Mosquitofish can learn to discriminate small quantities, even when non-numerical indicators of quantity are unavailable, hence providing the first evidence that fish, like primates, can use numbers. As in humans and non-human primates, genuine counting appears to be a ‘last resort’ strategy in fish, when no other perceptual mechanism may suggest the quantity of the elements. However, our data suggest that, at least in fish, the priority of perceptual over numerical information is not related to a greater cognitive load imposed by direct numerical computation.

Here we used a training procedure to investigate whether mosquitofish could learn to discriminate between two and three objects even when denied access to non-numerical information. In the first experiment, fish were trained to discriminate between two sets of geometric figures. These varied in shape, size, brightness and distance, but no control for non-numerical variables was made. Subjects were then re-tested while controlling for one non-numerical variable at a time. Total luminance of the stimuli and the sum of perimeter of figures appeared irrelevant, but performance dropped to chance level when stimuli were matched for the cumulative surface area or for the overall space occupied by the arrays, indicating that these latter cues had been spontaneously used by the fish during the learning process. In a second experiment, where the task consisted of discriminating 2 vs 3 elements with all non-numerical variables simultaneously controlled for, all subjects proved able to learn the discrimination, and interestingly they did not make more errors than the fish in Experiment 1 that could access non-numerical information in order to accomplish the task.

Research on human infants, mammals, birds and fish has demonstrated that rudimentary numerical abilities pre-date the evolution of human language. Yet there is controversy as to whether animals represent numbers mentally or rather base their judgments on non-numerical perceptual variables that co-vary with numerosity. To date, mental representation of number has been convincingly documented only for a few mammals.

The first experiment aimed to determine which cues mosquitofish used spontaneously when both numerical information and continuous physical attributes are available. Subjects learned a discrimination between 2 and 3 objects in the absence of any manipulation of the stimuli; after animals had achieved learning criterion they were tested without reward while controlling for one perceptual non-numerical variable at a time. In the second experiment we trained fish to discriminate between 2 and 3 objects while we simultaneously controlled for non-numerical variables, in order to determine whether fish could discriminate quantities by using only numerical information as shown for mammals.

In the present work we investigated whether fish can discriminate between two quantities when access to non-numerical cues was prevented. The procedure used in our previous studies with fish did not easily permit a fine-grained manipulation of stimuli and an efficient control of continuous perceptual variables that correlate with number. Therefore we adopted a procedure modelled on carefully-controlled experiments conducted on non-human primates, which consisted of training the subject to discriminate between sets containing different numbers of geometric figures while controlling for the perceptual non-numerical variables [3] , [21] .

Nonetheless, before concluding that the same systems are involved, it is necessary to understand whether similar limits really reflect identical underlying mechanisms. In particular, it has been extensively demonstrated that both humans and nonhuman animals can discriminate between two quantities without necessarily counting the number of objects. Numerosity normally co-varies with several other physical attributes, and organisms can use the relative magnitude of continuous variables such as the total area of the stimuli or the sum of their contour, to estimate which group is larger/smaller [16] – [18] . Discriminations based on number or on continuous extent often yield comparable results and therefore carefully controlled experiments are necessary to show that an animal is really using numerical information. Experiments of this type demonstrate that, when selecting the larger shoal, mosquitofish (Gambusia holbrooki) spontaneously use non-numerical cues, namely the sum of areas of the shoals and the overall quantity of movements of the individuals within the shoal [15] . This does not necessarily imply that fish are unable to discriminate two groups on the basis of the numerosity alone. Overall perceptual cues may simply be the easiest indicators of numerosity in this task. Indeed, there is persuasive evidence that even humans and non-human primates, which have the capacity to represent number, in many circumstances base their quantity judgment primarily on proxy measures such as area, contour or density of elements and that they use number as a last resort, when there are no other available cues [19] , [20] .

Recently we found [15] that fish, seeking safety from predators, display a rudimentary numerical ability in selecting the largest shoal. Interestingly, the limits shown by fish in this task closely resemble those that have been reported for primates. These data in particular suggest the existence, as in primates, of two independent pre-verbal systems: one for counting a small quantity (≤4) precisely, and the other for estimating large quantities (>4) approximately. These findings suggest the possibility that all extant vertebrates share similar quantificational mechanisms, which may have an ancient phylogenetic origin, at least predating the divergence of the tetrapod lineage.

McComb and co-workers [2] using playback experiments found that wild lions based the decision whether or not to attack a group of intruders on a comparison of the number of roaring intruders they had heared and the number and composition of their own group. Over the last two decades or so, extensive laboratory research carried out on monkeys and apes [3] – [6] has revealed the existence of non-verbal systems of numerical representation that non-human primates apparently share both with human infants and with human adults tested in comparable conditions [7] , [8] . In recent years, rudimentary numerical abilities have been reported in several other mammalian and avian species, among others, elephants, dolphins, dogs, cats, robins and chicks [9] – [14] .

Abilities such as recording the number of events, enumerating items in a set, or comparing two different sets of objects, can be adaptive in a number of ecological contexts. Lyon [1] , for instance, reported a spontaneous use of numerical information (egg recognition and counting) in a natural context as a strategy to reduce the costs of conspecific brood parasitism in American coots.

When we compared the number of trials necessary to reach criterion in Experiment 1 (when all numerical and non-numerical cues were available) and Experiment 2 (where only numerical cues were available), we found no difference between experiments (trials in Experiment 1: 25.2±11.7; trials in Experiment 2: 29.14±9.7; F(1,33) = 1.064, p = 0.170; power = 0.170).

As a by-product of controlling for three perceptual variables, stimuli differed for two other non-numerical variables that the fish could have used instead of number to learn the discrimination. The by-product of pairing the cumulative surface area between sets with two and three elements was that in the latter sets smaller-than-average in area figures were more frequent. The by-product of pairing the overall space occupied by configuration was that figures were more spaced out in the sets containing two figures. After reaching criterion, fish were thereby subjected to a test phase without reinforcement using pairs of stimuli composed of figures of identical size and similarly spaced.

In this experiment we trained fourteen fish to discriminate between 2 and 3 objects while we simultaneously controlled stimuli for their non-numerical variables in both the pre-training and the training phase, with the aim of determining whether fish could learn the discrimination using only numerical information. Using the same geometric figures as the previous experiment, we designed pairs of stimuli in which the total luminance, the cumulative surface area, and the overall space occupied by the arrays were paired between the groups with two and three elements. We found no difference in the proportion of correct choices between fish trained with three (mean±std. dev.: 0.690±0.037) and those trained with two figures (0.651±0.070; t(12) = 1.328, p = 0.209). All 14 fish reached the criterion (chi square test, p<0.05), proving thus able to select the trained numerosity. Overall the choice for the trained stimuli is highly significant (t(13) = 11.103, p<0.001).

On the whole, results of Experiment 1 showed that fish were found to base their discrimination on the cumulative surface area occupied by figures and on the overall space occupied by the arrays, while they did not use the sum of perimeter, the total luminance of the stimuli or the number of items. Interestingly, in Experiment 1.a we observed a negative correlation between the proportion of correct choices when cumulative surface area was paired and when the overall space of the arrays was paired (Spearman test, ρs = −0.734, p = 0.024) indicating that there was an individual variability in the cues used with some subjects relying on the cumulative surface area for discriminating and not being affected by the overall space, while others used the overall space but not area to solve the task.

We reported no difference in the proportion of correct choices between fish trained with three (mean±std. dev.: 0.788±0.066) and those trained with two figures (0.753±0.065; t(8) = 0.832, p = 0.429). When the relative ratio of the areas (but not the perimeter) was predictive of the numerical ratio, we observed a significant choice toward the trained quantity (t(9) = 3.786, p = 0.004) whereas no significant choice was observed in the condition in which the perimeter, but not the area, could be used to distinguish between two quantities (t(9) = −0.653, p = 0.530). The difference between the two conditions was significant (paired t-test, t(9) = 2.865, p = 0.019, Fig. 3b ).

We accordingly set up another experiment with ten subjects, using the same procedure as before in pre-training and training phases, whereas in the test phase fish were presented with only two different sets of stimuli: one set in which the cumulative surface area was paired whereas the sum of perimeter was not (i.e. the perimeter could suggest the exact ratio between the quantities), and one set in which the sum of the perimeter was paired whereas the cumulative surface area was not (i.e. the area could suggest the exact ratio between the quantities).

However, since area and perimeter of the figures are strictly related to each other, in this experiment by controlling one variable we inevitably affected the other, so that it was not possible to conclude whether one or both variables were important in the discrimination.

All ten subjects reached the learning criterion in the training phase, but one was excluded from the subsequent test phase due to poor health, and hence nine started the test phase. We reported no difference in the proportion of correct choices between fish trained with three (mean±std. dev.: 0.753±0.065) and those trained with two figures (0.678±0.028; t(7) = 2.337, p = 0.052). In the test phase a significant discrimination was observed when no perceptual cue was controlled for (t(8) = 2.449, p = 0.020) and when the total luminance was controlled for (t(8) = 2.310, p = 0.025); no significant choice toward the trained quantity was found when the sum of perimeter of the figures (t(8) = 1.316, p = 0.225), the cumulative surface area (t(8) = −1.512, p = 0.169), and the overall space occupied by the arrays (t(8) = −0.373, p = 0.719) were controlled for ( Fig. 3a ).

Subjects were singly placed in the middle of a test chamber provided with two doors (one associated to three and the other associated to two elements) placed at two opposite corners. Subject could pass through the reinforced door to rejoin shoal mates in the outer tank (not shown).

Ten female mosquitofish were placed in an unfamiliar tank and trained to discriminate between two doors in order to re-join their social group ( Fig. 1 ). Doors were associated with a pair of stimuli consisting of two or three small figures ( Fig. 2 ). These figures were randomly selected with replacement from a pool of approximately 100, and no control for non-numerical variables was operated in the learning phase. Subjects were given six trials per day for a maximum of ten days. Once a subject had reached the learning criterion, it was admitted to the test phase and was examined in the same apparatus without reward (no possibity to re-join the conspecifics) while controlling for one perceptual non-numerical variable at time. We controlled those variables that were shown to be relevant in previous studies with vertebrates, namely the total luminance of the two stimuli, the sum of perimeter of the figures, the cumulative surface area, and the overall space occupied by the arrays. Since operant conditioning is normally a stressful procedure for fish, we adopted a pre-training procedure that consisted of exposing the subjects, in the seven days preceding the training, to the choice of similar pairs of stimuli in order to move from one compartment to the other of their home-tank.

Discussion

Our experiments show that the ability of mosquitofish to discriminate among sets containing a different number of elements is not limited to the socio-sexual context [15], [22], [23], but also applies to sets of abstract elements. They also indicate that mosquitofish can accomplish this task when all non-numerical perceptual variables are matched between the stimuli, thus strongly suggesting that teleosts, like mammals, possess true counting abilities, at least in the domain of small numbers.

Cues spontaneously used by fish to discriminate between quantities The first experiment showed that during the extinction phase a good performance was maintained when no control of perceptual variables was operated or after the total luminance of the stimuli and the sum of perimeter of figures were matched. Conversely mosquitofish were unable to select the learned numerosity when stimuli were matched for the cumulative surface area or for the overall space occupied by the arrays, thus suggesting that these two cues had been used during the learning process. Previous works have already demonstrated that such variables play an important role as proxies of numerosity in humans and other mammals [12], [18], [19]. It is interesting to notice that the finding of Experiment 1.a, that fish are influenced by two different non-numerical cues, is somehow a paradox. In fact, because this experiment only controlled for one variable at a time, the fish should have succeeded in all conditions. That is, when the cumulative surface area was controlled for, the overall space of the arrays was not, and vice versa. One explanation for this apparent conflict is that fish combined two different non-numerical cues to learn the numerical discrimination, and that, consequently, the absence of one of the two cues was sufficient to worsen their performance. A second possibility is that fish used only one non-numerical cue, but that there was individual variation in the cue they adopted to learn the discrimination. Our data are more in accordance with the latter hypothesis. In Experiment 1.a the performance of subjects in trials when the cumulative surface area was controlled for correlated negatively with the performance in trials when the overall space occupied by the arrays was controlled for, suggesting that fish that were more influenced by the cumulative surface area were unaffected by manipulation of the overall space of the arrays, whereas subjects that relied on this latter cue did not use the cumulative surface area during the learning process. However, this evidence is based on the examination of a small number of subjects, and caution should be exercised before drawing firm conclusions on this question.

Discrimination of small quantities using only numerical information Results of the second experiment showed that fish can use numbers when perceptual variables that correlate with numerosity were excluded. To date, this is the first evidence that a lower vertebrate can really represent and compare numbers. The capacity to discriminate among sets containing different numbers of objects by using numerical information only, previously reported for six month old infants [24], primates, dolphins and dogs [4], [10], [11], [25], is here extended to include a species, the Eastern mosquitofish, which is phylogenetically very distant from mammals and has a much smaller brain size compared with the former species. Observations made in this study were limited to a single quantity discrimination, 2 vs 3. This was shown to be the upper limit in the capacity of discrimination of six-month old infants [18]. In experiments with continuous variables controlled for, non-human primates successfully discriminate between 3 and 4 objects [4], and non-verbal counting abilities of human adults can be even better [26]. Mosquitofish have been shown to discriminate a shoal of three fish from one of four in two different contexts [15], [22], but in these experiments access to continuous extent of the stimuli could not be prevented. In one of these studies [22], shoals were in two distinct compartments of the apparatus so that they could not be seen and compared simultaneously. This implies that whatever information, numerical or continuous, the mosquitofish encoded, they were able to maintain it temporarily in working memory. Further experiments are necessary to determine whether the upper limit of discrimination of fish also matches that of mammals when access to continuous extent of the stimuli is prevented. Many authors now agree that there are two distinct non-verbal systems for representing numerosity in animals, adults and human infants [7], [17]. The first mechanism proposed is the one most likely investigated by us in this study. It is an object-tracking system that operates on a small number of items by keeping track of individual objects [27], [28]. It is precise but, due to the limited number of available indexes, it is supposed to allow for the parallel representation of up to 3–4 elements only [29]. The second is an analog magnitude system of numerical representations that allows approximate discrimination of large quantities. It obeys Weber's Law, which holds that as numerical magnitude increases, a larger disparity is needed to obtain the same level of discrimination [8], [30]. Fish have shown to rival primates in their ability to discriminate large quantities approximately [15], [22], [31], [32]. However, while controlled experiments have shown that six month babies and non-human primates can perform large number discrimination using only numerical information, no such evidence exists for fish. Future research should assess if the analog magnitude system can operate in fish when access to continuous extent of the stimuli is prevented.