[FOM] Choice of new axioms 1
Andrej.Bauer at andrej.com
Thu Feb 16 09:18:47 EST 2006
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 :-)
More information about the FOM