I draw some connections between realist structuralism about mathematics and higher-order metaphysics. I first revisit two problems for realist structuralists: the incompleteness objection and the identity problem. I argue that Shapiro's responses to these problems are inconsistent with certain assumptions about ground, namely Fine's rule of lambda introduction. I show the consistency can be avoided by rejecting lambda introduction and taking on a rule that has received much attention in highe…
Read moreI draw some connections between realist structuralism about mathematics and higher-order metaphysics. I first revisit two problems for realist structuralists: the incompleteness objection and the identity problem. I argue that Shapiro's responses to these problems are inconsistent with certain assumptions about ground, namely Fine's rule of lambda introduction. I show the consistency can be avoided by rejecting lambda introduction and taking on a rule that has received much attention in higher-order metaphysics called "beta-identification". I argue that taking on beta-identification allows realist structuralists to have absolutely and structurally indiscernible places without inferring their identity. I close by considering some objections to my account in the form of puzzles of indiscernibility and reference.
