My guess would be that we actually want to view there as being multiple basic/intuitive cognitive starting points, and they’d correspond to different formal models. As an example, consider steps / walking. It’s pretty intuitive that if you’re on a straight path, facing in one fixed direction, there’s two types of actions—walk forward a step, walk backward a step—and that these cancel out. This corresponds to addition and subtraction, or addition of positive numbers and addition of negative numbers. In this case, I would say that it’s a bit closer to the intuitive picture if we say that “take 3 steps backward” is an action, and doing actions one after the other is addition, and so that action would be the object “-3″; and then you get the integers. I think there just are multiple overlapping ways to think of this, including multiple basic intuitive ones. This is a strange phenomenon, one which Sam has pointed out. I would say it’s kinda similar to how sometimes you can refactor a codebase infinitely, or rather, there’s several different systemic ways to factor it, and they are each individually coherent and useful for some niche, but there’s not necessarily a clear way to just get one system that has all the goodnesses of all of them and is also a single coherent system. (Or maybe there is, IDK. Or maybe there’s some elegant way to have it all.)
Another example might be “addition as combining two continuous quantities” (e.g. adding some liquid to some other liquid, or concatenating two lengths). In this case, the unit is NOT basic, and the basic intuition is of pure quantity; so we really start with R.
My guess would be that we actually want to view there as being multiple basic/intuitive cognitive starting points, and they’d correspond to different formal models. As an example, consider steps / walking. It’s pretty intuitive that if you’re on a straight path, facing in one fixed direction, there’s two types of actions—walk forward a step, walk backward a step—and that these cancel out. This corresponds to addition and subtraction, or addition of positive numbers and addition of negative numbers. In this case, I would say that it’s a bit closer to the intuitive picture if we say that “take 3 steps backward” is an action, and doing actions one after the other is addition, and so that action would be the object “-3″; and then you get the integers. I think there just are multiple overlapping ways to think of this, including multiple basic intuitive ones. This is a strange phenomenon, one which Sam has pointed out. I would say it’s kinda similar to how sometimes you can refactor a codebase infinitely, or rather, there’s several different systemic ways to factor it, and they are each individually coherent and useful for some niche, but there’s not necessarily a clear way to just get one system that has all the goodnesses of all of them and is also a single coherent system. (Or maybe there is, IDK. Or maybe there’s some elegant way to have it all.)
Another example might be “addition as combining two continuous quantities” (e.g. adding some liquid to some other liquid, or concatenating two lengths). In this case, the unit is NOT basic, and the basic intuition is of pure quantity; so we really start with R.