tarskian notion of
interpretation requires that the symbols be only finitely 'deep', or
at best countably infinitely so (considered as set-theoretic
structures, they need to be hereditarily finite/countable); but if we
say that the map surface is continuous, let along differentiable, it
is rather tricky to locate any suitable hereditable structures in it.
And yet, the continuity of the map surface sems to be crucial in it
being characteristically map-like rather than text-like. As I say,
this is a real tension. I think there is a way around it, but its not
by any means trivial.
Pat Hayes
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