[FOM] Epistemology for new axioms

[Timothy Y. Chow]

Joe Shipman wrote:

> I can foresee the "pluralism" possibly leading in an unhealthy 
> direction, with the kind of fruitlessness that is characteristic of the 
> endless debates about "interpretations" of quantum mechanics. If some 
> mathematical communities concentrate on developing set theory in 
> incompatible directions than others do (V=L vs Large Cardinals, for 
> example), they might not have much to say to each other, especially if 
> the methods and techniques are different enough that it will be uncommon 
> to be expert in both "schools".

I like your analogy with quantum mechanics, but I don't see anything 
"unhealthy" about your scenario, or at least nothing more unhealthy than 
the usual perils of specialization.

Suppose set theorists split into two camps as you suggest.  I'm not sure 
that it will be any "worse" than, say, the split between finite group 
theorists and infinite group theorists.  The axioms "the ground set is 
finite" and "the ground set is infinite" are incompatible, and so finite 
group theorists and infinite group theorists "might not have much to say 
to each other" and there might not be many who are experts in both 
subjects.  But so what?

It's true that set theory is closer to philosophy than most other areas of 
mathematics are, and debates about the foundations of mathematics may be 
similar in flavor to the debates about the interpretations of quantum 
mechanics.  But I don't see what is so harmful about such debates.  They 
don't prevent anyone from getting on with their technical work, since 
there is still a consensus about what constitutes a technically correct 
argument or calculation.  Once in a long while, the philosophical debates 
may even inspire some fresh technical idea.  So what's the problem?  If 
people want to engage in philosophical debates, why not let them?  If you 
don't like the lack of "progress" (whatever that means) then just ignore 
the debates.

Tim