Various authors have proposed that humans and animals integrate multiple magnitudes (number, area, density, etc.) when comparing numerosities; proposed mechanisms range from the idea that numerical information is more cognitively demanding than continuous magnitudes, and it is processed as a last-resort strategy when no other information is available, to the idea that the number system increases its precision by integrating available non-numerical information in the process of estimation and comparison (Agrillo et al. Reference Agrillo, Piffer and Bisazza2011; Davis & Perusse Reference Davis and Perusse1988; Meck & Church Reference Meck and Church1983).
What is new in the model proposed in the target article is the idea that humans and nonhuman species are born with a quantitative system that holistically processes numerosity and continuous magnitudes, and that a “sense of number” would gradually develop during ontogeny from understanding the correlation between numerosity and continuous magnitudes.
In the Introduction, the authors acknowledge the importance of animal studies for understanding the mechanisms of numerical discrimination. However, the evidence of such studies, whether in favor of or against their hypotheses, is not discussed. Here we argue that (1) experiments on animals, specifically fish research, can be more appropriate than research on humans to test some of the model's assumptions, especially to examine the hypothesis that number sense is not innate, but rather stems from individual experiences; and (2) evidence from animal studies that would be useful for evaluating the proposed model is already available.
Humans, other mammals, and most birds are extremely immature at birth, and the procedures commonly used to study number sense with adults (e.g., training procedures or free choice tests) cannot be employed; conversely, procedures used with young individuals (e.g., habituation or violation of expectancy) are usually complex to adapt to testing adults. This prevents researchers from comparing the different developmental stages with the same paradigm. Poor sensory acuity can further prevent testing for numerical abilities at very young ages (see Leibovich et al.). On the contrary, most fish species produce offspring that are completely independent at birth. Newly born fish generally face the same ecological challenges as adults (i.e., evading predators, selecting an appropriate diet and catching prey, orienting themselves in space, and interacting with conspecifics). This allows researchers exploiting a number of spontaneous behaviors, such as preferring the largest amount of food or the largest group of conspecifics, the same preferences studied in adults. Recently, we found that guppies can be trained to have numerical discrimination within their first week of life, which makes it a tool available for cross-age comparisons (Piffer et al. Reference Piffer, Miletto Petrazzini and Agrillo2013).
Innate cognitive abilities often appear later in life, not because they need experience to develop, but because the maturation of the nervous system is required or because a given cognitive ability is not necessary for survival in early life (a fact that was not considered by Leibovich et al.). In these cases, answering the question of number sense innateness requires the manipulation of experience to disentangle the relative contribution of cerebral maturation and individual experiences on the development of numerical abilities. For both practical and ethical reasons, in higher vertebrates it is difficult, if not impossible, to devise experiments that dissociate the role of these two factors (e.g., LeVay et al. Reference LeVay, Wiesel and Hubel1980; Ridley & Baker Reference Ridley and Baker1982). Such research is more feasible in fish. For example, one experiment found that in guppies that are prevented from experiencing different numbers of objects, the discrimination of large numerosities appears spontaneously at around 40 days of age; this capacity can, however, be anticipated at 20 days of age if guppies are reared in an environment that offers such experiences from birth (Bisazza et al. Reference Bisazza, Serena, Piffer and Agrillo2010).
Though none has been specifically designed to test the hypotheses of Leibovich et al., several fish studies provide information relevant to the present debate. Concerning the existence of a holistic system for processing numerical and continuous magnitudes, data on fish research generally support this view. For example, Agrillo et al. (Reference Agrillo, Piffer and Bisazza2011) found that mosquitofish routinely integrate numerical information and continuous magnitudes. Their performance was more accurate when both pieces of information were simultaneously available, compared with when only numerical information or only continuous information was provided. However, not all continuous magnitudes are equally important. Mosquitofish appear to rely on the total surface area and convex hull, whereas total luminance and contour length appear to be irrelevant. Interestingly, interindividual differences in the use of continuous magnitudes were observed in this species (Agrillo et al. Reference Agrillo, Dadda, Serena and Bisazza2009). Likewise, fish can rapidly discriminate four from five companions (Lucon-Xiccato et al. Reference Lucon-Xiccato, Dadda, Gatto and Bisazza2017), but their performance drops dramatically when prevented from using the total surface area or total activity of the stimulus fish (Agrillo et al. Reference Agrillo, Dadda, Serena and Bisazza2008).
Regarding the question of whether number sense is innate, available data on fish seem to contradict the authors' core hypothesis. In one experiment, 1-day-old fish were able to select the larger group of companions even when they were tested in an apparatus that allowed them to see only one fish at time, thus precluding the possibility of summing up areas or contours and gauging the density or convex hull of the groups (Bisazza et al. Reference Bisazza, Serena, Piffer and Agrillo2010). In another study in which guppies were trained in their first week of life to discriminate either between two numerical quantities controlled for the total surface area and other continuous magnitudes or between two figures differing in area by the same ratio (a condition in which numerical information was made irrelevant), only fish trained with numbers learned how to discriminate (Piffer et al. Reference Piffer, Miletto Petrazzini and Agrillo2013). Therefore, if a temporal mismatch between the number sense and the discrimination of continuous magnitudes does exist, in fish this appears to be opposite to that predicted by the model.
We acknowledge that the aforementioned data were not collected with these working hypotheses in mind and that alternative explanations are available in some cases. Nonetheless, we believe the cited examples convincingly demonstrate the possibility of investigating in fish the interesting issues raised by Leibovich et al. in a way that cannot as easily be done in higher vertebrates.
Various authors have proposed that humans and animals integrate multiple magnitudes (number, area, density, etc.) when comparing numerosities; proposed mechanisms range from the idea that numerical information is more cognitively demanding than continuous magnitudes, and it is processed as a last-resort strategy when no other information is available, to the idea that the number system increases its precision by integrating available non-numerical information in the process of estimation and comparison (Agrillo et al. Reference Agrillo, Piffer and Bisazza2011; Davis & Perusse Reference Davis and Perusse1988; Meck & Church Reference Meck and Church1983).
What is new in the model proposed in the target article is the idea that humans and nonhuman species are born with a quantitative system that holistically processes numerosity and continuous magnitudes, and that a “sense of number” would gradually develop during ontogeny from understanding the correlation between numerosity and continuous magnitudes.
In the Introduction, the authors acknowledge the importance of animal studies for understanding the mechanisms of numerical discrimination. However, the evidence of such studies, whether in favor of or against their hypotheses, is not discussed. Here we argue that (1) experiments on animals, specifically fish research, can be more appropriate than research on humans to test some of the model's assumptions, especially to examine the hypothesis that number sense is not innate, but rather stems from individual experiences; and (2) evidence from animal studies that would be useful for evaluating the proposed model is already available.
Humans, other mammals, and most birds are extremely immature at birth, and the procedures commonly used to study number sense with adults (e.g., training procedures or free choice tests) cannot be employed; conversely, procedures used with young individuals (e.g., habituation or violation of expectancy) are usually complex to adapt to testing adults. This prevents researchers from comparing the different developmental stages with the same paradigm. Poor sensory acuity can further prevent testing for numerical abilities at very young ages (see Leibovich et al.). On the contrary, most fish species produce offspring that are completely independent at birth. Newly born fish generally face the same ecological challenges as adults (i.e., evading predators, selecting an appropriate diet and catching prey, orienting themselves in space, and interacting with conspecifics). This allows researchers exploiting a number of spontaneous behaviors, such as preferring the largest amount of food or the largest group of conspecifics, the same preferences studied in adults. Recently, we found that guppies can be trained to have numerical discrimination within their first week of life, which makes it a tool available for cross-age comparisons (Piffer et al. Reference Piffer, Miletto Petrazzini and Agrillo2013).
Innate cognitive abilities often appear later in life, not because they need experience to develop, but because the maturation of the nervous system is required or because a given cognitive ability is not necessary for survival in early life (a fact that was not considered by Leibovich et al.). In these cases, answering the question of number sense innateness requires the manipulation of experience to disentangle the relative contribution of cerebral maturation and individual experiences on the development of numerical abilities. For both practical and ethical reasons, in higher vertebrates it is difficult, if not impossible, to devise experiments that dissociate the role of these two factors (e.g., LeVay et al. Reference LeVay, Wiesel and Hubel1980; Ridley & Baker Reference Ridley and Baker1982). Such research is more feasible in fish. For example, one experiment found that in guppies that are prevented from experiencing different numbers of objects, the discrimination of large numerosities appears spontaneously at around 40 days of age; this capacity can, however, be anticipated at 20 days of age if guppies are reared in an environment that offers such experiences from birth (Bisazza et al. Reference Bisazza, Serena, Piffer and Agrillo2010).
Though none has been specifically designed to test the hypotheses of Leibovich et al., several fish studies provide information relevant to the present debate. Concerning the existence of a holistic system for processing numerical and continuous magnitudes, data on fish research generally support this view. For example, Agrillo et al. (Reference Agrillo, Piffer and Bisazza2011) found that mosquitofish routinely integrate numerical information and continuous magnitudes. Their performance was more accurate when both pieces of information were simultaneously available, compared with when only numerical information or only continuous information was provided. However, not all continuous magnitudes are equally important. Mosquitofish appear to rely on the total surface area and convex hull, whereas total luminance and contour length appear to be irrelevant. Interestingly, interindividual differences in the use of continuous magnitudes were observed in this species (Agrillo et al. Reference Agrillo, Dadda, Serena and Bisazza2009). Likewise, fish can rapidly discriminate four from five companions (Lucon-Xiccato et al. Reference Lucon-Xiccato, Dadda, Gatto and Bisazza2017), but their performance drops dramatically when prevented from using the total surface area or total activity of the stimulus fish (Agrillo et al. Reference Agrillo, Dadda, Serena and Bisazza2008).
Regarding the question of whether number sense is innate, available data on fish seem to contradict the authors' core hypothesis. In one experiment, 1-day-old fish were able to select the larger group of companions even when they were tested in an apparatus that allowed them to see only one fish at time, thus precluding the possibility of summing up areas or contours and gauging the density or convex hull of the groups (Bisazza et al. Reference Bisazza, Serena, Piffer and Agrillo2010). In another study in which guppies were trained in their first week of life to discriminate either between two numerical quantities controlled for the total surface area and other continuous magnitudes or between two figures differing in area by the same ratio (a condition in which numerical information was made irrelevant), only fish trained with numbers learned how to discriminate (Piffer et al. Reference Piffer, Miletto Petrazzini and Agrillo2013). Therefore, if a temporal mismatch between the number sense and the discrimination of continuous magnitudes does exist, in fish this appears to be opposite to that predicted by the model.
We acknowledge that the aforementioned data were not collected with these working hypotheses in mind and that alternative explanations are available in some cases. Nonetheless, we believe the cited examples convincingly demonstrate the possibility of investigating in fish the interesting issues raised by Leibovich et al. in a way that cannot as easily be done in higher vertebrates.