# Counting

Rosencrantz and Macduff, who are at Columbia University in New York know how many beans make five. In fact, they can do a lot better than that. They have passed a series of practical tests that show they can distinguish the more numerous of two groups of objects containing up to nine objects of any type.

There are not many universities where such meagre numerical abilities would be considered remarkable. Even so, the test results made headlines around the world two weeks ago when they were published by Elizabeth Brannon and Herbert Terrace of Columbia’s Psychology department. The reason is that Rosencrantz and Macduff are not students; they are rhesus monkeys. The tests show that monkeys have an innate sense of number which has several features in common with our own.

Psychologists have been interested in – and sceptical about – the mathematical abilities of animals for decades. The celebrated case of “Clever Hans” a horse in turn-of-the-century Germany that appeared to be able to solve complex arithmetic problems, tapping out the answers with his hoof, has made them wary. Hans turned out to be responding to signals given to him – probably unconsciously – by his trainer. When the trainer was fooled by giving him a different problem from that given to Hans, the horse always answered the trainer’s problem, indicating that he was probably able to sense his trainer’s excitement as the number of his hoof-taps approached what the trainer thought was the correct answer.

The tests on Rosencrantz and Macduff were carried out in ways that make such cheating extremely unlikely. Psychologists like Norman Freeman of Bristol University are impressed with the results. The testing was done by a computer, which eliminates possible influences of the expectations of the experimenters, and the ingenious design of the tests allowed the monkeys to demonstrate a genuine numerical ability.

They were trained to touch different parts of a screen, showing groups of 1, 2, 3 or 4 objects, in numerical order. They were then tested using groups of up to nine objects. Even when they were presented with groups containing numbers they had not been trained on and objects they had never seen before, they scored well on the task of touching the groups in ascending order of numerosity. “We have known for some time that animals can respond differently to different numerosities, but this shows that they know that one number is bigger than another” Freeman says.

The monkeys’ errors show a pattern also found in numerosity judgements made by humans. Errors are more common when comparing numbers that are close together. This suggests that numerosities are represented on a sort of mental sliding scale, an idea known as the “accumulator model”. Registering a number on the scale is slightly imprecise, so errors in comparing two numbers are likely to happen if they are close together, but not if they are far apart. Curiously, a similar pattern of errors occurs even when adult humans are asked which of two written down numbers is the larger, suggesting that whenever we think about the size of numbers we tend to use our internal accumulator, even though we can put a symbol and a name to the exact number.

Rosencrantz and Macduff could not speak, so naming numbers is not an option for them. Even so their abilities are superior to any shown so far in human infants. One of the most fundamental abilities shown by human infants is that they expect numbers to stay the same. Even six-month old babies will show surprise if two puppets walk behind a screen and only one is revealed when the screen is removed. Similarly they will be surprised if there is an extra puppet behind the screen

However, although they can tell that two numbers are different, there is no evidence that infants can understand which of two numbers is larger. The fact that the monkeys can do this explodes the myth that language, particularly the ability to name numbers, is a prerequisite for learning anything about what numbers mean.

Once they have learned to speak, children quickly go further than the monkeys, and learn to use numbers in ways that cannot be explained by the accumulator model. The mathematical principal known as cardinal extension or abstraction – knowing that if you have seven policemen each carrying a truncheon and a pair of handcuffs then you must also have seven truncheons and seven pairs of handcuffs – is within the reach of 50 per cent of three-year olds. When Catherine Sophion, of Hawaii University, told children a story about a group of frogs going to a party, each in his own boat, half the children were subsequently able to use the number of boats moored outside the party to calculate the number of frogs inside.

Monkeys have not passed a test of the abstraction principle, but they have not failed one either. And if a test that did not depend on language could be devised, I would be reluctant to bet against their passing it.