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> but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now I think it's an arbitrary perspective, whether you trea

> but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary. People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context. I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?

pdonis 6y ago
> I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary. Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases. > the generalization can be done in multiple ways There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.