ONT Re: Category Theory
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CAT. Discussion Note 4
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John,
Yesterday's discussion of the "lattice of theories" brings back
to mind a number of recurring issues, and these in turn lead me
to think of a generic criticism that I would have about most of
the projects, formal or informal, advanced in this group so far.
Let us call it the "Forgetfulness Of Semiotic Computability".
The lattice is a fine and proper place to store our ideas.
But lattices and their elements are mathematical objects.
This means that we are not given these objects directly,
but only the signs of them. When I was a mathematician
I spake as a mathematician, which means that I remained
bissfully oblivious most of the time as to what it might
take to connect a concrete sign with its abstract object.
But when I put aside the larger part of that heavenly
paradise and started to focus on what was computable,
the world began to look a whole lot different, and
a lot of what I'd taken for granted as solid ground
became pure blue sky to my new computational and
sign-theoretic eyes. I still see a whole lot
of blue sky being kited in this group. And
one of the reasons for this is that nobody,
but me, of course, is even thinking to ask
what the computational properties of all
their high-flown expresssiveness might
actually turn out to be in the end.
I already know what happens to
projects when people do this.
You can read about it in
the Bible, under Babel.
In case it isn't clear, I'm talking about something over and above parsing syntax --
I'm talking about computing denotations and interpretants of signs, and what it
actually takes in practical computational terms to get from a theory, which is
just a set of syntactic elements called sentences, to its proper place in the
object lattice. This is the really big gap that will have to be crossed in
order to realize any of the envisioned plans for a lattice of theories.
So, don't get me wrong -- I'm all on board with this lattice of theories stuff.
Have been for a long time. But the next questions that I have to ask all have
to do with what it would take to make it real, and I'm still getting a lot of
the kind of hand-waving that I already know won't cut it.
To Contemplate for Next Time -- The Great Mandala or The Web Of Maya?
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| Language 1 Object Domain Language 2 |
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| o-----------o o-----------o |
| /| s s s ... |\~~~~~~~~~~~~~o~~~~~~~~~~~~~/| s s s ... |\ |
| / o-----------o \ / \ / o-----------o \ |
| / \ / \ / \ |
| o-----------o \ / \ o-----------o \ |
| | s s s ... |~~~~~~~~\~~~~~o~~~~~~~\~~~~~~| s s s ... | \ |
| o-----------o \ \ \ o-----------o \ |
| \ \ \ \ \ \ |
| \ o-----------o \ \ \ o-----------o |
| \ | s s s ... |~~~~~~\~~~~~~~o~~~~~\~~~~~~~~| s s s ... | |
| \ o-----------o \ / \ o-----------o |
| \ / \ / \ / |
| \ o-----------o / \ / \ o-----------o / |
| \| s s s ... |/~~~~~~~~~~~~~o~~~~~~~~~~~~~\| s s s ... |/ |
| o-----------o o-----------o |
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Figure 1. Lattice of Objects Inducing a Diversity of Sign Partitions
Jon Awbrey
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