I understand your point—it’s akin to the Box quote “all models are wrong but some are useful”—when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful—of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.
My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I’d be happy.
My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I’m happy to agree that this secondary claim is not mainstream.
My primary point is actually that I don’t care if math is useful. Math is awesome. This is obviously an extremely rare viewpoint, but very common among.
But I do agree with that quote, more or less. I think that potentially some models are true, but those models are almost certainly less useful for most purposes than the crude and easy to work with approximations.
I agree that second-order logic is not necessary to work with the integers. Second-order logic is necessary to work with the integers and only the integers, however. Somewhat problematically, it’s not actually possible to work with second-order logic.
I understand your point—it’s akin to the Box quote “all models are wrong but some are useful”—when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful—of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.
My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I’d be happy.
My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I’m happy to agree that this secondary claim is not mainstream.
My primary point is actually that I don’t care if math is useful. Math is awesome. This is obviously an extremely rare viewpoint, but very common among.
But I do agree with that quote, more or less. I think that potentially some models are true, but those models are almost certainly less useful for most purposes than the crude and easy to work with approximations.
I agree that second-order logic is not necessary to work with the integers. Second-order logic is necessary to work with the integers and only the integers, however. Somewhat problematically, it’s not actually possible to work with second-order logic.
What sort of practical tasks are you thinking of?