FOM: instrumentalist dodge

[Penelope Maddy]

Dear Colleagues,

I'd like to add a brief word about what John's been calling the
'instrumentalist dodge'.  While I agree with him that it's a pointless
and harmful move, I think I'd argue for that position somewhat
differently.

Suppose, in the simplest case, we're to choose between ZFC + MC and
ZFC + Con(ZFC + MC).  John suggests that choosing the second is like
adopting the physical theory that the world behaves as if there are
atoms or that the world came into existence five minutes ago with all
the fossils, etc., that lead us to believe it's older than that, etc.
We might add Descartes evil demon to this list: there is an evil demon
who fools my senses into thinking the world is as it seems.  Some of
these hypotheses, like the evil demon, fall in the category that
philosophers call 'hyperbolic doubt'; many, perhaps most, philosophers
these days think there's nothing you can say to this radical skeptic,
but that this doesn't much matter.  The hypothesis about the atoms is
a little different, in that some respectable scientists believed it
until the early 1900s, for respectable scientific reasons.  It was
overthrown in the usual scientific way, by various theoretic
calculations and experiments.  Only philosophers like Bas van Fraassen
continue to think there's something to it, and the arguments against
them seem to me very strong.  My worry about all this is that I'm not
sure the analogy between the physical and the mathematical can bear
the weight of transporting the particular considerations relevant to
the physical case over to the mathematical.

In fact, the mathematical case strikes me as fairly simple.  We have
two proposed theories, as above.  We want our theories to be
consistent, and measurable cardinals are in some (remote) danger of
being inconsistent, but both candidate theories take the step of
assuming that they are consistent with ZFC.  After that, what's to
worry?  What seems to motivate a preference for the weaker theory is a
concern that 'measurable cardinals might not exist, really!', in other
words, an ontological worry; related worries could be phrased in terms
of realism or anti-realism, in semantic terms (e.g., about meaning) or
in other metaphysical terms (e.g., about truth).  My naturalist takes
such all worries to be irrelevant to methodology, to the question of
which axioms to adopt.  (Please notice that I'm *not* saying
'consistency implies existence'.  I'm saying that consistency is a
goal of the practice, and that existence in the philosophical sense is
irrelevant to the methodological decision.)

What *is* relevant, according to the naturalist, is: which theory is a
better mathematical tool for the given mathematical goals?  One goal
here is the one John mentions: we want a framework capable of
interpreting all our theories in mathematically natural ways.  (John
might sometimes say 'in a meaning-preserving way', but I'd prefer to
stick to the more concrete mathematical desiderata that I suspect lie
behind such phrasing.)  It usually goes without saying, but we also
want our theory to be easy to work with, easy to extend, not
needlessly ad hoc, etc.  I'd make the simple suggestion that ZFC + MC
meets these criteria better than ZFC + Con(ZFC+MC).  If nothing else,
it's just plain more convenient and straightforward, both completely
legitimate reasons to prefer it (according to the naturalist).

Anyway, there's my two cents.

Yours,

Pen