Body parts, tally marks and counting in twos

The nearest counting tools are the ones you were born with. Many peoples have counted on their hands and body parts; a group in Papua New Guinea used (and still uses) a fixed order of body parts as their standard sequence.

One of the oldest written methods is notching: cutting marks on a bone or a cave wall. These are tally marks, one mark for each thing counted. It's the stick method, with scratches instead of sticks.

Archaeologists have found bones more than 20,000 years old that seem to carry tally marks. The two oldest known are the Ishango bone and the Lebombo bone.

The Lebombo bone is one of the oldest mathematical objects we have, and may have been a tally stick or a lunar calendar: 29 notches, and a lunar month is about 29½ days.

Now a different idea. The Gumulgal of Australia counted like this:

They counted in twos. Anything above 6 they simply called ras.

Here's the puzzle. The Bakairi of South America counted tokale, ahage, ahage tokale, ahage ahage, ahage ahage tokale, ahage ahage ahage. The Bushmen of South Africa counted xa, t'oa, 'quo, t'oa-t'oa, t'oa-t'oa-t'a, t'oa-t'oa-t'oa. Three peoples on three continents, with no sign of contact, built systems that work the same way. One theory is that they shared distant ancestors who counted like this, and their descendants carried it with them as they moved.

Even though the Gumulgal names stop at 6, an important idea has appeared: count in groups of one size, and use the name of the group to build bigger numbers. Across history the favourite group sizes have been 2, 5, 10 and 20; you can see 5 at work in the Roman system.

Why those sizes? Perhaps because of our eyes. Glance at a few objects and you can tell 2, 3 or 4 at once, but most people can't see 5 or more in one glance without counting. That may be exactly why people bundled every 5 tally marks into one group.