Adrian Riskin 🇵🇸🍉 on Nostr: This is a common but fairly pernicious misconception about mathematics. Axioms are an ...

This is a common but fairly pernicious misconception about mathematics. Axioms are an aesthetic standard for defining mathematical concepts agreed to by a community that uses them. Acceptable means of deduction are community-designed epistemological standards. Deductions aren't verifiable without entering into the community's ways of knowing and in that strong sense are not at all objective, but instead are intersubjective. It's been known since Gödel that it's not possible to capture mathematical truth via axiomatization and checkable deduction outside of human insight. All deduction checking in mathematics is a community process.

This is the same way that communities can define any concepts that are essential for their needs, including health and security. They can then proceed to use the community's epistemological standards for drawing conclusions. Mathematics is exactly as concrete as any other intersubjective community knowledge. It's crucial to recognize this because the false idea that mathematics provides externally checkable concrete truths leads to its use by politically dominant communities to discard the conclusions of the people they dominate.

For an example of the failure of axiomatization and deduction it's not even necessary to look to Gödel. Just try to deduce Lagrange's theorem from the axioms of group theory -- that in finite groups the order of a subgroup divides the order of a group.

For an example of differences in community epistemology even among mathematicians compare constructive analysis to standard analysis.

For an example of the use of putative mathematical objectivity to dismiss subordinate people's understanding of their world compare the US government's insistence that the economy is doing well based on some formulas compared to the understanding of rent-burdened impoverished people who can't afford the necessities of life

To get back to OP's point, there are communities of computer users that spend a great deal of effort understanding what they mean by the security of the services they use. Through discussion and shared knowledge they can and do arrive at understandings that are as concrete and as intersubjectively rigorous as mathematics, and not to be dismissed from the outside.