[FOM] The boundary of objective mathematics

[Eray Ozkural]

On Sat, Mar 14, 2009 at 6:19 PM, Paul Budnik <paul at mtnmath.com> wrote:
> Einstein, Feynman, a recent Noble Prize winner Gerard 't Hooft and many
> others have come to suspect that physics is ultimately discrete or
> digital (see http://www.mtnmath.com/digital.html for quotes and
> additional names). Thus it is at least a respectable philosophical
> position to take. Potential infinity is very useful in developing
> mathematics and our universe could be potentially infinite.

This is an interesting point. And I think it goes right back to the
Aristotelian tradition. The point that Paul makes is that mathematical
theories have an objective (or non-fictitious) meaning only if they
can be given a physically plausible interpretation. (I tried to
advance such a point on sci.math previously) So, for instance, if our
universe is indeed made up of  finite space as current cosmology
suggests, then it might not make sense to focus on abstraction of
infinite space (such as the axiom of infinity in ZFC). However, I do
not believe that all those possible combinations have ever been
analyzed rigorously. Famously, Aristotle argued that time may be
potentially infinite (unbounded). However, of course, he argued
against infinitely extended bodies, and in general completed/actual
infinity. Current cosmology seems similar, there is no limit to time
in theory; and the universe is finite in extent. Although, of course,
our universe is expanding as well (according to current theory), thus
one could say that space is also potentially infinite.

On the other hand, it would be interesting to see what the
consequences of alternative cosmology models would be (such as "many
directions of time" theory). Thus, I think we should note the deep
interplay of astrophysics/cosmology and philosophy of mathematics.

Best Regards,

PS: To give another example, it makes sense to think of integers as
"objective" because if quantum theory is true (and by all appearances
and experiments it looks that way), the fundamental resources of the
universe are quantized. There is even quantization of time in new
work.

-- 
Eray Ozkural, PhD candidate.  Comp. Sci. Dept., Bilkent University, Ankara
Research Assistant, Erendiz Supercomputer Inc.