I think this is roughly pointing to what I'm calling the layer for "useful maths conventions," and I agree that they're socially constructed, but don't find that a helpful lens to understand them much like I don't find that helpful for understanding e.g. typing or data structures
If you convince yourself that "math" is just whatever the current fallible mathematicians persuade each other it is, and there's nothing deeper, you have to explain why rockets care what mathematicians think.
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Engineering and math are different things, and rockets don't care about computable numbers or the axiom of choice or the excluded middle, but mathematicians think about these things. Do you define truth based on what rocket scientists care about?
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Math is integral to physics, engineering, etc. Not all mathematical concepts will be forced to contend with reality anytime soon, but when they do, nobody gets to say "my math was right, the rocket was wrong."
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FWIW I just don't think one's understanding of what math is has to imply anything about social ideology