17 November 2020, EXPRESS Seminar, Julien Murzi and Brett Topey

Abstract: On a
widespread naturalist view, the meanings of mathematical terms are determined,
and can only be determined, by the way we use mathematical language -- in
particular, by the basic mathematical principles we're disposed to accept. But
it's mysterious how this can be so, since, as is well known, minimally strong
first-order theories are non-categorical and so are compatible with countless
non-isomorphic interpretations. As for second-order theories: though they
typically enjoy categoricity results -- for instance, Dedekind's categoricity
theorem for second-order PA and Zermelo's quasi-categoricity theorem for
second-order ZFC -- these results require full second-order logic. So appealing
to these results seems only to push the problem back, since the principles of
second-order logic are themselves non-categorical: those principles are
compatible with restricted interpretations of the second-order quantifiers on
which Dedekind's and Zermelo's results are no longer available. In this paper,
we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly
more basic problem -- Carnap's categoricity problem for propositional and
first-order logic -- and show that our solution generalizes, giving us full
second-order logic and thereby securing the categoricity or quasi-categoricity
of second-order mathematical theories. Briefly, the first-order quantifiers
have their intended interpretation, we claim, because we're disposed to follow
the quantifier rules in an open-ended way. As we show, given this
open-endedness, the interpretation of the quantifiers must be
permutation-invariant and so, by a theorem recently proved by Bonnay &
Westerståhl, must be the standard interpretation. Analogously for the
second-order case: we prove, by generalizing Bonnay & Westerståhl's
theorem, that the permutation invariance of the interpretation of the
second-order quantifiers, guaranteed once again by the open-endedness of our
inferential dispositions, suffices to yield full second-order logic.