Mathematicians choose what to keep according to aesthetics. Mathematics can only be defined as what mathematicians like.
For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Zero took a long time to be accepted, starting in India. Negative solutions of quadratics were illegitimate until astonishingly recent days. Complex numbers were accepted even more recently. Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently, when it turned out interesting theorems could live there.
People worked outside these boundaries all along, but what they wrote didn't catch on. Laplace's transforms were ignored and forgotten until they were needed to shore up Heaviside's extremely practical D operator. Complex numbers turned out to be needed to for electromagnetics. Once people got deeply into the topic, they discovered beauty and then mathematics accepted them.
Mathematics is the world's largest and longest-running effort to produce a collective work of sublime beauty. What is beautiful goes in, what isn't dies with its creator. New forms come to be seen as beautiful as they are shown to open new vistas to explore, but very slowly.
> For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics.
Note that this is more or less a myth.
> Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently
Astronomers did a huge amount of sophisticated spherical geometry, from Mesopotamians through e.g. Hipparchus and later Ptolemy, then Arabs/Persians, Indians, medieval Europeans, right down to the present.
For a long time, they didn't like irrationals, so anything involving those didn't count as mathematics. Zero took a long time to be accepted, starting in India. Negative solutions of quadratics were illegitimate until astonishingly recent days. Complex numbers were accepted even more recently. Greek geometers knew they lived on a sphere, but spherical geometry was too unpleasant to contemplate until quite recently, when it turned out interesting theorems could live there.
People worked outside these boundaries all along, but what they wrote didn't catch on. Laplace's transforms were ignored and forgotten until they were needed to shore up Heaviside's extremely practical D operator. Complex numbers turned out to be needed to for electromagnetics. Once people got deeply into the topic, they discovered beauty and then mathematics accepted them.
Mathematics is the world's largest and longest-running effort to produce a collective work of sublime beauty. What is beautiful goes in, what isn't dies with its creator. New forms come to be seen as beautiful as they are shown to open new vistas to explore, but very slowly.