Even if we can’t currently prove certain axioms, doesn’t this just reflect our epistemological limitations rather than implying all axioms are equally “true”?
It doesn’t and they are fundamentally equal. The only reality is the physical one—there is no reason to complicate your ontology with platonically existing math. Math is just a collection of useful templates that may help you predict reality and that it works is always just a physical fact. Best case is that we’ll know true laws of physics and they will work like some subset of math and then axioms of physics would be actually true. You can make a guess about what axioms are compatible with true physics.
This is an appealingly parsimonious account of mathematical knowledge, but I feel like it leaves an annoying hole in our understanding of the subject, because it doesn’t explain why practicing math as if Platonism were correct is so ridiculously reliable and so much easier and more intuitive than other ways of thinking about math.
For example, I have very high credence that no one will ever discover a deduction of 0=1 from the ZFC axioms, and I guess I could just treat that as an empirical hypothesis about what kinds of physical instantiations of ZFC proofs will ever exist. But the early set theorists weren’t just randomly sampling the space of all possible axioms and sticking with whatever ones they couldn’t find inconsistencies in. They had strong priors about what kinds of theories should be consistent. Their intuitions sometimes turned out to be wrong, as in the case of Russel’s paradox, but overall their work has held up remarkably well, after huge amounts of additional investigation by later generations of mathematicians.
So where did their intuitions come from? As I said in my answer, I have doubts about Platonism as an explanation, but none of the alternatives I’ve investigated seem to shed much light on the question.
It doesn’t and they are fundamentally equal. The only reality is the physical one—there is no reason to complicate your ontology with platonically existing math. Math is just a collection of useful templates that may help you predict reality and that it works is always just a physical fact. Best case is that we’ll know true laws of physics and they will work like some subset of math and then axioms of physics would be actually true. You can make a guess about what axioms are compatible with true physics.
Also there is Shoenfield’s absoluteness theorem, which I don’t understand, but which maybe prevents empirical grounding of CH?
This is an appealingly parsimonious account of mathematical knowledge, but I feel like it leaves an annoying hole in our understanding of the subject, because it doesn’t explain why practicing math as if Platonism were correct is so ridiculously reliable and so much easier and more intuitive than other ways of thinking about math.
For example, I have very high credence that no one will ever discover a deduction of 0=1 from the ZFC axioms, and I guess I could just treat that as an empirical hypothesis about what kinds of physical instantiations of ZFC proofs will ever exist. But the early set theorists weren’t just randomly sampling the space of all possible axioms and sticking with whatever ones they couldn’t find inconsistencies in. They had strong priors about what kinds of theories should be consistent. Their intuitions sometimes turned out to be wrong, as in the case of Russel’s paradox, but overall their work has held up remarkably well, after huge amounts of additional investigation by later generations of mathematicians.
So where did their intuitions come from? As I said in my answer, I have doubts about Platonism as an explanation, but none of the alternatives I’ve investigated seem to shed much light on the question.