It used to be called *arithmetic* but it sounds so much more modern and scientific when it is called *number theory*. It is the branch of mathematics which deals with the *integers* and the relationships between them. Its origins (whether one wants to call it a discovery or an invention) lie with the invention of *counting* itself. It is from where all the various branches of mathematics derive. The origin of counting can be said to be with the naming of the integers, and is intimately tied to the development of language and of writing and perhaps goes back some 50,000 years (since the oldest known tally sticks date from some 30,000 years ago).

How and why did the naming of the integers come about? Why were they found necessary (necessity being the cause of the invention)? Integers are whole numbers, indivisible, complete in themselves. Integers don’t recognise a continuum between themselves. There are no partials allowed here. They are separate and discrete and number theory could as well be called quantum counting.

Quite possibly the need came from counting their livestock or their prey. If arithmetic took off in the fertile crescent it well may have been the need for trading their live goats among themselves (integral goats for integral numbers of wives or beads or whatever else they traded) which generated the need for counting integers. Counting would have come about to fit their empirical observations. Live goats rather than carcasses, I think, because a carcass can be cut into bits and is not quite so dependent upon integers. Quanta of live goat, however, would not permit fractions. It might have been that they needed integers to count living people (number of children, number of wives …..) where fractions of a person were not politically correct.

The rules of arithmetic – the logic – could only be discovered after the integers had been named and counting could go forth. The commutative, associative and distributive properties of integers inevitably followed. And the rest is history.

But I wonder how mathematics would have developed if the need had been to count raindrops.

After all:

2 goats + 2 goats = 4 goats, and it then follows that

2 short people + 2 short people = 4 short people.

But if instead counting had been inspired by counting raindrops, they would have observed that

2 little raindrops + 2 little raindrops = 1 big raindrop.

They might then have concluded that

2 short people + 2 short people = one tall person

and history would then have been very different.