To make the "target" clear, here's my summary:
Since the incompleteness of formal systems can be extended to
Turing machines [with the addition of a single, well accepted
axiom], the incompleteness shows that Turing machines cannot
"capture" the full range of mathematical insight.
This "excess" of mathematical understanding over that of
computational "knowledge" operates as an "existence proof"
for noncomputational capacities in humans. The case of
mathematical insight is NOT seen as preeminent, but because of
the resources of 100 years of work on math foundations, we are in
a better position to "make the case" than in other areas.
On to my question: IF we accept the argument, the implications for
human understanding more broadly are supported by a Platonist under-
standing of mathematical rationality. Human capacities to function
in mathematics reflect a crucial access to an important order of
reality: sets and other mathematical objects. IF, on the other hand,
we are a bit more sparse in ontology, say in a formalist move, and
view the nature of mathematical insight as grounded in the rules which
define the "game" of marks on papers, etc. that we call mathematics,
then there is no AUTOMATIC implication from mathematical insight to
other forms of human understanding. The formalists, though, are left
with the "Unreasonable reasonableness" of mathematics that raises
questions about the applicability of non-Euclidean geometries to physical
systems, complex numbers in systems of equations describing physical
systems, etc.
So, jointly, WHY do our mathematical systems, however obstruse, find
"application"? What is the relationship between mathematical insight
and human understanding in other domains? I am likely to want to
jump on the evolutionary epistemology bandwagon at this point, but
I would appreciate others' reactions.
Peace,
Marvin (MAC) McDonald mcdom@augustana.ab.ca