I don't think that's what Matt was saying, but you'd have to ask him. I think the field of math, and the definitions it uses and the consensus of say, using ZFC are all social constructions.
The fact that simple-seeming axioms are actually sometimes very contentious is like the rigor issue — we can not know the answer, or the field can be mistaken about the answer, but that does not mean there is not an actual ground truth
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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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FWIW I just don't think one's understanding of what math is has to imply anything about social ideology