[FOM] "Mathematician in the street" on AC

[Paul Budnik]

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.

Paul Budnik
Mountain Math Software
www.mtnmath.com