What Animals Think About Numbers

The British philosopher Bertrand Russell once said that "it must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two." That discovery, however, was made not by the brace of pheasants, but the philosopher himself, presumably as an adult human being. And what of the pheasants? Are they capable of understanding that as a pair they represent the number two? Birding wisdom holds that to watch most birds without disturbing them, it is best to hide behind a blind. If the bird sees you enter, however, you're not much better off because it is now aware of the blind. One way around this problem is for two people to enter the blind together. Some time later, one person leaves and the bird, apparently assuming the coast is clear, goes back to business as usual. Why? Because most birds observed in this situation are incapable of computing a simple subtraction: 2 - 1 = 1! It would seem that, if birds are any indication, animals are far from the most astute of mathematicians. But we do know that animals must have at least limited numerical capabilities. Species with widely different life cycles, ecologies and mating systems have been known to engage in varied forms of mental calculus designed to maximize energetic intake: They calculate average rates of return in food patches, use information about search costs and search speed to assess optimal rates of return, obey Bayes's theorem (a calculation of the probability of future returns based on prior experience) and hide seeds over a broad swath of turf, returning months later to retrieve their stash. These calculations show that animals are indeed equipped with some form of powerful "number-crunching" device. click for full image and caption Figure 1. Counting termites? . . . When facing competition within and between groups, social animals often adhere to the dictum "there is strength in numbers." In chimpanzees, for example, attacking and killing a member of another community occurs only if the intruder is alone and there are at least three adult males in the attacking party. Within social groups, individuals may form coalitions of two or more members to increase their relative dominance over a third individual. In bottlenose dolphins, such coalitions can reach exceptional levels of sophistication. Two to three males in one coalition join up with a second coalition to defeat a third. Occasionally, a large group of 14 male dolphins forms a team that readily overpowers smaller groups. And for what? A single, sexually receptive female. It remains to be seen, however, whether a group's superiority arises from overall numerical superiority or from some other, as yet undiscovered factor. For example, what if the total number of individuals in the two united coalitions is four and the number of individuals in the single coalition is five? Here, two coalitions are greater than one, but five is greater than four. If dolphins are truly counting the number of individuals, then such differences matter. It is clear that animals have need for certain basic arithmetic calculations. We know little, however, about how they represent those calculations, and the corresponding numbers, in their heads. There are two popular experimental designs from which we hope to gain insight into the conceptual representation of numbers in both animals and people: those that explore spontaneous representations of number and those that involve training. Magic with Numbers One recent approach to understanding animal cognition takes advantage of a technique originally developed for human infants. Called the expectancy-violation technique, it uses the twisted logic of a magic show to explore spontaneous representation in animals. Here's the basic principle: Imagine a magic show in which the magician walks on stage, saws a human body in half, separates the pieces and, with a wave of the wand, brings them together again. The victim sits up, perfectly aligned and good as new. The audience stares in amazement. Why? Because adult humans know that the magician has violated a fundamental physical principle: Human bodies can't be sawed in half and then brought back together again. But would human infants watching the same show be equally amazed? Do they respond differently to "magical" deviations of physical principles than to events consistent with those principles? If so, the logic goes, then they have detected a violation and have expressed some level of understanding of physical principles. Developmental psychologist Karen Wynn of Yale University used the expectancy-violation technique to explore whether five-month-old human infants can compute simple math problems, such as 1 + 1= 2. To remove the effects of novelty, the experimenter must first familiarize the infant with the key objects and non-magic events. In this particular study, the experimenter showed an infant either one, two or three Mickey Mouse dolls on a stage, as well as a screen that moved up and down. Test trials started once the infant was bored, looking away from the stage. In the "expected" test (1 + 1 = 2), an infant watched as an experimenter lowered one Mickey Mouse doll onto an empty stage. A screen was then placed in front of the doll. The experimenter then produced a second Mickey Mouse doll and placed it behind the screen. When the screen was removed, the infant saw the expected outcome: two Mickeys on the stage. No magic. In the "unexpected" test, the infant watched the same sequence of actions, involving the same two Mickey Mouse dolls but with one crucial change—a bit of backstage magic. When the experimenter removed the screen, the infant saw either one doll (1 + 1 = 1) or three (1 + 1 = 3). Wynn found that five-month olds consistently look longer (about two seconds more) when the outcome is one or three Mickeys than when the outcome is two. And precisely the same kind of result emerges from an experiment involving subtraction (2 - 1 = 1) instead of addition. Wynn concluded that infants have an innate capacity to do simple arithmetic. By simple, she meant addition and subtraction with a small number of objects. By innate, she meant that the general capacity to track objects and perform arithmetical operations on them comes standard as part of our genetic equipment.