Mathematically speaking, what is an ‘Axiom’?
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I’ve given an answer before that might help:
To put it simply, math is about creating abstract “systems”. The rules that govern these systems are the axioms. You can create (again, to put it simply) __any__ abstract system with arbitrary rules, and as long as those rules are consistent and not contradictory, it is a valid mathematical system.
To give an example, suppose i create abstract system to “count”. I give this system a set of axioms, for example whether i have one amount and add another, or have the other amount and add the first, i should get the same result. These axioms (the fundamental axioms of algebra) let us create a system that I can, simultaneously, use to count apples, as i can to count distances, something completely different! How crazy is that!
When the real world gives us examples where this system breaks, (I went 3 miles, then another 4 miles, but the distance is only 5 miles from the beginning not 7!), is when we create a new abstract system with its own axioms – in the above example, we deal with vectors instead of numbers directly.
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