[ahoho]

This was exactly my undergrad research topic! My knowledge is a bit out of date (perhaps this is in TFA), but the running theory is that there are two number systems. The first operates noiselessly and immediately for numbers under 4. The second works in the exact way you are describing, where the magnitude of errors are linear in the size of the set (the “analog magnitude” system). As I recall, there is substantial evidence supporting the idea of two separate systems rather than one.

I don’t really follow your point about integers. We know the brain does object detection at a low level, so it makes sense that the number systems will operate over countable elements.

[pixl97]

I was watching a NOVA episode about the human mind recently, and in the episode they were showing that the eyes only have a very limited 'high bandwidth' optical viewing area. Anything outside of this high bandwidth area has to be committed to memory. I wonder if anyone has researched if that corresponds with our ability to count.