Wild arithmetic: an insight from comparative studies

In early 90-th development psychologists revealed that very young children can represent numerical quantities without the use of language. Even more importantly, they can understand that addition increases the numerosity of the set of items, while subtraction does the opposite. Wynn (1992) explored whether five-month old infants can solve addition and subtraction problems taking advantage of the human infant’s capacity for understanding object permanence (see details in Chapter 13). As many development psychologists who work with pre-linguistic infants, she used “looking time” as a relevant measure to judge about subject’s understanding of a problem. Wynn showed an infant one, two or three identical dolls (Mickey Mouse) on a stage, as well as a screen moving up and down. Test trial started when an infant was bored, looking away of the stage. In the " expected" test (1 + 1 = 2), an infant watches as an experimenter lowers one Mickey Mouse doll onto an empty stage. A screen is then placed in front of Mickey. The experimenter then produces a second doll and places it behind the screen. With the screen removed, the infant sees two Mickeys on the stage, an outcome that should be expected. In the " unexpected" test, the infant watches the same sequence of actions, involving the same two Mickey Mouse dolls, but with one crucial change. When the experimenter removes the screen, the infant sees either one Mickey (i.e., 1 + 1 = 1) or three (i.e., 1 + 1 = 3). Infants consistently look longer when the outcome is one or three Mickeys than when the outcome is two. The same kind of result emerges from an experiment involving subtraction instead of addition. Wynn concluded that infants have an innate capacity to do simple arithmetic.

Starkey (1992) found even more advanced arithmetic in young children. She used a box where a child could search for tennis balls without being able to look inside. Children were shown a small set of balls put into the box, and then asked to retrieve that set of balls. An assistant, in anticipation of every child’s retrieval, secretly put balls inside the search box so that their number remained constant. 36 to 42 month-old children were able to perceive numerosities of up to 4. When children were shown an additional placement or removal of 1 to 3 balls and then asked to search for the set of balls, the question was whether they would retrieve the number of balls placed originally, or whether they would search for the number of balls after the addition or subtraction. Nearly all 18-24 month-old children searched for the set of balls after the addition or removal, signifying that they could understand the result of a simple addition or subtraction of up to 4. The above experimental results do not principally contradict Piaget's experiments that suggest poor number sense and arithmetical efficiency of children until the age of 7, moreover, they demonstrated a deficiency for more complex addition and subtraction problems, at the preverbal stage of children life.

Hauser et al. (2000) set up Wynn’s 1 + 1 = 2 task for the rhesus monkeys living on the Puerto Rican island of Cayo Santiago. In the first series of experiments subjects watched as an experimenter placed two eggplants behind a screen and then removed the screen. Subjects looked longer when the test outcome was one or three eggplants than when it was the expected two. Like human infants, rhesus monkeys appear to understand that 1 + 1 = 2. Rhesus monkeys also appear to understand that 2 + 1 = 3, 2 -1 = 1, and 3 -1 = 2. They seem, however, not to understand that 2 plus 2 equals 4. Comparable results have been obtained with adult cotton-top tamarins.

In the second series of trials the subjects observed experimenters place pieces of fruits, one at a time, into each of two opaque containers. The experimenters then walked away so that monkeys could approach. The monkeys chose the container with the greater number of apple slices when the comparison were one versus two, two versus three, three versus four and three versus five slices. It is worth to note that this technique differs from many others in that it reveals what animals think, spontaneously, in the absence of training. With respect to numerical abilities, adult rhesus monkeys and tamarins are comparable with one-year-old human infants when it comes to summing objects. Spontaneous representation of number in these primate species is limited to a small number of objects. With training, animals surpass this limitation to sometimes great extent (Uller et al., 2001).

Boysen and co-authors conducted elegant experiments with chimpanzees’ arithmetic. One of their pupils, Sheba, was at first shown a tray beside three placards. The tray contained up to three objects and the placards each portrayed either one, two, or three discs. The task was to select the placard with the same number of discs as objects in the tray. In the next stage of the experiment, discs were replaced by the appropriate numerals. At the end of this training, she was able to select correctly the numerals 0, 1, 2, 3, and 4. In the next experiment Sheba was allowed into the room where oranges were hidden in as many as three different places. To gain reward, Sheba was expected to inspect three locations, and then select the numeral that corresponded to the total number of oranges that she had seen. Finally, numerals were placed in two of the locations. Sheba responded correctly by selecting the numeral that corresponded to the sum of the two numerals that she just inspected (Boysen, 1993; Boysen et al., 1996).

The game played between Sheba and the second chimpanzee, Sarah, displayed how hedonistic interests overshadow arithmetic skills but the use of numerals restored to chimps their prestige as mathematicians that they had nearly lost. In that game (Boysen and Hallberg, 2000) Sheba played the role of selector and the other, Sarah, the role of receiver. Sheba was presented with a tray containing two containers filled with candies; one container always had more candies than the other. Sheba could points to one container. The amount of candies in this container goes to Sarah, and Sheba is left with candies in the other container. Logically, Sheba, the selector, should always point to the container with fewer treats. In so doing, she guarantees receiving the container with more treats. In reality, the chimpanzees could not inhibit their tendency to select the larger of two candy piles, even though a reversed reinforcement contingency dictated that choosing the larger pile resulted in gaining access to smaller pile and vice versa. Sheba displayed signs of excitement such as touching, moving and rearranging the items in the array. Fortunately, Sheba knew numerals. In the key experiment, instead of presenting Sheba with a tray of candies, researchers presented a tray where each set of candies was covered by a card representing a distinctive numeral in dependence of how many items the set contained. Under these conditions, Sheba picked the card with the number “1”. Consequently, she received six treats while Sarah received one treat.

Results of the game experiment, although very expressive, can be considered an evidence of chimpanzee’s awareness of ordinality rather than of their arithmetic skills. In experiments of Beran (2001, 2004) chimpanzees successfully coped with tasks based on addition and removal of items within sets of 10 items. Each of two subjects watched how experimenters place food items into opaque cups. The quantities in each cup were presented as two (2+4; 3+2) or three (2+3+2; 3+1+4) sequentially presented sets. The task was to choose a desirable box in conformity with chimpanzee’s hedonistic interest. They performed at high levels in selecting the largest of two and even of three sets. Subtraction seems to be more complex for chimpanzees: they performed at above chance level for the task of removal of one, but not more than one item.

Reznikova and Ryabko (2000) elaborated a new experimental paradigm of studying ants “arithmetic” skills that in principle can be extended to other social species possessing flexible behaviour, individual recognition and necessity to pass and memorise complex “messages”. The paradigm is based on a fundamental idea of Information Theory that proposes that in a “reasonable” communication system the frequency of the use of a certain message and the length of that message must correlate. This correlation is described by the equation l = - log p, were l is the length of a message and p is its frequency of occurrence. The informal pattern is quite simple: the more frequently a message is used in the language, the shorter is the word or the phrase coding it. For example, even in official documents, the words “White House” are used instead of “The Executive Branch of the Government of the United States of America”. The professional slang, abbreviations, etc. serve the same purpose. This phenomenon is manifested in all known human languages. The second idea is that when using a complicated numerical system, one has to add and subtract small numbers. As it has been noted in 19.4, when using Roman figures, VII = V + II, IX = X - I etc.

The main experimental procedure was described in 19.4. The scheme of the new set of experiments and the main ants’ achievements were the following. Ants were offered a horizontal trunk with 40 branches. The trough with syrup was placed on different branches with different frequencies. On the preliminarily chosen two “special” branches, say, number 10 and number 20, it was placed much more frequently than on the others (i.e. in 2 cases out of 3). When the ants had learnt this, they changed their way of transmitting the information about the branch containing food. The time required for transmission of the message “the trough with food is on the branch number 10” or “…number 20” by the ants was considerably reduced. This enables the researchers to suggest that group retrieving ants have a communication system with a great degree of flexibility. Furthermore, in those cases when the trough with food was placed on branches close to the “special” ones (in the described example - numbers 11, 12, 9, 8, 21, 22, 19, 18), the time required for transmitting the information about them by the ants also decreased considerably. The analysis of time duration the ants spent for transmission of information about different branches suggests that the ants use a mode of representing numbers similar to the Roman numerals in which the “special” numbers (10 and 20 in this case) play the same role as the “special” Roman figures V, X, L etc. Thus, the ants were shown to be able to add and subtract small numbers.

Nowlet us consider plasticity of the ants’ “number system” in more details. In fact, in the cited experiments the researchers examined the ants’ aptitude to change the length of messages in correspondence with their frequency in ant's communication. The experiments were divided into three stages, and at each of them the regularity of placing the trough on branches with different numbers was changed. At the first stages, in selecting the choice of the number of the branch containing the trough, a table of random numbers was used. So the probability of the trough being on a particular branch was 1/30 because only branches 1 - 30 were used. At the second stage the experimenters chose two “special” branches A and B (N 7 and N 14; N 10 and N 20; and N 10 and N 19 in different years) on which the trough with syrup occurred during the experiments much more frequently than on the rest - with a probability of 1/3 for “A” and “B”, and 1/84 for each one of the other 28 branches. In this way, two “messages” of the ants - “the trough is on the branch A” and “the trough is on the branch B” - had a much higher probability than the remaining 28 messages. In one of series of trials we used only one “special” point A (the branch N 15). On this branch the food appeared with the probability of 1/2, and 1/ 58 for the other 29 branches. At the third stage of the experiment, the number of the branch with the trough was chosen at random again.

Now let us consider the relationship between the time which the ants spent to transmit the information about the branch containing food, and its number. The data obtained at the first and third stages of the experiments are shown on the graphs (Fig. VI-15) in which the time of the scout's contact with foragers (t) is plotted against the number (i) of the branch with the trough. At the first stage the dependence is close to linear. At the third stage, the picture was different: first, the information transmission time was very much reduced, and, second, the dependence of the information transmission time on the branch number is obviously non - linear: depression can be seen in the vicinities of the “special” points. So, the data obtained demonstrate that the patterns of dependence of the information transmission time on the number of food - containing branch at the 1-st and 3-d stages of experiments are considerably different. It means that the ants have changed the mode of presenting the data about the number of the branch containing food and rearranged their communication system. Moreover, in the vicinities of the “special” branches, the time of transmission of the information about the number of the branch with the trough is, on the average, shorter. For example, in the first series, at the first stage of the experiments the ants spent 70- 82 sec. to transmit the information about the fact that the trough with syrup is on the branch N 11, and 8-12 sec. to transmit the information about the branch N 1. At the third stage it took 5-15 sec. to transmit the information about the branch N 11.

What about ants’ ability to add and subtract small numbers? An analysis of the time duration of the information transmission by ants raises a possibility that at the third stage of the experiment the messages of the scouts consisted of two parts: the information about which of the “special” branches is the nearest to the branch with the trough, and the information about the distance from this branch with the trough to this definite “special” branch. In other words, the ants, presumably, passed a “name” of the “special” branch nearest to the branch with the trough, and then the number which is necessary to add or subtract in order to find the branch with the trough.

In order to verify this statistically, the experimenters calculated the coefficient of correlation between the time of transmission of information about the trough being on the branch i and the distance from i to the nearest “special” branch. The results confirmed the hypothesis that the time of transmission of a message about the number of the branch is shorter when this branch is closer to any of the “special” ones. The high values of correlation coefficients showed that the dependence is close to linear. This, in turn, suggests that at the third stage of the experiment the ants used a “number system” reminding of Roman numerals, and the numbers 10 and 20, 10 and 19 in different series of the experiments, played a role similar to that of the Roman figures V and X. All this allow suggesting that the ants of highly social group retrieving species are able to add and subtract small numbers (Reznikova and Ryabko, 1996, 1999, 2000).

CONCLUDING COMMENTS

In this Part we have considered many elegant classic and modern experiments aimed to investigate to what extent animals are able to touch the ground of solved problem. In order to examine limits of animal intelligence, researchers confront members of different species with problems that demand, somehow or other, mental reorganisation of cumulative experience based on rule extraction. The ability to form learning sets, that is, to grasp a general rule of solving a concrete problem and then apply this rule to solve new sets of similar problems is one of the necessary prerequisites for insightful behaviour. Another is exploratory activity that extends an area for gaining information and premises to hidden learning in many species.

There is much work to be done to extend our understanding of whether at least some species share advanced characteristics of intelligence with human beings, or all animals think about the world in a way radically different from our own. Working with tool using animals as well as with counting beasts serves as a good background for development of comparative studies of cognition.

It is still discussible whether profiling tool users such as chimpanzees and New Caledonian crows can take into account imperceptible physical forces or they only are capable of reasoning about perceptible things. Some experiments revealed a great deal of flexibility and rationality, whereas others established the distinction between performance and competence in wild users. To clear this problem to a lesser or greater extend, development studies are needed that allow extraction of inherited and acquired behaviour. This approach and obtained data will be considered in Part VII.

Although our knowledge about advanced forms of animal intelligence have been extremely enlarged by applying new experimental paradigms including studies on counting in animals, we are still so far from integrative measure of animal intelligence, just as a teacher asking a little child to what limit can he or she count. May be ants are more competence than chimpanzees in numerosities; pigeons and crows are equal to our close relatives in applying number related skills; may be Darwin finches are not less and New Caledonian crows are even more advanced than chimps in tool manufacture; may be not. We will see in Part VII that some species are predisposed to intellectual feats within narrow limits of solving life-or-death problems. It is possible in principle that group retrieving ant species together with honey bees share with humans (within modest limits of course) such fundamental and interrelated properties of intelligence as predisposition to mathematics and symbolic language. What allows humans to have their heads in the clouds, allows social insects to search for food in complex situations and treat their collective reason well, from small groups of individuals, to huge inter-communities. We will consider an intriguing problem concerning inter-relations between intelligence, language and sociality in Part VIII.

In the course of the book, we have considered dozens of experiments conducted in boxes, more or less comfortable for tested subjects. Only several outdoor experiments have been considered in this Part at last. The next Part VII will be devoted to the nature and specificity of intelligence in different species. We will examine members of species in the context of their natural life.