[hackinthebochs]

>And in fact mathematics is a great place to dive in, because one very important question -- namely, whether mathematical realism is true¹ -- is really a metaphysical question, and one which can never be answered by empirical observations about the world.

The important distinction between math and metaphysics is that the substantive content of the field of mathematics does not turn on whether mathematical realism is true. Mathematics tells us something about the world by telling us what can be true about the world. We know that there are prime numbers because we know that no such world exists where prime numbers are finite. But this is just saying that a system with certain axioms have certain relationships, or necessarily exclude certain relationships (e.g. whatever axioms of arithmetic you take).

On the other hand, there is no substantive content in metaphysics (at least as far as ontology goes) if the statements do not pick out real entities.

[ubernostrum]

On the other hand, there is no substantive content in metaphysics (at least as far as ontology goes) if the statements do not pick out real entities.

You've just committed the fallacy of begging the question -- you assumed the truth of mathematical realism in this line, and arguably in a couple others.

If mathematical realism is false, various parts of your argument start to crumble.

[hackinthebochs]

I fail to see how the quoted portion of my comment assumes mathematical realism. I could see how other parts of my comment might suggest I'm assuming MR. Assuming you meant to reference the portions where I talk about how the results of mathematics tells us something about the world, modal structural realism[1] is an underappreciated interpretation of mathematics.

[1] https://plato.stanford.edu/entries/nominalism-mathematics/#M...