Does anyone know if there is there a name for the idea that new concepts (such as negative numbers, complex numbers or type constructors in a type system) are only ever introduced into a preexisting "system" so that they can be eliminated later, after helping make some progress?
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Society no longer has a pressing need to find the roots of quintic polynomials but Galois theory is of ongoing importance. Bombelli, who introduced complex numbers in the 16th century, would be astounded by the developments of his idea e.g. complex manifolds, quantum theory, ...
"Construction" is actually a very simple answer that basically describes it. Maybe it just seems a bit unsatisfactory because it also describes other things too. Anyway, I completely agree about the "constructions" often being more interesting than the original problem.
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Of course individual constructions have specific names. Adding negatives to the naturals or fractions to the integers is the Grothendieck group construction (free abelian group on a commutative monoid).
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Including the rationals in the reals or the reals in the complex numbers are examples of two different kinds of "completion" (Cauchy completion, algebraic closure).
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