Is it observable? If yes it's provable. If not, its unprovable.
How do you prove "the only things we can prove are those things we can observe?"
Ehm no. I used weaseling out the indispensability argument from the oxford journal. From the colection of arguments against mathematical platoism.
From the abstract, it looks like she uses "sets of sentences." Whoa there, what's a set and are we using concrete sentences or types of sentences? This is me being lazy and not wanting to read the paper but if she touches on these objections I'll stand corrected.
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.
How do you know about your thoughts though?
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.
I certainly can't prove it by your definition of prove but I'm okay with that because I don't think your definition is tenable. I think I can prove that the claim that "math is essentially a description of the natural world" is untrue at the very least.
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.
Sure, but that's a different claim from "if you can't observe it you can't know about it."
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 ?
Didn't we observe it in the wild? I see 5 trees and 2 dogs and there's a logic to how these things can be split up and combined. This logic isn't something I invented, it's something we can clearly see in nature. But I can't see this logic-in-itself at all, there's nowhere in space-time I can locate the rules about how arithmetic works. Is there this kind of logic at all?
The way that mathematics has developed is that we figured out how general laws about how certain mathematical fields work from observation (disjoint unions of sets = addition) and from there we found out more things. But in doing so we've become unmoored from the empirical base: we can no longer justify our claims because recourse to experience because we have never seen 300 things being combined with 40030 things and yielding 40330 things. Appeals to "just continue on as before" beg the question of why we're allowed to continue on as before - I see a person enter a building and then leave a building every 5 minutes over the course of an hour but I'm certainly not justified in thinking that this will always happen. But we are always justified in thinking that 5 cows and 5 cows makes 10 cows.