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.
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.