[FOM] "Mathematician in the street" on AC
paul at mtnmath.com
Mon Aug 17 12:07:18 EDT 2009
Timothy Y. Chow wrote:
> For a concrete example, if a student asks a professor, "My classmate said
> that there exist non-measurable sets. Is that true?" Almost surely, the
> professor will say yes without hesitation. However, if a philosopher
> approaches the same professor and asks, "Do you believe that
> non-measurable sets truly exist?" then it is likely that the response
> will be completely different.
Almost every mathematician would agree that finite mathematical
statements are objectively true or false. Most would agree that some
infinite mathematics, such as the computer halting problem, are
objective. Where is the boundary between mathematically true and
philosophically true and why? What is the epistemological status of
mathematics that is not philosophically true?
I have argued that only mathematical statements that are logically
determined by a recursively enumerable sequence of events are
objectively true or false. (See www.mtnmath.com/pom.html). The rational
is that these are statements that are objectively determined by events
each of which can occur in an always finite but unbounded universe. Thus
they have a connection with physical reality as we experience it. There
is no other known source for mathematical intuition other than personal,
cultural or genetically inherited experience with physical reality.
Gödel's proof suggests that mathematics is both creative and objective.
Something similar is suggested by the combination of Cantor's proof that
the reals are not countable and the Lowenheim-Skolem theorem. We can
assume the reals are a fixed set because those provably definable in a
given formal system are. However we can always expand the system so more
reals are defined. Statements like AC and CH are true, false or
undecidable only relative to a particular formal system.
Mountain Math Software
More information about the FOM