Mathematically speaking, what is an ‘Axiom’?
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An axiom is a logical statement that you decide you’re going to just assume is true. For most people, these will be obvious and well-established things, like
a=a
and
two parallel lines do not intersect
You need these kinds of assumptions for logic to have something to build on. It’s possible to logically prove these statements, but only by taking *other* statements as axiomatic – and so on, forever.
Your everyday life is built on axioms like “there is a reality external to my mind” and “my senses are able to perceive information from that outside reality” and “my mental model of reality, is reasonably accurate”. You have to assume *something* to get anywhere.
Notably, axioms do not have to be true. The geometry you learned in school is Euclidean geometry. Euclidean geometry takes it as axiomatic that planes are flat, and lines are straight. You may have heard that space is curved, and Earth is a sphere. In real life, [parallel lines frequently *do* intersect](https://notesychs.weebly.com/uploads/1/8/5/4/18542518/8163074_orig.png), and the interior angles of a triangle [don’t have to add up to 180°](https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/03/ShapeU_Sphere08.jpg).
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