unreasonably effective

Some years ago I bought a book called Algebra by Michael Artin. In the preface there was a warm conversation:

"One, two, three, five, four ..."
"No Daddy, it's one, two, three, four, five."
"Well if I want to say one, two, three, five, four why can't I?"
"That's not how it goes."

The conversation instantly found a place in my long-term memory. When I think about it now, number and counting seem so basic that I fail to find an analogy that does not end up being circular.

Number is one of the earliest abstractions a child learns. There is something in common between two dreams, two dogs, two rain drops. To explain it, all I have is example. My words fail. By pointing to more and more examples that an adult has somehow learned to recognize as "two" and uttering the word "two" or showing two fingers simultaneously, we hope a child will also learn that abstract concept we call "two", and other numbers.

Counting (by one) is like associating members of a set to an ordered, official set. The official set can be the fingers. We can associate or map or put straight lines between sets of two things say two apples and two fingers. After that neither the set of apples, nor the set of fingers will have lonely or left-over member. The map is exact in some sense.