Mathematically speaking, what is an ‘Axiom’?
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Axioms are fundamental assumptions that we propose, in order to make a logical system sound.
If I propose that 1+1=2, and if I propose that 2+2=4, then it follows that 1+1+1+1=4 as well. That last step is a deduction that follows other evidence, but where did that first evidence come from? Who says 1+1=2 in the first place?
In most sciences we can simply look at the world and answer that “we’ve measured it.” But in maths we can’t do that; it’s abstract. There’s nothing in the world that forces the assumption that 1+1=2. You could look at one apple and then another, but there’s no force in existence that defines those as _2_ and not _aX7;h_.
Because maths is abstract, there are a lot of logical trails you can follow to get to basic rules of it, and when you do, at some point you’ll simply have no more deductions to make — only fundamental assumptions to propose, without which your mathematics wouldn’t work.
And those are axioms; proposed fundamental assumptions, without which a logical system wouldn’t be sound.
Formal logic, and maths, has about a dozen of them, IIRC. Literally all else in maths can be derived from those few axioms. Everything. Every single computation you’ve ever made or will ever make in your life.
If you want to read more on this, look into Russell as a starting point. Very gifted logician, and a very clear communicator too.
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