That continuity, Dr. Pagel added, "should astonish us."?
The earliest formal number systems, complete with those all-important place values, date back only about 3,500 years, to ancient Mesopotamia, but quantitative record keeping is much older. Francesco D’Errico, an archaeologist at the University of Bordeaux, described his team’s analysis of a 70,000-year-old hyena femur from the Les Pradelles site in France.
The leg bone fragment is incised with nine parallel, nearly identical notches, which look far too regular to be the incidental result of butchery with stone tools or an early attempt at artful decoration. “This fits with the idea that the incisions are a form of numerical notation,” Dr. D’Errico said. “We see the beginnings of externalizing our numeric sense, of taking it outside the body.”
Attitudes about animal numerosity have changed drastically since the mid-twentieth century when many researchers believed only humans had enoughgray matter to think quantitatively.
They cited as an object lesson the 1907 case of Clever Hans, the horse that supposedly could solve arithmetic problems and would tap out his answers by hoof; as it turned out, he was responding to unconscious cues from the people around him.
Since then, researchers have approached the field with caution and rigor, seeking to identify the specific evolutionary pressures that might spur the need for numeric judgments in any given species.
Social carnivores like spotted hyenas, for example, live in fission-fusion societies, collectively defending their territories against rivals but in ever-shifting groups of widely roaming members. “You can never predict who you’ll find in which group,” said Sarah Benson-Amram, an assistant professor of zoology and physiology at the University of Wyoming. “You might be alone or in a group of 10.”
Because hyena jaws can pulverize zebra bones, encounters between competitors can prove deadly; every spotted hyena must have a good sense from moment to moment of how many there are of us versus how many of them. Dr. Benson-Amram and her colleagues tested hyena numerosity skills in the field, playing back the recorded whoops of spotted hyenas living in South Africa and Namibia to hyenas in Kenya.
The Kenyan carnivores reacted to the whoops of starngers as predicted, approaching the hidden source of the sound when they had a home team advantage, retreating when they heard a few too many distinctive voices in a row. Or sometimes, when the local hyenas were outnumbered, they’d call for back up.
“I always loved it when the hyenas would whoop and others would come running in,” Dr. Benson-Amram said. “It was very dramatic, very aggressive, and I appreciated this ability to recruit other group members to the cause.”
Chimpanzees are social scorekeepers, episodic warriors and number ninjas, too. They can be taught to associate groups of objectives with corresponding Arabic numerals up to the number 9 and sometimes beyond ? three squares on a computer screen with the number 3, five squares with 5, and so on. They can put those numbers in order.
The numeric working memory of young chimpanzees is astonishing: Flash a random scattering of numerals on a screen for just 210 milliseconds ? a half an eye blink ? and then cover the numbers with white squares, and a numerically schooled young chimpanzee will touch the squares sequentially to indicate the ascending order of the numbers hidden beneath.
Don’t bother trying to do this yourself, Tetsuro Matsuzawa, a primatologist at Kyoto University, said at the scientific meeting in London on which the themed journal edited by Professor Butterworth was based. “You can’t.”
By the look of it, cerebral property in humans once dedicated to numeric memory has, in the six million years since we diverged from chimpanzees, been co-opted for grander purposes, like the ability to judge whether a sentence like this is true: “There is no non-vanishing continuous tangent vector field on even dimensional spheres.”
Stanilas Dehaene, a psychologist at the University of Paris, and his colleagues presented evidence from brain scans of professional mathematicians that the neural circuitry for advanced mathematical thinking is an elaboration on the archaic number sense we share with other animals.
It is distinct from our everyday language pathways, even when that math-related challenge involves words rather than numbers ? like the true statement above.
Yet our inborn numerosity hardly guarantees mathematical proficiency, and can sometimes work against us. Psychologists Rochel Gelman of Rutters University and Jennifer Jacobs Danan of the University of California, Los Angeles, have studied how often reasonably well-educated people miscalculate percentages.
We hear that the price of something rose by 50 percent and ten fell by 50 percent, and we reflexively, mistakenly conclude, “Oh good, we’re back to where we started.” Our natural number sense adds and subtracts whole numbers, whole chucks, guppies and hyenas. But it really abhors a fraction, and so has led us astray.
