A child doesn't have to know what "weight" *is* — that it's a function (and measure) of gravity — in order to put it to use deciding whether they'll be able to pick something up. It's useful because of facts about the world unknown to the child. The map is not the territory.
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What on earth are you talking about? I don't see engineers beating down anyone's doors; I see them using established maths to predict real-world outcomes *all of the time.* Seriously, Aaron, do you actually think the features of the universe are decided in committee?
If you believe that physics has nothing to do with maths, we're done. That's just patently absurd. Listing a bunch of topics/subfields in math that don't have any known applications in physics is the kind of red herring that makes me wonder just how stupid you must think I am.
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I don't think it's a red herring, though? I think that's precisely my point, physics doesn't inform mathematics, though sometimes physicists/engineers/etc discover some math is useful, it's a vanishingly small portion of the output of mathematics.
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And what the practical users pull keeps changing, the foundations of that mathematics have changed many times over since Euclid, and will continue to change again. Believing in the axiom of choice isn't a matter of rigor: it's a choice, is it useful? Does it feel right?
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