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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are you thinking of something like https://www2.clarku.edu/faculty/djoyce/complex/cubic.html … where in solving cubics you need complex numbers but you have real solutions? counterpoint is lots of equations have complex solutions and complex numbers do more for us besides
Yes, this is the idea. Complex numbers were introduced as a means of solving cubic equations which could not be evaluated without them, but the solutions eliminate all the complex terms from the answer. The idea is that if they were not eliminated, the answer wouldn't be useful.
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I think this fact is interesting, I think it shows the utility of complex numbers (you can solve things you could not do before, interesting special case is when you go via complex to get real solutions) But this is not all they do
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It might be that over time, the new concept proves itself so useful for this purpose that it's established as something so fundamental that the original system (the natural numbers without negative numbers, or the reals without complex numbers) seems less useful.
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