I honestly don’t think the tradeoff is real (but please tell me if you don’t find my reasons compelling). If I study category theory next and it does some cool stuff with the base map, I won’t reject that on the basis of it contradicting this book. Ditto if I actually use LA and want to do calculations. The philosophical understanding that matrix-vector multiplication isn’t ultimately a thing can peacefully coexist with me doing matrix-vector multiplication whenever I want to. Just like the understanding that the natural number 1 is a different object from the integer number 1 peacefully coexists with me treating them as equal in any other context.
I don’t agree that this view is theoretically limiting (if you were meaning to imply that), because it allows any calculation that was possible before. It’s even compatible with the base map.
I honestly don’t think the tradeoff is real (but please tell me if you don’t find my reasons compelling). If I study category theory next and it does some cool stuff with the base map, I won’t reject that on the basis of it contradicting this book. Ditto if I actually use LA and want to do calculations. The philosophical understanding that matrix-vector multiplication isn’t ultimately a thing can peacefully coexist with me doing matrix-vector multiplication whenever I want to. Just like the understanding that the natural number 1 is a different object from the integer number 1 peacefully coexists with me treating them as equal in any other context.
I don’t agree that this view is theoretically limiting (if you were meaning to imply that), because it allows any calculation that was possible before. It’s even compatible with the base map.