The mechanism of counting

Imagine you live ten thousand years ago and own a herd of cows. Three questions come up naturally:

• Have all the cows come back from grazing?
• Do I have fewer cows than my neighbour?
• If so, how many more would I need to match?

Now the hard part: answer them without any number names or written numbers. How?

Method 1: a stick for every cow. One stick, one cow, and no two cows sharing a stick. This pairing is called a one-to-one mapping. In the evening, as each cow comes back, put one stick aside. A stick left over means a cow is missing. You've answered the question without saying a single number.

The same sticks answer the other two questions. Lay your sticks beside your neighbour's, one against one. Whoever has sticks left over has more cows, and the leftover sticks are exactly how many more the other would need.

Method 2: a fixed list of sounds or names, say the letters a, b, c, … Count by pairing each object with the next letter. It's quicker than carrying sticks, but it runs out: with a to z you can't count more than 26 things.

Method 3: a sequence of written symbols: I, II, III, IV, V, VI, …, XX. That's the Roman system, used in Europe before the Hindu one replaced it.

Put the three together. To count anything you need a standard sequence (of objects, names or written symbols) in a fixed order. Such a standard sequence is called a number system. Counting is then just making a one-to-one mapping between the things and the sequence, in order.

Since numbers never end, the challenge is a sequence that never runs out but is still easy to use. Sticks never run out but are hopeless for big counts; letters are easy to say but stop at 26; Roman numerals do much better but keep needing new symbols for bigger numbers. The symbols of a written system are called numerals (0, 1, 5, 36 and 193 are numerals of the Hindu system), and a written system always came hand in hand with a system of number names.