ONT Re: Differential Logic A -- Discussion
DLOG A. Discussion Note 8
Perforce a necessity that custom deems practical, empirical researchers
and not just applied mathematicians are accustomed to approach the more
refractory objects of inquiry by strategic orders of approximation, and
thus to lay down the successive strata of a representation that renders
every object, that may be simple enough in its own right, complex to us.
We come to the question of whether the analysis is terminable, and thus
convergent to a definite result, or interminable, and thus inconclusive
in its indications of the object. It is in the setting out of analytic
representational series that we see the importance of closure operators,
because they clue us in to how an otherwise infinite representation may
wrap up in a finite term.
Here is one way to see this. A typical form of analytic expansion will
generally be conducted with respect to an operator Q in such a way that
the successive levels of analysis correlate with increasing powers of Q,
as Q^0, Q^1, Q^2, Q^3, ..., and so on. If the operator Q is subject to
a law that makes all higher powers redundant after some point, then one
has the power to sum up what is logically an infinite series in what is
computably a finite term.
Some of the simpler operator laws that might turn up,
at least, among those that are not entirely trivial,
are those of the form Q^2 = 0, Q^2 = 1, Q^2 = Q.
A "closure operator" C is one that obeys a rule of the last mentioned shape,
since requiring C(C(x)) = C(x) for all x is the same as saying that C^2 = C.
In algebraic language, one refers to such an operator as being "idempotent".
Cf. Kelley, 'General Topology'. http://suo.ieee.org/ontology/msg03874.html
I'll discuss this further, later on,
in the context of concrete examples.