Mathematically speaking, what is an ‘Axiom’?
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When we say that a mathematical fact is “true” or “false”, that’s not really a statement about the real world. Circles and triangles and numbers and equations don’t really “exist” outside of textbooks and the minds of mathematicians. Mathematics is *useful* for getting things done in the real world, like building bridges, but there aren’t any actual mathematical objects in a bridge, just bits of metal and stuff. We’ve figured out ways to do mathematics and then relate it to real life in such a way that it works out well.
Instead, mathematical “facts” only exist in the context of systems that are built on axioms. For example, everyday geometry is built on axioms like “you can draw a straight line between any two points” and “all right angles are the same”. Using these axioms, you can build up lots of theorems like “the angles at the base of an isosceles triangle are equal”. And that’s very useful for real life, because real life happens to correspond quite well to these axioms. But those theorems are only “true” in the sense that they can be derived from that particular set of axioms. If you use a different set of axioms, then different theorems are “true”.
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