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Peterdjones comments on Logical Pinpointing - Less Wrong
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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?