> For example, the way dual numbers are used to extend the reals for automatic differentiation doesn't have a deep connection to duality in vector spaces.
Yes, the right way to think about dual numbers (esp once you generalize them beyond just the single e^2=0), is to think of them as tangent vectors (sections of the tangent bundle). I've never really liked the "dual number" terminology here. That's why I deliberately chose to use the duality of forward and reverse mode AD, because that notion of duality agrees with the underlying linear algebra (or in general differential geometry). I do agree it's a mess of terminology.
Yes, the right way to think about dual numbers (esp once you generalize them beyond just the single e^2=0), is to think of them as tangent vectors (sections of the tangent bundle). I've never really liked the "dual number" terminology here. That's why I deliberately chose to use the duality of forward and reverse mode AD, because that notion of duality agrees with the underlying linear algebra (or in general differential geometry). I do agree it's a mess of terminology.