Animals and human infants are even capable of some simple arithmetic, such as addition and subtraction (reviews in Gallistel 1990; Dehaene 1997; Spelke & Dehaene 1999; Brannon & Roitman 2003).

Some of these studies employed choice paradigms involving very little or no specific numerical training. Chimpanzees (Pan troglodytes) presented with two sets of two food wells, each of which contained a number of chocolates, selected the pair of quantities whose sum was greater than the sum of the other pair (Rumbaugh et al. 1987). To choose the sets of larger overall quantity, the chimpanzees had to add up the number of chocolates in each set and then compare the two resulting values. Chimpanzees solved the task even when, on the critical trial, the correct set of wells did not contain the larger single value, demonstrating that performance was not based on choosing the largest single amount (Rumbaugh et al. 1988).

Other studies employed specific training on symbols representing numbers. Washburn & Rumbaugh (1991) trained two rhesus monkeys (Macaca mulatta) to choose between two Arabic numbers presented on a touch-sensitive screen. As a reward, the animals received the corresponding number of pellets to the number they had selected. Monkeys always chose the larger number even when they were presented with new combinations of numbers. A similar paradigm was used to test squirrel monkeys (Saimiri sciureus) on tasks requiring them to choose between pairs or triplets of Arabic numerals. Monkeys chose the larger sum and their performance could neither be attributed to choosing the largest single value nor to avoiding the single smallest value (Olthof et al. 1997). Evidence on symbolic training has been reported for apes, e.g. chimpanzees (P. troglodytes; Beran & Rumbaugh 2001).

Using the methodology of the violation of expectancy, Wynn (1992) showed that five-month-old infants can solve some simple arithmetic operations. The idea was that, if infants keep track of the numbers of toys they see being placed behind a screen, they should look longer at the screen that, once lowered, violates their expectations revealing an unexpected outcome. For example, in the 1+1=2 task, one object was placed on a stage, covered by the screen, and then another object was introduced behind the screen. When the screen was removed, infants looked longer at the ‘impossible’ outcomes of either one object or three objects, suggesting that infants expected to see exactly two objects. In the original study, continuous variables such as volume or surface area were not controlled for, thus infants might well have attended to the volume, area or contour length of the objects rather than to their actual number. Wynn's results, however, were then replicated by subsequent studies in which continuous variables were at least partially controlled for. Simon et al. (1995) replicated Wynn's paradigm, but used ‘Elmo’ dolls in the initial phase of the trials and then surreptitiously replaced ‘Elmo’ dolls with ‘Ernie’ dolls. The infants' expectations were not violated by this identity switch. They looked longer only at the numerically unexpected outcome and not at the identity of the outcome. This indicates that, in some sense, infants represent the number of objects stripped of their non-numerical features. Moreover, they do so even across sensory modalities. Five-month-old infants underwent preliminary familiarization trials in which tones were paired with objects, thereby, they learned to associate the two events. At test, the infants were presented alternatively with two types of arithmetic events: the expected, correct outcome operations (one object+one tone=two objects; and one object+two tones=three objects) or the unexpected, incorrect ones (one object+two tones=two objects; and one object+one tone=three objects). Infants looked longer at the unexpected outcomes rather than at the expected ones (Kobayashi et al. 2004).

Wynn's paradigm was also adapted to test arithmetic reasoning in rhesus monkeys (M. mulatta). Subjects viewed food items. A screen was then raised to obscure the items on the stage. Some items were then added or removed from behind the screen. Finally, the screen was lowered to reveal an expected or unexpected number of objects and looking time was measured. Monkeys looked longer when the unexpected outcome was revealed for 1+1=1 or 2 and 2−1=1 or 2 operations (Hauser et al. 1996), even when continuous variables, such as volume and surface area were partially controlled for (Hauser & Carey 2003). Cotton-top tamarins (Saguinus oedipus) and lemurs (Eulemur fulvus, Eulemur mongoz, Lemur catta and Varecia rubra), if presented with the 1+1 operation, looked longer at the unexpected outcome of 3 or 1 compared with the expected value of 2, demonstrating that subjects expected ‘exactly 2’ (Uller et al. 2001; Santos et al. 2005). Also, using a choice measure, expressed by reaching rather than looking time, Sulkowski & Hauser (2001) have shown that rhesus monkeys are remarkably precise at calculating the outcome of subtraction operations for small numbers up to 3. Even domestic dogs have been shown to be capable of solving simple additions such as 1+1=2 rather than 1 or 3 (West & Young 2002). In all these studies, regardless of the paradigm employed, participants were able to compute the correct outcome of simple arithmetic operations whenever both sets were small (upper limit of 4 for adult rhesus monkeys and upper limit of 3 for human babies), but failed whenever any of the sets exceeded such limit: rhesus monkeys failed at three versus eight (Hauser et al. 2000); infants failed at 2 versus 4, 3 versus 6, 1 versus 4 and even 0 versus 4 (Feigenson & Carey 2005). Nevertheless, several other studies showed that animals succeed when dealing with large as well as small sets. Rhesus monkeys (Brannon & Terrace 1998, 2000), socially housed hamadryas baboons, socially housed squirrel monkeys (Papio hamadnjas, Saimiri sciureus; Smith et al. 2003) and brown capuchin monkeys (Cebus apella; Judge et al. 2005) trained to order numerosities from 1 to 4, could then generalize to numbers from 5 to 9. Monkeys trained to respond (in ascending or descending order) to pairs of numerosities (1–9) spontaneously ordered in the same direction new pairs of larger values (i.e. 10, 15, 20, 30; Cantlon & Brannon 2006b). Chimpanzees were able to summate different sets of food items and to select the larger quantities in the comparison 5 versus 8; 5 versus 10 and 6 versus 10 (Beran & Beran 2004) and 2+2+3 versus 3+4+1 (Beran 2001). Monkeys could also sum different numbers of dots and select the correct response in a matching-to-sample task even when large numerosities were employed (Cantlon & Brannon 2007).

Quite interestingly, the discrimination held for both the simultaneous disappearance of all the elements of a set of stimuli as well as for the one-by-one (i.e. sequential) disappearance of the single elements of the set. Indeed, the performance tended to be slightly better in the latter case, although this difference could not be due to cues provided by differences in the time of disappearance of the two sets as this variable had been equalized in the first two experiments.

The results of experiment 2 showed that discrimination was based on the number, and not on the continuous physical variables that may covary with number, such as area, contour length or time of disappearance during stimuli presentation. Evidence has been reported that when the number is pitted against continuous dimensions that correlate with number (e.g. area and contour length), infants sometimes seem to respond to continuous physical dimensions (Feigenson et al. 2002a,b). Nonetheless, evidence has been collected showing that human infants also encode discrete numbers as well as continuous extent (Feigenson & Carey 2003). It seems that when the objects in an array differ in their individual characteristics (such as colour, shape, texture), infants rely on the number rather than spatial extent, whereas the reverse is true for sets of homogeneous objects (Feigenson 2005). Somewhat similar results were obtained in the study by Rugani et al. (submitted; see §2) in which chicks, were directly facing the two sets of objects while choosing between them at test, so that encoding them as a homogeneous or heterogeneous array of elements was perceptually apparent. In the task studied here, by contrast, summation and subtraction required an updating of stored memory representations of small arrays that were no longer visible.

Updating of stored representations by chicks is particularly impressive in experiment 3. Here, after the initial disappearance of the two sets of objects behind both screens, some of the objects were again visibly transferred from one screen to the other before the chick was allowed to search. Thus, an initial addition and a subsequent subtraction of elements were needed in order to determine the screen hiding the larger number of elements at the end of the two events. Chicks correctly chose the screen hiding the larger number of elements irrespective of the directional cues provided by the initial or final movement of the elements.

Human infants seem to show a failure when an exact numerical representation has to include more than three objects (review in Hauser & Spelke 2004). A signature limit similar to that described for young infants has been reported for adult monkeys (Hauser & Spelke 2004), and it was also observed in chicks, although in a different task and on animals a few days older than those used here (Rugani et al. 2008). In the present study, however, chicks seem to go beyond such a limit, for, in order to perform the arithmetic discriminations described in experiment 3, they must have represented five distinct individuals exactly.

It can be speculated that altricial and precocial species may differ in signature limits (or in their timing of appearance in life) in exact numerical representations. Moreover, the signature limit may be task specific, in the sense that it is shaped by the specific demands of the ecological conditions in which a certain numerical computation has evolved. If so, we could perhaps expect that chicks' signature limits would be defined by their typical brood size (i.e. approx. 8–10 siblings).

The capacities exhibited by young chicks appear to be really noteworthy, particularly considering that, in the very brief period of rearing that preceded testing, the animals had no possibility to experience the sorts of events they were faced with at test, since the chicks were reared singly with the suspended balls (and without any experience of occluding surfaces in their home cage). We cannot exclude of course that, over their first days of life, chicks experience (in their home cage) would contribute to some familiarization with quantity manipulation. Besides the imprinting objects, chicks in their home cages were exposed to food, water and the dishes. For example, chicks may learn that the level of chicks' starter food decreases over the time spent eating. Nevertheless, we regard it unlikely for the chicks to experience precise computation of actual subtraction of food elements (we see no possible way for them to experience additions) while eating, as their food (i.e. standard chicks' starter crumbs) was not made of discrete elements (as would occur, for example, if the chicks were fed seeds), but rather of clumped aggregate of grain flour. Moreover, although food amount in the dish could somewhat diminish daily, new food was constantly added so that its level was kept constant.

Chicks could apparently maintain a record of the number of hidden objects comprising two distinct sets, updated by arithmetic operations of addition and subtraction computed on the basis of only the sequential visual appearance and disappearance of single separate objects. Models based on the idea of an operating ‘accumulator’ are common in the theoretical analyses of how animals can perform counting in the absence of verbal tags (Meck & Church 1983). However, to the best of our knowledge, this is the first evidence showing that sequential addition and subtraction can be successfully performed by animals on the same sets.