[FOM] Epistemology for new axioms

[Timothy Y. Chow]

Joe Shipman wrote:

> Tim, the difference is that finite group theorists' results and infinite 
> group theorists' results are not logically incompatible, they can use 
> each others' work. Those who favor V=L and those who favor large 
> cardinals (large enough to contradict V=L) can build up independent and 
> incompatible developments of mathematics, and both will not like being 
> forced to always reframe their results as implications of unproven 
> axioms when in their own world and that of all their colleagues those 
> axioms are always assumed.

Okay, I can just barely imagine mathematics bifurcating into a "V=L camp" 
and a "large cardinal camp" where in each camp, axioms that are 
incompatible with the other camp so thoroughly permeate the entirety of 
mathematics that nobody wants to keep track of which theorems rely 
essentially on such axioms.  (As an analogy, the vast majority of 
mathematicians accept the law of the excluded middle and don't have much 
interest in keeping track of when it is really needed.)  I agree that this 
would be a somewhat annoying state of affairs.

But I'm not going to lose sleep over this possibility.  Currently, it's 
only set theorists who find themselves using these axioms, and they're 
accustomed to keeping track of which axioms are needed for which theorems. 
One day, perhaps, it won't just be Harvey Friedman finding applications of 
large cardinal axioms outside of infinite set theory.  But by that time, 
the landscape of mathematics may have changed in ways that we can't 
foresee today, so I feel that it is premature to worry about how the 
mathematical community might cope in that hypothetical future world.

Tim