Mathematically speaking, what is an ‘Axiom’?
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An axiom is a basic assumption that underpins all your other reasoning, and that you aren’t likely to give up in different situations.
So in a way, it is ‘just’ an assumption, but it is one that you think is not likely to be very context dependent, and that you’ll stick with much of the time.
Most people want to live, and so for most people, a desire to live might be an axiom. You don’t make a logical argument that you want to live, you *just do* want to live (hopefully). (Maybe we can make some argument about evolution or something resulting in a self-preservation instance, but that begins outside of you, rather than internal to your logical reasoning.)
Many people might have some axioms in their ethics/morality. Maybe some people care about equality. Maybe some people care about suffering (or preventing suffering). Maybe some people care about the word of God. If they believe them as the foundation of their ethics, then we might be able to phrase those ethical principles as axioms. If someone thinks equality is important, it can be hard for them to give a *reason* they think that – it just is what they think matters.
For mathematics, axioms are the foundational assumptions of a branch of mathematics. I’ll give an example (now, it turns out that most mathematical axioms are more estoeric and weird than this example, but I think it is a decent example just to get the vibe here). Consider the assumption that “The number 1 exists.” It is hard to give *reasons* that the number 1 exists. We just assume that it exists, because well, chances are most people doing maths just need that to be the case, no questions asked. I think actual modern mathematics actually go to some deeper or more abstract level, so eventually the existence of the number 1 I think becomes something you need to prove once you get pedantic enough, but as a general idea, I think this example is ok.
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