Marvin McDonald <mcdom@AUGUSTANA.AB.CA> writes:
>The interest in Penrose's new book, SHADOWS OF THE MIND, hits as I am
>working through it. One issue I would appreciate reaction to is the
>linkage between mathematical rationality, the Goedelian incompleteness
>result, and consciousness.
If one supposes that the primary requirement for any thought at all
involves the phenomenology of consciousness (e.g. as described by writers
such as Merleau-Ponty in his Phenomenology of Perception, as well as by
Whitehead, Popper, Jaspers and others), than rationality, including the
mathematical variety and its theorems, are necessarily dependent upon
consciousness, not the other way around.
>This "excess" of mathematical understanding over that of
>computational "knowledge" operates as an "existence proof"
>for noncomputational capacities in humans.
O.K., but from within the conceptual framework of mathematical thought.
>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.
Insofar as Platonism implies an order of reality which is independent, in
some ultimate sense, of the mind and brain which is the foundation for the
required conceptualizations, then this confuses the question of
noncomputational capacities in humans, which are not to be equated with
Platonic concepts of mind.
>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.
Questions of ontology aside, if mathematics involves intellectual tools for
rigorous logical thinking, its insights cannot be automatically applied to
other areas of intellectual inquiry in any case, perhaps especially at very
high levels of abstraction where the specifics of implications may be
totally obscure.
> 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.
Perhaps mathematicians would feel they are in a better position to make the
case ("for no automatic implication from mathematical insights to other
forms of understanding") but, as I understand them, not only Merleau-Ponty
and the phenomenologists but also Whitehead and Popper (who of course also
qualify as mathematicians) have already made the case in effect very well.
>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 would suggest there is no general answer to this question. I have been
reading John L.Casti's _Alternate Realities: Mathematical Models of Nature
and Man, recently recommended to PCP readers by Don Mikulecky, and it seems
to me that Casti's descussions demonstrate very clearly the step by step
and topic by topic approach required. Analytic thinking is a tool, not the
whole game!
Cheers and best wishes.
Bruce B.