Book reviews: Not much is known about the 12th-century German prelate Hugo of Regensburg, even though he left behind 78 pages of writing in a great messy bundle. It was filled with notes on all sorts of stuff – family history, monastery records, letters, the Hebrew alphabet, lunar eclipses, spiritual musings.
What made Hugo’s papers extraordinary weren’t his ideas, but the way he dated them. Instead of using Roman numerals, such as MCLXVII, he wrote 1167. Instead of MCCI, he wrote 1201. “It is hard to convey just how eccentric Hugo was to date his annals in a way that now seems the natural choice,” writes Benjamin Wardhaugh, who has written a contemplative history of our fascination with numbers that isn’t just for maths nerds.
“No more than a handful of [Hugo’s] contemporaries in the city would even have recognised the symbols as numbers, never mind be able to read them. He became in his small way a pioneer, the first in Europe to use the Arabic numerals outside the realm of astronomical tables or arithmetical treatises.”
Arabic numerals might today seem the obvious choice, but Wardhaugh argues they were hardly inevitable. The counting system of 1 to 9 was just one of a thicket of numbering systems, and their journey from Sanskrit to Arabic-speaking mathematicians to European manuscript writers like Hugo was slow and circuitous.
Like every counting system, Arabic numerals had their drawbacks, and their detractors. Unlike physical counting boards with moveable counters, you needed to memorise addition and multiplication tables, and learn algorithms – tricks to help work out sums. Other systems, from Roman numerals to counting machines like the Japanese Soroban (a form of abacus), are still used today for good reason.
The gong for endurance really belongs to cuneiform, a writing and counting system using clay tablets that arose in the early Bronze Age. For more than 3000 years, scribes used it to keep track of taxes paid in wheat and goats, wine deliveries, armies, war loot and house prices in 15 different languages, from Turkey to Egypt and from Lebanon to Iran.
Although it is hard to imagine gathering data or statistics or doing accounts with a clay tablet and reed stylus, counting systems are still changing. When was the last time you added anything without resorting to a calculator app? Binary computer codes are rapidly encroaching on the everyday use of Arabic numerals.
Wardhaugh, an English academic who has written several books about maths and history, has a good grasp of his subject, and covers vast swathes of time and cultural landscape, conveying nuance without being dull about it.
Counting, he says, is not the same as measuring, or maths. Humans have counted in every way you can possibly imagine, plus a few thousand more – mentally, using fingers, wrists, toes, or pointing to other body parts, by notching bone and wood, with words, symbols and a rich variety of counting machines.
It may not feel like it, but your innate number ability is so powerful that Wardhaugh calls it a sense, like vision, hearing or smell. A few hours after birth, babies can apparently distinguish three items from one. By four days old, he says, they can tell a two-syllable word from a three-syllable one.
Every human culture has a tradition of counting – although a few small groups seem to have lost theirs, including the much-studied Pirahã people of South America. To understand why and how people count, it helps to know what they count, as their systems suit a wide range of different purposes.
Portable bamboo or ivory tally sticks helped the Chinese tax assessors of Ming dynasty China. The languages of the Polynesian family, including Māori, used complex base-10 numeral systems, often counting specific objects like bananas or pandanus leaves with specialised counting units based on groups, usually of two, four, 10 or 20. The Anindilyakwa (Warnumamalya) Australian First Nations people use one of the most complex of all grammar systems ever recorded. They not only have counting words, but variations like “once” or “twice”, “by ones” and “by twos” with prefixes that need to agree with nouns in the same sentence. Many of these spoken numbers also come with particular gestures. Their main reason for counting, says Wardhaugh, “seems to have been sharing mid-sized food items like fish, eggs or wild apples”.
Far from patting ourselves on the back for slick modern numeracy, Wardhaugh’s respectful inquiries into the complexities of other counting systems is humbling, and this is one of the greatest pleasures of the book. We are not, it turns out, the only species who can count. An “approximate sense of number” is present in nearly every species that has been tested for it, from cuttlefish to chickens. Fish can tell 50 objects from 25; rhesus monkeys can tell the difference between 12 items and 10, although Wardhaugh cautions that some fish, and some monkeys, are better counters than others.
Although humans seem to know how many objects are in a group of four or fewer without counting or estimating, in the fascinating process called “subitising”, no species on Earth, Wardhaugh says, can tell 100 items from 101. There is a significance to small numbers compared with large ones, for every species tested.
It is not hard to imagine that we are all descended from the apes who noticed if three bears entered a cave but only two came out, those who chose the tree with the most fruit, or who fled from a band of invaders when slightly outnumbered.
You don’t need to be good with numbers to enjoy this engrossing dip into the countless ways humans count – and why they do it.
Counting: Humans, History And The Infinite Lives Of Numbers by Benjamin Wardhaugh (William Collins, $39.99) is out now.
