Axioms can be demonstrated. They don’t have to be purely theoretical.
Mass and Energy are axiomatic to the study of physics, for instance. The periodic table is axiomatic to understanding chemistry. You can establish something as self-evident that’s also demonstrably true.
One could argue that mathematics is less a physical thing than a language to describe a thing. But once you have that shared language, you can factually guarantee certain fundamental ideas. The idea of an empty set is demonstrable, for instance. You can even demonstrate the idea of infinity, assuming you’re not existing in a closed system.
You can posit axioms that don’t fit reality, too. And you can build up features of this hypothetical space that diverge from our own. But then you can demonstrate why those axioms can’t apply to this space and agree as such with whomever you’re trying to convey ideas.
When we talk about “absolute truth”, we’re talking about a point of universal rational consensus. Mathematics is a language that helps us extend subjective observation into objective conclusion. That’s what makes it a useful tool in scientific inquiry.
The way I see it, axioms and notation are made up but everything that follows is absolute truth
I’d say if your axioms don’t hold you wouldn’t go far in your quest for truth.
The thing that is absolute is a predicate of the form “if [axioms] then [theorems]”.
And the fun thing about if statements is that they can be true even when the premise is false.
Of course in boolean algebra “if [false] then p” is always true no matter “p”, but it’s not telling us much.
That’s not a gotcha. It’s basically just the definition of an axiom.
Axioms can be demonstrated. They don’t have to be purely theoretical.
Mass and Energy are axiomatic to the study of physics, for instance. The periodic table is axiomatic to understanding chemistry. You can establish something as self-evident that’s also demonstrably true.
One could argue that mathematics is less a physical thing than a language to describe a thing. But once you have that shared language, you can factually guarantee certain fundamental ideas. The idea of an empty set is demonstrable, for instance. You can even demonstrate the idea of infinity, assuming you’re not existing in a closed system.
You can posit axioms that don’t fit reality, too. And you can build up features of this hypothetical space that diverge from our own. But then you can demonstrate why those axioms can’t apply to this space and agree as such with whomever you’re trying to convey ideas.
When we talk about “absolute truth”, we’re talking about a point of universal rational consensus. Mathematics is a language that helps us extend subjective observation into objective conclusion. That’s what makes it a useful tool in scientific inquiry.