Mathematically speaking, what is an ‘Axiom’?
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Humans discover physics. Physics exists out there already and we stumble upon it, test it, and try to understand it. The line between discovery and invention in mathematics is much more fuzzy. Negative numbers didn’t always exist. At some point, somebody thought “wouldn’t it be cool if there were numbers less than zero to signify owing money” (I don’t even know which came first, 0 or negatives, but just go with it.) And that person said “let there be negative numbers” and thus, negative numbers now exist. That’s an Axiom. Once someone decided negative numbers exist, we have a brand new playground of math to probe and explore.
Same thing happened with imaginary/complex numbers and infinity and a host of other things. What does it mean for 2 things to be equal. Defining certain operations that can be done like a union or replacement. Things like that. They are the way they are because we said so. Mathematicians throw a bunch of axioms in a blender together to discover what math drops out the other side… More or less, it’s obviously much more difficult than that.
Now, sometimes, we assume too many things at once or not specific enough. This gets us into trouble because we may end up being able to prove paradoxical inconsistencies like 1=2. At which point, we need to go back to the axioms and reword them, add prohibitions for certain activities (like dividing by 0), or throw them out. Doing so over and over for the hundreds of years modern algebra has been around has made math more and more eerily efficient at describing the natural world around us.
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