It’s odd, when you think about it, that mathematics ever got going. We have no innate genius for numbers. Drop five stones on the ground, and most of us will see five stones without counting. Six stones are a challenge. Presented with seven stones, we will have to start grouping, tallying and making patterns.
This is arithmetic, ‘a kind of “symbol knitting”’ according to the maths researcher and sometime teacher Paul Lockhart, whose Arithmetic explains how counting systems evolved to facilitate communication and trade, and ended up watering (by no very obvious route) the metaphysical gardens of mathematics.
Lockhart shamelessly (and successfully) supplements the archeological record with invented number systems of his own. His three fictitious early peoples have decided to group numbers differently: in fours, in fives, and in sevens. Now watch as they try to communicate. It’s a charming conceit.
Arithmetic is supposed to be easy, acquired through play and practice rather than through the kind of pseudo-theoretical ponderings that blighted my 1970s-era state education. Lockhart has a lot of time for Roman numerals, an effortlessly simple base-ten system which features subgroup symbols like V (5), L (50) and D (500) to smooth things along. From glorified tallying systems like this, it’s but a short leap to the abacus.
It took an eye-watering six centuries for Hindu-Arabic numbers to catch on in Europe (via Fibonacci’s Liber Abaci of 1202). For most of us, abandoning intuitive tally marks and bead positions for a set of nine exotic squiggles and a dot (the forerunner of zero) is a lot of cost for an impossibly distant benefit. ‘You can get good at it if you want to,’ says Lockhart, in a fit of under-selling, ‘but it is no big deal either way.’
It took another four centuries for calculation to become a career, as sea-going powers of the late 18th century wrestled with the problems of navigation.