[FOM] Choice of new axioms 1

[Andrej Bauer]

On Thursday 16 February 2006 13:02, you wrote:
> OK, maybe I was reading you too uncharitably. Do you put this forward as a
> general strategy for mathematics, or limit it just to the strong set
> theoretical principles (say, the ones that go beyond ZFC)? If the latter,
> what is the principled reason for this selectivity? That is, why not to
> treat e.g. the axioms of power-set, separation, infinity, or even the
> axioms of PA, and their negations, in the same way?

I put this forth as a general strategy for mathematics, not just ZFC. And 
since I am not advocating total mathematical anarchy, yet, there should be 
some principle for selecting certain axioms (and logics) over others. Here I 
do not have an answer, other that such a principle will lie outside 
mathematics and logic proper. Perhaps good mathematics is useful mathematics, 
or beautiful mathematics, or profitable mathematics, or publishable 
mathematics, or terror-preventing mathematics, or whatever mathematics makes 
one look cool, or whatever one's laptop can compute, I just don't know.

I am just going on my gut feeling that absolutism is a bad thing.

A visible consequence of what I am suggesting is this: young mathematicians 
should _not_ be taught a single "standard" kind of logic and axiom system. 
Instead, students should be taught how to think with or without classical 
logic, axiom of choice, large cardinals, powersets, etc. In effect I am 
proposing that f.o.m.ers make themselves indispensable for the forseeable 
future. Surely you must agree :-)

Andrej