Well isn't that true of all abstractions? They give us tools that we couldn't use before, and some of those tools don't make sense. I mean, in real physics it d

Well isn't that true of all abstractions? They give us tools that we couldn't use before, and some of those tools don't make sense. I mean, in real physics it doesn't make sense to add different dimensions, eg time and distance, but math lets us do that, because they are 'just numbers'. We need to know that dimensions can only be multiplied and divided, and dimensional analysis needs to be done afterwards to check the unit of the result. However, something like the dot-product here does make sense, since you can use it to determine similarities of vectors.