David McAllester (
Wed, 2 Nov 94 10:05:21 EST

I agree that "it is too narrow a view of mathematics to consider
it as only concerning the consequences of an initially chosen set
of axioms". This is only the first (and the main) component of
mathematics. The second one is the art of inventing interesting,
powerful, wonderful etc. sets of axioms.

The point I have been trying to make (perhaps too often now) is
that thinking seems to require thinking Platonically, even for
syntacticists who, I claim, think Platonically about syntax. It may be possible
to "compile" different axiom systems into one's subconscious so that
one can shift one's Platonic thinking between different axiom systems.
I am skeptical however. It seems more likely that we think more effectively
if we commit our Platonic thinking to a single system. For example, even
set theorists who spend their time considering a wide variety of axiom systems
are generally fierce Platonisists in their commitment to the existence of V.
When they think about other systems then think Platonically (presumably in the same
fixed metatheory) about various MODELS of the alternate systems (such as L where
the axiom of choice is indeed true).