Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of being able to make one-to-one correspondences). They cite (p. 8) a young shepherd boy from Sicily in the 1950's:
I can't count, but even when I was a long way away, I could see if one of my goats was missing. I knew every goat in my herd - it was a big herd, but I could tell every one of them apart. I could tell what kid belonged to what mother... The master used to count them to see if they were all there, but I knew they were all there without counting them.
[from: Dolci Danilo (1959), Report from Palermo]
I wonder if this is meant as a joke (in the sense of "the poor boy just misunderstood 'counting'. In fact he's actually counting, i.e. making unconscious mental one-to-one correspondences"), or if there is something more in it, not easy to capture in mathematical terms?
Thinking about what the shepherd boy actually does mentally (as described somehow precisely and comprehensible in his own words) lets me seem it possible that there is a essentially different way of telling whether "all goats are there: no one is missing, and no one doesn't belong to the herd (= all belong to the herd)".
Has anybody thought seriously about this? Can there be made mathematical sense out of it?
[Related question: Some people are able to tell if two areas of size $n_1 \times m_1$ and $n_2 \times m_2$ are the same without being able to multiply. How do they manage?]