> "I would respond that omega is a _given_, an object of our
> immediate pre-rational experience. And to use the tools of symbolic logic
> to investigate an _empirically_ existing phenomenon..."
>
If you are a Platonist, all mathematical constructs actually exist in
their entirety. Penrose discusses the Mandelbrot Fractal in this
context, as an example of something which cannot even approximately be
rendered within a finite context, such as computer generated imagery.
I am on the other side of the fence. Omega is the name of a mathematical
object, and we can generate a finite number of finite statements about
the properties of Omega. Obviously, enumerating Omega is not physically
possible, and I don't believe in the existence of some mystical space
where one can enumerate Omega. Similarly, with the Mandeobrot Fractal,
we may explore admittedly inexact representations to the degree that our
computational resources allow. However, this does not say the Mandelbrot
Set exists as a complete object somewhere, only that we can explore some
aspects of it in a finite manner.
Mathematical theorems and conjectures are statements of how to generate
and investigate strings finite in length, grammar and syntax which
describe the mathematical objects and operations in relation to same
(the strings representing the objects only!)
Shades of Wittgenstein (I hope I spelled it right).
Such is MHO...
O---------------------------------O
| Hara Ra <harara@shamanics.com> |
| Box 8334 Santa Cruz, CA 95061 |
O---------------------------------O