One, two, many: the prehistory of counting

The Victorian idea that “primitive” tribes can’t count has cast a long shadow over efforts to understand the origins of mathematics

LOOKING back, Francis Galton would call it “our most difficult day”. It was 4 March 1851, and the young English explorer was beginning to appreciate the obstacles confronting his attempts to map out the Lake Ngami region of south-western Africa. Struggling to navigate a narrow ridge of jagged rock, his wagon had “crashed and thundered and thumped” while his oxen “charged like wild buffaloes”.

To make matters worse, Galton had little faith in his local guides from the Damara tribe, who appeared to lack even an understanding of basic arithmetic – a situation Galton found “very annoying”. He recounts that having established an exchange rate of one sheep for two sticks of tobacco, he handed four sticks to a local herdsman in the expectation of purchasing two sheep. Having put two sticks in front of the first sheep, the man seemed surprised that two sticks remained to pay for the second. “His mind got hazy and confused,” Galton reported, and the transaction had to be abandoned and the sheep purchased separately.

As further evidence of the apparent ignorance of the Damara, Galton wrote that they “use no numeral greater than three” and that they managed to keep track of their oxen only by recognising their faces, rather than by counting them. At a most inopportune time for his expedition, Galton seemed to have stumbled into a world without numbers.

To a modern reader, these tales in Galton’s 1853 Narrative of an Explorer in Tropical South Africa seem little more than pithy anecdotes that reflect his prejudices as a gentleman of the growing Victorian empire. (His preoccupation with the supposed inferiority of other peoples persisted in his later work in eugenics.) Within 10 years, however, those same reports of primitive

innumeracy were being used by the finest scientific minds of Victorian Britain to glimpse the savage condition of prehistoric humans.

The reports’ influence did not stop there. As I have traced this trend over the ensuing decades, it has become clear just how important these speculations were in shaping the anthropological study of numbers, with ramifications for psychology, linguistics and the philosophy of mathematics. Its legacy still lingers, 100 years after Galton’s death. So just how did his account become so central to such a broad swathe of 19th-century science?

Narrative of an Explorer in Tropical South Africa initially launched Galton to prominence as an explorer and travel writer, thrilling readers with tales of strange worlds occupied by ignorant peoples. Among his many fans, his cousin Charles Darwin wrote to profess “how very much I admire the spirit and style of your book”. He became the toast of the Royal Geographical Society, though few scientists at that point read his tales for anything more than entertainment.

All that would change in 1859. In September of that year, leading geologist Sir Charles Lyell addressed the British Association for the Advancement of Science at its annual meeting in Aberdeen. Lyell had been prominent among those who were sceptical of the idea that humans may have roamed the Earth tens or even hundreds of thousands of years ago, but that day he finally declared before a rapt audience “that the date of man must be carried further back than we had heretofore imagined”. Among his evidence, Lyell cited a forthcoming book that would become one of the most influential tomes ever written: Darwin’s On the Origin of Species.

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One, two, many: the prehistory of counting