Counting without one-to-one correspondence? [closed]

Question

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?

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[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?]

Answer 1

"Can there be made mathematical sense out of it?"

Perhaps as follows: given a finite set $X$ labeled with bits $y\in\{0,1\}^X$, indicating whether the corresponding element is present or missing, decide whether $\sum_{x\in X}y(x)=|X|$ or $<|X|$. This is a strictly weaker operation than counting. Certainly a counter that evaluates $\sum_{x\in X}y(x)$ can be used to solve the decision problem, but the latter cannot be used to count (since it returns a bit rather than an integer in $[0,|X|]$).

Since the connection to psychology was mentioned in the comments, I can easily imagine a dedicated neuron for each child/goat, which is activated iff the relevant object is present. There is then a very simple schematic (and even biological) implementation of a "summation" neuron, which fires iff all of the basic inputs do. This is certainly a simpler neural circuit than a counter.

Answer 2

The answer given by Robert Israel seems worth elaborating on. Knowing if everyone is present at a large family gathering is like going through the family history, or family tree: grandmother is here, her daughters A, B, and son C are here. Of course, son D isn't here, since he died in childhood. Daughter A's son E and his wife F are here with their baby son G ... et cetera. Once you know how to count, it's hard not to apply it, but it doesn't seem required if you have a form of narrative or visual memory.

Answer 3

In the case of rectangles, and more generally with some other shapes too, there are basically two well-known methods--cut the two given items and match the congruent pieces or complement the two items by two congruent shapes so that the two results will be congruent. These methods can be applied to simple finite arithmetics so that 1- and 2-grade children can apply them to a great advantage of the education around the world.

A general answer was already indicated in this thread. Sets may appear together with certain relations. When an element is missing then a pattern is violated. These relations/patterns don't have to be geometric or mathematical in a narrow sense, there can be--for instance--certain emotional connections, some subsets can be special (friends or enemies or a complete collection of skills (e.g. when a person is missing then the skill they possess is missing too).