"But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world"
This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.
They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.
I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.
This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.
They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.
I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.