Axioms can be demonstrated. They donât 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 thatâs 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 youâre not existing in a closed system.
You can posit axioms that donât 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 canât apply to this space and agree as such with whomever youâre trying to convey ideas.
When we talk about âabsolute truthâ, weâre talking about a point of universal rational consensus. Mathematics is a language that helps us extend subjective observation into objective conclusion. Thatâs what makes it a useful tool in scientific inquiry.
luciole@beehaw.org âš2â© âšmonthsâ© ago
Iâd say if your axioms donât hold you would go far in your quest for truth.
Malgas@beehaw.org âš2â© âšmonthsâ© ago
The thing that is absolute is a predicate of the form âif [axioms] then [theorems]â.
And the fun thing about if statements is that they can be true even when the premise is false.
luciole@beehaw.org âš2â© âšmonthsâ© ago
Of course in boolean algebra âif [false] then pâ is always true no matter âpâ, but itâs not telling us much.
lolcatnip@reddthat.com âš2â© âšmonthsâ© ago
Thatâs not a gotcha. Itâs literally the point of stating axioms.