> One issue I would appreciate reaction to is the
> linkage between mathematical rationality, the Goedelian incompleteness
> result, and consciousness. As I understand his thesis, Penrose wants
> to draw out the implication of formal incompleteness for the capacity of
> human insight into mathematical realities. Since he has received some
> "attacks" for his ontological commitments to mathematical structures,
> he seems to be trying to make the thesis plausible, espcially in
> responses of chapter 2 to a series of objections.
I will respond more fully after I have read chapter 2...I've just finished
chapter 1.
> 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 believe this is precisely the point Rosen is trying to make in his book
Life Itself. Rosen uses the term simulation where Penrose uses the term
computation. They are both talking about the same thing...a formalization
which only operates on syntax rules....where there is not any semantics
that is not represented by syntax. Rosen makes the point that in a
simulation (computational model) some of the necessary causal information
has been stripped out (removed) so therefor the simulation (in general)
cannot tell you anything about the physical reality. This would be the
"sparse ontological" case you mentioned.
Rosen would (I think) imply that the Platonist possition of the link
between mathematical rationality and the physical world (which he calls
Natural Law) holds only in general if we go back and re-consider the
four Aristelian causes...thus expanding our ontology. If we adopt the
formalist position (of only using "the rules which define the "game" of
marks on papers, etc") then we are limited to only understand simple
machines...and definately never to understand consciousness.
Jeff Prideaux