Can we count without knowing numbers? It seems to be impossible. However, our ancestors were able to do it. They did not name any numbers so how was it possible?
Yes, it is possible. As a matter of fact counting is an action of comparing. When we use numbers we compare counted elements with certain symbols said or written. Those symbols represent certain properties of sets called quantity of elements or cardinal numbers.
When we are comparing our set with another set we are creating pairs where each element from one set has its compagnion from another. This way we can make the second one (stones) equal to the first (sheeps) in the sense of quantity.
In the set of stones every stone represents exactly one sheep it is not important which one but only one exlusive sheep.
Did shepherds count their sheeps? Yes they did as they created a second set with the same property. This common property of both sets is the quantity of the elements each one has. This property of a set is callled a cardinal number.
Of course by giving to every quantity a specific name or symbol they would do much better but they had to organise those symbols by putting them in order (so as to know for any given one which is the previous one an what is the next)
The sets of sheeps and stones are much different phisically but they have one thing in common, their cardinal numbers are the same.
the cardinal numbers had made the concept of natural numbers as we know it.
The only remaining problem was to attribute some symbols to different quantities an put them in order.
The results of this attempt were different. Here are some of them:
As we can see some symbols were designed in a way that they could represent the quantity visualy.
Of course it was not possible to do that for bigger numbers. Hindu an Arabs were more imaginative as they invented a positional way to represent any quantity and only ten figures were necessary to do the job. We call those figers decimal digits.
Now, Can you work out this puzzle?
A driver has a brother, but thi one has no an brothers. What is the driver?