SUO: Re: Monoclonal Antidotes
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Robert,
Here is how I think about the relationship between reality --
elements chemical and set-theoretical -- and the coordinate
maps of it that are generated by the observing eye, whether
the ordinary orb or the mind's eye. Models in general are
things of which a theory is true. In the real world folks
begin with models and populations and instances and such,
the things of which they hope their theories will someday
be true. Formal models were invented all the better to
armchair navigate the relationship between real models
and useful theories -- it's pretty clear from reading
Frege that he intended and understood this all along --
and if they fail at that purpose then they are not
much good for anything.
Here is sample piece of reasoning:
Let's say that we have a universe of discourse X that can be described in
terms of the predicates Green and Hot. We may want to express the fact:
"Green implies not Hot". In Peirce's Alpha, "P => Q" is represented as
"(P (Q))" and "not Q" is represented as "(Q)", so "P => ~Q" is written
"(P ((Q)))", but ((Q)) = Q, so we have have "(P Q)", which is to say,
"not both P and Q". Applying this abstract fact to the concrete case:
"Green implies not Hot" says the same thing as "Not both Green and Hot".
Here is how how I would think about just a couple of the levels of abstraction
that are involved in this piece of reasoning, and this is even before we bring
the machine into the business.
Predicates like Green and Hot are functions from a domain of interest, say X,
to the boolean space B = {0, 1}, and so their concrete types can be indicated
as Green : X -> B, and Hot : X -> B. The following diagram comes into play:
F
X o------>o B
\ ^
M \ / f
v /
o
B^2
What this signfies is the way that we use a coordinate representations M
to talk about a universe of discourse X. The coordinate map M classifies
or codifies everything in X according to its values with respect to the
predicates Green and Hot, for example, coding a Green & Hot thing with
the coordinates <1, 1>, and so the map M has the type M : X -> B^2.
Now suppose we define a new predicate Fui in terms of the basic predicates
Green and Hot, and say that Fui is defined this way: Fui = Green & Hot.
Fui is now a function from the "real world" X to the indicator domain B,
and so it has the "concrete type" Fui = F : X -> B.
But we can factor Fui into a composition of maps, F = f o M,
where M : X -> B^2 is the coding map and f : B^2 -> B is the
proxy of F that gets computed in terms of binary codes alone.
I would normally gloss over this mess of gory detail by just saying
that F (really its code representative f) has the "abstract type"
f : B^2 -> B.
Jon Awbrey
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