SUO: Re: Computable Manifolds & Discrete Topologies
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| And then sweeping up the jokers
| that he left behind
| you find he did not leave you very much
| not even laughter
| Like any dealer he was watching for the card
| that is so high and wild
| he'll never need to deal another
|
| Leonard Cohen, 'The Stranger Song'
Pat Hayes wrote:
>
> > 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.
I see that I should have written: 'Any application [i o u^-1](x) of
what is usually called the "transition map" [i o u^-1] : U -> I 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".'
> > 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, ...
It is a rather persistent mannerism of mine that I use the word "arbitrary"
to imply the existence, the operation, and the relevance of an "arbiter",
so when I say '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', all that I mean to imply is that people
in communication about an objective space, where "objective"
includes but is not limited to the notion of an intentional,
and thus potentially "inexistent, yet" object, have choices
about the particular form of "nicety" that they think might
be appropriate to their objective situation. Of course, to
say that they have choices, as when their options are in the
nature of abductive hypotheses about what is true, or what is
to do, or what is to hope about a real situation, is not to say
that the choice is facetious, or that every choice will turn out
to have been, after the fact, equally admirable, advisable, feasible,
felicitous, or just plain good. Still, the facts of the matter may be
something that no mortal folk will be able to discover until after a more
or less lengthy accumulation of efforts, errors, experiences, and experiments.
> mere matter of semiotics:
I realize that it is something of an automatic cliche in some circles
to speak of "mere semantics" or "mere semiotics", but I no longer run
in those circles, rendering this a mannerism of which I am past cured.
> it refers to the spaces themselves.
Alas! Here is where you eternally find me at a disadvantage --
with no direct access to the nature of the spaces themselves --
I can only sweep up the jokers that it and thee leave behind --
| Please understand, I never had a secret chart
| to get me to the heart of this
| or any other matter
> (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'.
What is a generalization that stretches from one shore to the other,
if not just such a bridge? Did you expect it to identify the banks?
| Let's meet tomorrow, if you choose
| upon the shore, beneath the bridge
| that they are building on some endless river
But here, again, you must possess some special insight
into what it is that qualifies the "natural open sets",
that is I say, from amidst all the conceivable choices.
And there are those who probably say, with discernment
like that, you ought to turn your eyes upon the matter
of "natural kinds", what marks them among all the sets.
| and you say ok the bridge or someplace later.
> 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.
Now you understand just how many people view set theory
in relationship to the task of describing natural kinds.
| Then he leaves the platform
| for the sleeping car that's warm
| You realize, he's only advertising one more shelter
| And it comes to you, he never was a stranger
| and you say ok the bridge or someplace later.
> 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;
| And then taking from his wallet
| an old schedule of trains
| he'll say, I told you when I came I was a stranger
| I told you when I came I was a stranger
> 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).
By some people.
> 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?
Do I dare suggest a source? 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. But, of course, the subspecies of
folks who worked out all of these fine subtleties did
it way back before they had science or bicameral brains
or anything, or could tell their logical pons asinorum
from a psychological hole in the ground, so I guess that
you are really on virgin territory here, "real maps" or not.
> 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.
Hey, be my guest -- you are quite obviously the
first person who has ever thought of such things.
No really! -- I am a collector of isolated sources.
Now & then they do come up with the strangest truths.
Jon Awbrey
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