Mathematically speaking, what is an ‘Axiom’?
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An axiom is a logical statement that we accept as true, and can then deduce things from.
For example, we could create a set of axioms that describe how the real numbers work (saying things such as “There is an operation called the addition that has some properties”, “There is an element that we note 0 such that for every real number x, x + 0 = x” “There is an operation called the multiplication that has other properties”, etc….). Once we are satisfied with our description of the numbers, we can actually start proving things.
The key idea is that we don’t try to prove the axioms. We decide them to be true, and then math is deducing theorems from these axioms that we chose as ground-truth. With a different set of axioms, we would deduce different things.
The reason why we spend way more time studying the truths from one set of axioms over a different one, is that that set of axioms seems to match our understanding of reality well and we are able to create useful models using it. You could study what happens if you choose as ground truth 1 + 1 = 3, but you won’t get very useful results from it.
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