SUO: Re: Computable Manifolds & Discrete Topologies
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Pat Hayes wrote:
>
> > > 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.
What is that? --
Proof by cognitive dissonance?
> > Do I dare suggest a source?
>
> Please do.
CP, CE, NEM, ... I can be a mite more specific,
given time, or a clue as to the sticking point.
> 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.
Okay, so maybe you have already read all of the stuff
about signs and symbols in Peirce, and maybe you just
did not think enough of it to keep it in your head --
tastes differ -- but I, for one, believe it to be the
best stuff written on the subject of signs, bar none.
> If you have any actual thoughts on these matters yourself,
> I would be delighted to discuss them with you. However,
> your next paragraph:
I am sorry that I must defend myself with a flippant tone,
but I have experienced your particular style of "drive-by"
critique before, and I have invested many hours responding
to stuff that you have not stuck around to consider at all,
and the next time you drive-by, it is always another thing.
> > Or do you really prefer to reinvent for yourself the wheels
> > of this car that you do not see spinning all the while?
Yet another attempt at exegesis follows, interlinearly below:
> > Where I come from, being a symbol is
> > not an absolutely essential category
> > but a relatively interpretive category,
This means that I use the word "symbol" to describe
a specific function or role that a thing may serve.
In this connection, I use the word "sign" to describe
a more generic function or role that a thing may serve,
of which function or role a symbol is a special species.
> > 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.
This means that the best way that I know to make sense
of what is a sign (or any species: icon, index, symbol)
and what is not is to re-place the thing in question in
the context of the "relevant" sign relation -- which is
most likely rather closely related to the sign relation
that the person asking the question is "thinking in" at
the moment in question -- and just to examine it there.
> ... 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?)
Thanks for the career advice, but I do not understand
your sudden conversion to psychologism -- The Horror! --
in regard to a subject matter that I thought we were
discussing from a formal and a logical point of view.
> 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?
Assuming what is probably not yet true, if it ever will be,
that we are reading the symbol "symbol" in compatible ways,
I would opt to rephrase the question in the way that makes
the best kind of sense to me:
"How many symbols do 'you' interpret this map 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); ...
I am happy to work within these bounds,
but only because I have come of an age
where I respect practical necessities.
But you appear to still be talking about some feature or property
that you feel is "absolutely" or "essentially" given in the nature
or the substance of the map, as inherent to it, all by its own self,
while I am thinking more of the variety of "finity" that comes to be
toward the end of a process of picking out (finding or making) symbols
from the superficial appearance of a map, so is a constructed condition,
subject to constraints, of course, but what constraints are matters about
which we can only gather information within the same or a similar process.
> but if we say that the map surface is continuous,
We could 'say' it, Cap^n, but does saying it really make it so?
As far as I know, the whole Cosmos is one big finite automaton.
> let along differentiable, it is rather tricky to
> locate any suitable hereditable structures in it.
> And yet, the continuity of the map surface seems
> to be crucial in it being characteristically
> map-like rather than text-like.
That, I do not see.
> As I say, this is a real tension.
But do you really have any capacity to tell this tension from one that stretches
between the underwhelmingly small and the overwhelmingly large, but still finite?
> I think there is a way around it, but its not by any means trivial.
Maybe, but maybe I still don't get the question yet.
Jon Awbrey
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