>>13628411Historically, an "axiom" was essentially a truism, formalised as to make arguments rigorous.
The golden example of such a system was Euclids 'The Elements'. Wich laid out all the axioms wich were thought to be "obvious" about geometry.
One of the axioms, the parallel axiom, is equivallent to the statement that given a line and a point outside the line, there exists one and only one line through the point wich doesn't intersect the line.
This axiom looked like it could be proved from the other axioms, and hence mathematicians in Greece, later the arab world and finally Europe tried to prove this axiom from the others, this effort turned out to be in vain however, as every attempt failed.
Turns out that the parallel axiom isn't provable from the others, and the one who first got a propper model for geometry where multiple "parallel" lines exist to a given line but the other axioms of Euclid are satisfied was Italian mathematician Eugenio Beltrami (although many came close before him). The model was built on projections from surfaces with constant negative Gaussian curvature onto parts of the plane, and was hence, ultimately, built from Euclidian geometry, wich seemed to lack contradictions.
The geometry obtained was called hyperbolic (the reason has to do with the hyperbolic trig functions being used in a simillar way to the circular ones in euclidean geometry).
Since it turns out that we can build other geometries if we change the axioms, axioms went from being truisms, to things that different mathematical structures may or may not satisfy, and we then study the consequenses of these axioms. We later prove that given structures satisfy the axioms, and we can then use the results obtained in the abstract.
>I cam say that some ""wrong"" math is correct by changing the axiomsNo, because if you change the axioms, you're no longer talking about the same structure.
