And other theories of the ontological status of mathematics aren't unprovable speculation? I'm not sure what you're looking for here.

Is it observable? If yes it's provable. If not, its unprovable.

Are you referring to the Field project? Because it's not particularly clear that his project actually succeeded (he had to use second-order logic, "set theory in sheep's clothing).

Ehm no. I used weaseling out the indispensability argument from the oxford journal. From the colection of arguments against mathematical platoism.

What are non-empirical objects? Are thoughts non-empirical? This seems like you're starting with some hardcore empiricism and expecting us to accept this as obviously true.

Empirical means knowledge acquired through observation and experimentation. But for the sake of argument, you can imagine it as something we can observe, test, work with,etc... I used it in the sense of Scientifical evidence.

"Non-empirical objects" Are immaterial, unobservable, non-testable claims.

Are thoughts non-empirical

Oh you won't go infinite regress on me now. Thoughts are results of the workings of our brain. They do not exist outside of brain. Which I also think about language and the human construct we call mathematics.

This seems like you're starting with some hardcore empiricism and expecting us to accept this as obviously true.

Ehm no. I'm myself hardcore empyricist. If you cannot prove your claim, certainly don't invoke it as fact. And don't build upon that shaking foundation is my approach. But take it reasonably.

When you have a wast scientific body of evidence describing a phenomen that we can't observe. Then it's relatively reasonable to believe in that.

This default position thing is intellectual laziness.

How come? Default position is : I don't know, whatever the issue may be. But, the important thing is the observation part. Did we or did we not observe mathematics somehow in the wild, beyond our brains ? I'm not even sure how would such evidence look like. And I'm pretty sure it is impossible. And when your argument is irrefutable, the default position is. I don't believe that.

Now I believe mathematics is this. But I cannot prove it, because I don't know how. Much like the other side of the argument.