*one-to-one correspondence*
. Primitive people would not have known of this concept; however, one-to-one correspondence underlies all primitive counting methods. Therefore we will use it here
in this explanation of primitive counting.

In
mathematical terms, a *one-to-one correspondence *
between sets S and T is a map such
that distinct elements in S have distinct images in T; also each element of T
is the image under the map of at least one element of S. In
simpler terms this just means that each element in the first set
corresponds to exactly one element in the second set and vice
versa. To see this clearly we can draw a one-to-one
correspondence between the objects illustrated below.

What we can see illustrated are two sets of objects. The first set consists of five butterflies and the second set consists of five flowers. Between these two sets we have drawn arrows to indicate the one-to-one correspondence. As we can see, each butterfly corresponds to exactly one flower, and similarly, each flower corresponds to exactly one butterfly. Therefore, we have a one-to-one correspondence between the set of butterflies and the set of flowers. |

It is this one-to-one correspondence, or reckoning, that enabled primitive man to begin to count. Once man was able to see this correspondence between objects he was able to use it to keep track of anything of importance to him – for example, a shepherd could keep track of the number of sheep in the flock. While this was not an abstract counting system and the actual number of objects was not known, primitive people were able to tell if there were any objects added to or subtracted from the group. This was done by comparing the group with whatever was being used to keep count. Consider again the example of the shepherd and his sheep; The shepherd may have had a pouch of pebbles by which he could tell if any sheep had gone astray. By making a correspondence between each pebble and each sheep, the shepherd could tell if any were missing without the use of numbers. This method also allowed for the death, loss or sale of any sheep by the removal of pebbles from the collection and the birth of any lambs by the addition of pebbles to the collection.

The understanding of the concept of one-to-one correspondence was the first step in the development of abstract counting. Man had made this first step towards an abstract counting system by increasing his number sense from recognising more or less, when changes were made to a small group of objects, to knowing exactly how many objects were present.

This correspondence was used in many ways
by primitive people and for many reasons, which will be discussed
in the section to follow. Eventually this method of counting lead to the
creation of number names and, in the end, number symbols.

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