[FOM] Choice of new axioms 1

[Andrej Bauer]

As an innocent onlooker to the current discussion about "choice of new 
axioms", I have been wondering all along why there needs to be one set of 
standard axioms that "normal" mathematicians use? Is this something that is 
good for mathematics, or are we all in search of aboslute truth, or what?

Proliferation of mathematically relevant statements which are independent of 
ZFC might lead to a form of relativism in mathematics, rather than adoption 
of any particular axioms that resolve those statements. I would in fact 
expect such relativism to be more fruitful than an absolutism of the 
"ZFC+my_holy_axiom" kind.

I do not deny that providing a single good foundation for mathematics was a 
great achievment of 20th century, but perhaps we're ready for the next step. 
Is it not the case that the lesson learnt from the story about euclidean vs. 
non-euclidean geometry is relevant at the level of foundations of 
mathematics? (The story is that once it was realized that there are many 
kinds of geometries, and that they are all "worthy", the result was very 
positive for development of geometry.)

Andrej Bauer