Mathematically speaking, what is an ‘Axiom’?
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One equals one. It’s both apparent and true, and doesn’t really benefit from deconstruction
The term axiom refers to a fundamental fact. This is the case in philosophy as well (and in ancient times many mathematicians were also philosophers).
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.
An axiom is a statement that you accept as true, so that you can move on with your life, and continue working on whatever it is that actually interests you. If you accept the axiom as true, then you can use that to prove a bunch of other stuff is true as well.
– Math is like games. Axioms are the rules of a game.
– You can design a game with with whatever rules you want. In math, you can develop a theory with whatever axioms you want.
– However, not all rules make fun games. Likewise, not all axioms make interesting or useful mathematical theories.
People tend to use [ZFC set theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) as an example of an axiomatic theory, but I think it gives the misleading impression that axioms are always these low level rules that are “fundamental” and set in stone.
In practice, axioms are much more fluid than that and they exist in higher level theories: [groups](https://en.wikipedia.org/wiki/Group_(mathematics)), [rings](https://en.wikipedia.org/wiki/Ring_(mathematics)), and [vector spaces](https://en.wikipedia.org/wiki/Vector_space) have way more relevance in practical applications.
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