That's a fair point and an interesting view. I would consider the axioms as necessary but incomplete knowledge. You need more knowledge to apply these axioms.

That's a fair point and an interesting view. I would consider the axioms as necessary but incomplete knowledge. You need more knowledge to apply these axioms. A bit like the axioms are the parts of a car. You can use the car schematic and the parts to create a car. Or you can generate the missing knowledge of the schematic by spending lots of time combining random parts. I am a bit out of my depth here, but maths can be reformulated. I presume like Minkowski did with Einstein's relativity. Or Riemann with integration. The result of the reorganized knowledge is deeper understanding. There is both compression and correction going on, and even correction can be seen as compression by removal of special cases.