The need to count

"How do you count if you have no words for numbers?" A herder by the Saraswati river sends cattle out each morning. He drops one pebble into a pot for each cow that leaves, and takes one out for each cow that comes back. An empty pot means every cow is home.

Matching one thing to one other thing like this is called one-to-one correspondence. It is how the natural numbers began: ℕ = {1, 2, 3, 4, …}.

The oldest counting records are notches on bone, from Africa:

• the Lebombo bone, from the Lebombo Mountains between South Africa and Eswatini (the book uses its older name, Swaziland), has 29 notches and may have tracked the phases of the moon;
• the Ishango bone, from near the start of the Nile in the Democratic Republic of Congo, has notches grouped 11, 13, 17 and 19: the prime numbers between 10 and 20. Another column seems to show doubling.

In India:

• the Indus Valley cities, such as Lothal and Harappa, needed standard weights and measures for trade;
• the Vedas named every power of 10 up to 10¹² (parārdha);
• the Lalitavistara names numbers up to 10⁵³ (tallakṣhaṇa).

Thinking in powers of 10 prepared the way for place value, and for zero.

Closed or not? Add two natural numbers and you always get another natural number: ℕ is closed under addition. Subtract and you may not: 3 – 5 = –2 and 4 – 4 = 0 are not natural numbers. So ℕ is not closed under subtraction. That gap is why the number line had to grow.