My impression (which might be partially a result of not understanding second order logic well enough) is that logical pinpointing is hopeless in at least two senses: (1) it’s not possible to syntactically represent sufficiently complicated structures (such as arithmetic and particular set theoretic universes) in some ways, and (2) trying to capture particular structures that are intuitively or otherwise identified by humans is like conceptual analysis of wrong questions, the intuitions typically don’t identify a unique idea, and working on figuring out which class of ideas they vaguely identify is less useful than studying particular more specific ideas chosen among the things partially motivated by the intuition. For example, in set theory there are many axioms that could go either way (like the continuum hypothesis) that as a result identify different set theoretic universes.
What’s left is to reason with theories that are weaker than is necessary to pinpoint particular models (and to introduce additional data to stipulate some properties we need, instead of expecting such properties to fall out on their own). On the other hand, physics, if seen as a model, seems to make the semantic view of mathematics more relevant, but it’s probably isn’t fully knowable in a similar sense. (Learning more about physical facts might motivate studying different weak theories, which seems to be somewhat analogous to this sequence’s “mixed reference”.)
My impression (which might be partially a result of not understanding second order logic well enough) is that logical pinpointing is hopeless in at least two senses: (1) it’s not possible to syntactically represent sufficiently complicated structures (such as arithmetic and particular set theoretic universes) in some ways, and (2) trying to capture particular structures that are intuitively or otherwise identified by humans is like conceptual analysis of wrong questions, the intuitions typically don’t identify a unique idea, and working on figuring out which class of ideas they vaguely identify is less useful than studying particular more specific ideas chosen among the things partially motivated by the intuition. For example, in set theory there are many axioms that could go either way (like the continuum hypothesis) that as a result identify different set theoretic universes.
What’s left is to reason with theories that are weaker than is necessary to pinpoint particular models (and to introduce additional data to stipulate some properties we need, instead of expecting such properties to fall out on their own). On the other hand, physics, if seen as a model, seems to make the semantic view of mathematics more relevant, but it’s probably isn’t fully knowable in a similar sense. (Learning more about physical facts might motivate studying different weak theories, which seems to be somewhat analogous to this sequence’s “mixed reference”.)