Here's Looking at Euclid: Chapters Zero–One

Pica is a linguist and, perhaps because of this, speaks slowly and carefully, with painstaking attention to individual words.

Then he set off down the Trans-Amazon Highway, a 1970s folly of nationalistic infrastructure that has deteriorated into a precarious and often impassable muddy track.

Jacareacanga is on the edge of the Munduruku’s demarcated territory.

The more I pushed Pica for facts and figures, the more reluctant he was to provide them. I became exasperated. It was unclear if underlying his responses was French intransigence, academic pedantry or simply a general contrariness.

It was unclear if underlying his responses was French intransigence, academic pedantry or simply a general contrariness.

The Munduruku are only subsistence farmers and money has only recently begun to circulate in their villages, and so they never evolved counting skills.

Numbers are so prevalent in our lives that it is hard to imagine how people survive without them.

When King David counted his own people he was punished with three days of pestilence and 77,000 deaths.

Counting people with numbers is considered a way of singling people out, which makes them more vulnerable to malign influences.

In the rainforest Pica conducts his research using laptops powered by solar-charged batteries. Maintaining the hardware is a logistical nightmare, because of the heat and the damp, although sometimes the trickiest challenge is assembling the participants.

Yet maybe in our dependence on linearity we have gone too far in stifling our own logarithmic intuitions. Perhaps, said Pica, this is a reason why so many people find math difficult. Perhaps we should pay more attention to judging ratios rather than manipulating exact numbers.

In the early 1900s, crowds gathered regularly in a Berlin courtyard to watch Hans's owner, Wilhelm von Osten, a retired math instructor, set the horse simple arithmetical sums. Hans answered by stamping the ground with his hoof the correct number of times. His repertoire included addition and subtraction as well as fractions, square roots and factorization.

Public fascination and suspicion that the horse's supposed intelligence was some kind of trick led to an investigation of his abilities by a committee of eminent scientists.

Her mindless gaze, the nonchalant tapping of a flashing, beeping computer and the mundanity of continual reward reminded me of an old lady doing the slots.

Ai’s competence in both the cardinality and the ordinality tasks meant that Matsuzawa could reasonably say his student had learned to count.

All creatures seem to be born with brains that have a predisposition for math.

Studies into the limits of animals’ numerical capabilities bring us naturally to the question of innate human abilities.

Starkey argued that this extra stare-time meant the babies had noticed something different about three dots compared with two dots, and therefore had a rudimentary understanding of number.

It's impossible. (Unless you are an autistic savant, like the character played by Dustin Hoffman in Rain Man, who would be able to grunt in a split second, “Seventy-five.”) Our only strategy is to estimate, and we’d probably be far off the mark.

I used to think that I was just being indecisive. I prefer to think that I am drawing on my inner number sense: an intuitive, animal propensity to deal in approximations.

Yet this study found a strong correlation between a talent at reckoning and success in formal math.

He pointed out to me that from one to four, the number of syllables of each word is equal to the number itself. This observation really excited him. “It is as if the syllables are an aural way of counting,” he said.

“This is amazing since it seems to corroborate the idea that humans possess a number system that can only track up to four exact objects at a time,” Pica said.

Yet dyscalculia is little understood. It can be thought of as the numerical counterpart of dyslexia; the conditions are comparable in that they both affect roughly the same proportion of the population and they appear to have no bearing on overall intelligence.

The words were simply the numbers five, ten and fifteen in a jargon used by shepherds when counting their sheep.

Much more evocative was the sign for a million, a self-satisfied gesture of achievement and closure: the hands clasped together, fingers intertwined.

Base ten systems have been used in the West for thousands of years. Despite their harmoniousness with our bodies, however, many people have questioned whether they are the most sensible base for counting. In fact, some have argued that their physical provenance makes them an actively bad choice.

King Charles XII of Sweden dismissed base ten as the product of “rustic and simple people” fumbling around with their fingers.

So, in 1716, he ordered the scientist Emanuel Swedenborg to devise a new counting system with a base of 64. He had arrived at this formidable number due to the fact that it was derived from a cube, 4 x 4 x 4. Charles, who fought—and lost—the Great Northern War, believed that military calculations, such as measuring the volume of a box of gunpowder, would be made easier with a cube number as a base.

The bellicose Swede was correct to ask which base best suits our scientific needs—rather than opting for the one that suits our anatomies.

In fact if you look at how 100 is divided by the numbers 1 to 12, base 12 provides more concise numbers.

Michael’s love of avant-garde bases did not stop at 12. He has toyed with 8, which he sometimes uses when doing DIY at home. “I use bases as tools,” he said. And he has gone up to base 60.

“[It] permits the Assayer to weigh all sorts of masses with few weights and could serve in coinage to give more value with fewer pieces,” he wrote in 1703.

He believed that the cosmos was comprised of being, or substance, and nonbeing, or nothingness. The duality was perfectly symbolized by the numbers 1 and 0.

When Leibniz was made aware of the binary nature of Fu Hsi, it gave him “a high opinion of [the I Ching’s] profundity.”

Since he thought that the binary system mirrored Creation, his discovery that it also underlay Taoist wisdom meant that Eastern mysticism could now be accommodated within his own Western beliefs.

Leibniz's panegyrics on base two were a rather eccentric preoccupation of the preeminent polymath of his day.

Leibniz's panegyrics on base two were a rather eccentric preoccupation of the preeminent polymath of his day.

The record, however, goes to the Cantonese, whose digits are spoken with even more brevity. They can remember ten of them in a two-second period.

An announcer stood at the front of the hall and dictated, with the intonation of an impatient muezzin, numbers to be added, subtracted or multiplied.