R3.1. Cross-modal studies

Some contributors suggested that cross-modal studies with infants bypass the inherent confound between numerosity and continuous magnitudes (Burr, Hyde & Mou, Libertus et al., and Opfer & McCrink). In such studies, infants are exposed to X and Y number of items in one modality (e.g., visual, tactile) and X number of items in a different modality (e.g., number of sounds). Some of these studies have demonstrated that infants look at mismatched trials (i.e., X items and Y sounds) and matched trials (i.e., X items and X sounds) for different durations, supporting the notion that numerosity is innate. Looking closer into the literature of cross-modal studies, however, reveals a complicated yet fascinating body of evidence that should be further examined before cross-modal studies can be the “smoking gun” (Hyde & Mou) for the innateness of number.

There are several issues with cross-modal studies that restrict the assertion that numerosity is innate. These caveats were put forward more than a decade ago by Mix et al. (Reference Mix, Huttenlocher and Levine2002a). We focus on two main caveats: confound of number and continuous magnitudes and mixed results in cross-modal studies.

First, the use of cross-modal designs does not disentangle numerosity and continuous magnitudes. To illustrate, it takes more time to play three drumbeats than two. To keep the duration constant, one must change the rhythm – three drumbeats in a faster tempo than two. Hence, the match and mismatch that infants detect can be explained by detecting other attributes, such as rhythm, and not necessarily numbers. Indeed, in these studies a lot of effort was put into “controlling” continuous magnitudes (Nieder), but is it at all possible? The influence of continuous magnitudes could be reduced but not excluded, as we discussed in the target article (see also Mix et al. Reference Mix, Huttenlocher and Levine2002a). Because in most of these studies the infants were 5 months old or older, it is possible that discreteness was noticed. To evaluate the role of continuous magnitudes in such studies, it is important to separate the data according to congruity between discrete and continuous magnitudes and investigate whether the same conclusions (significant difference between matched and mismatched looking time) hold in the different congruity conditions. The bottom line here is that controlling for continuous magnitudes by itself is not enough. It is important to properly demonstrate the effect of such control.

Second, the results of cross-modal studies are often contradictory. In some studies, infants are expected to look longer at matched trials (Izard et al. Reference Izard, Sann, Spelke and Steri2009; Jordan & Brannon Reference Jordan and Brannon2006; Starkey et al. Reference Starkey, Spelke and Gelman1990). In other studies, infants are expected to look longer at the mismatched trials (Feigenson Reference Feigenson2011; Féron et al. Reference Féron, Gentaz and Streri2006; Kobayashi et al. Reference Kobayashi, Hiraki, Mugitani and Hasegawa2004; Reference Kobayashi, Hiraki and Hasegawa2005). Feigenson (Reference Feigenson2011) suggests that the interpretation could go either way, depending on the design of the study; in spontaneous preference (without any habituation), infants were found to look longer at matched trials, whereas in studies that included habituation and tested for violation of expectations, infants looked more toward mismatched trials. However, this does not explain how using the same design and the same analysis produce mixed, often contradictory, results. Moore et al. (Reference Moore, Benenson, Reznick, Peterson and Kagan1987) used a violation of expectation design previously used by Starkey et al. (Reference Starkey, Spelke and Gelman1983). Although Starkey et al. found preference for matched trials, Moore et al. found the opposite: 7-month-old infants looked significantly longer at the mismatched trials. Using the same design as Moore et al. and the same analysis of Starkey et al. (Reference Starkey, Spelke and Gelman1990), Mix et al. (Reference Mix, Levine and Huttenlocher1997) found again that infants preferred mismatched trials. The difficulties of replication cast some doubt on the interpretation of the results of cross-modal studies. As is the case with any study that involves newborns and pre-verbal infants, interpreting the results is highly challenging. Pre-verbal babies and newborns are not small adults. Their vision, memory, attention span, and other cognitive skills are limited, which limits the design options of the study. The conclusions drawn from such studies should reflect the possible contribution of such cognitive differences between infants and adults to the results and take into account the difficulty of interpreting them.

In sum, there are many challenges in cross-modal studies: In addition to confounds of continuous magnitudes, it is difficult to interpret the results of studies that are nonreplicable under some conditions (Coubart et al. Reference Coubart, Streri, de Hevia and Izard2015; Mix et al. Reference Mix, Levine and Huttenlocher1997) or whose replication predicts an opposite significant effect. What we can learn from the cross-modal literature is that the ability to discriminate between different magnitudes exists from birth (Izard et al. Reference Izard, Sann, Spelke and Steri2009). This ability might include relying on both number and non-numerical magnitudes and representing abstractly the difference between “more” and “less.”

R3.2. Infant studies and the innateness of the number sense

Lourenco et al., Savelkouls & Cordes, and de Hevia et al. reviewed some studies conducted with infants as young as 4 months old that demonstrate early emergence of a number sense, suggesting that the number sense is innate. Putting aside confounds of continuous magnitudes that were already discussed in the target article, one cannot yet conclude that numerosity is innate. Rubinsten & Karni claim that it is impossible to separate between an ability being innate or an ability being learned to the point of automatic and fluent performance. Therefore, it is impossible to say that either number discrimination or continuous magnitude discrimination is innate; by the age of 4 months there are many interactions of an infant with the environment, and a substantial learning process could have taken place. Accordingly, what we measure at the age of 4 or 7 months could be the end result of such a learning process. Accordingly, Rubinsten & Karni rightfully suggest shifting the focus of the studies from talking about “innateness” to studying how the biology-environment interaction shapes number/magnitude representation.

R3.3. Animal studies

Beran & Parrish and Agrillo & Bisazza suggested that the question of innateness could be answered by studying nonhuman animals, and especially animals that have no experience with magnitudes at all. Imagine having an animal model that has never been exposed to any magnitude – not in audition, vision, touch, or any other modality. How would such an animal perform in a comparative judgment task? The initial preference of such an animal could tell us which magnitude is truly innate. However, is that at all possible? Can one eliminate an animal's access to all possible magnitude cues in all possible modalities? It is difficult to think of such a scenario because, for example, even eating for a longer time means that you will probably eat more and feel fuller, and the same goes for drinking. As long as it is impossible to divest an animal of all possible magnitudes, then the best we can do is to get as much information as we can from the process in which animals learn to use magnitudes.

Agrillo & Bisazza discussed a study in which 1-day-old fish were able to choose the larger group of fellow fish and considered it as evidence of an innate number sense. The claim is that because the fish were viewed serially, there was no confound with continuous magnitudes. However, it could be argued that the fish responded to the amount of time it took them to see all of the fish, or the rhythm in which the fish were viewed, not necessarily their quantity. Therefore, what can be concluded is that some magnitude discrimination is innate. However, we cannot determine which magnitude is innate.

R3.4. What does it mean to “control” for continuous magnitudes?

Rugani et al. suggested that continuous magnitudes could be controlled. They considered two continuous magnitudes – perimeter and area – and demonstrated how these two magnitudes could be controlled through an experiment. Rugani et al. described the relations between numerosity, area, and perimeter in an elegant and simple way. Accordingly, number and perimeter were positively correlated when the area was kept constant, whereas number and area were inversely correlated when the perimeter was kept constant.

However, dissociating perimeter and area from numerosity ignores the possibility that participants rely on other magnitudes or even switch between them. Salti et al. (Reference Salti, Katzin, Katzin, Leibovich and Henik2017) detailed an elaborate taxonomy showing a more complex relationship between continuous magnitudes and numerosity. According to this taxonomy, the inter-correlations between the different continuous magnitudes make the dissociation between numerosity and all continuous magnitudes far from trivial.

Importantly, Rugani et al. acknowledge the importance of continuous magnitudes in numerical perception, stating that human infants and nonhuman animals can solve complex numerical tasks “when both continuous magnitudes and numerical cues are available and consistent” (para. 7), or in other words, when they are highly correlated. We, of course, embrace this notion as it underlies the target article. Moreover, we put forward the notion that to develop an understanding of numerical perception one has to relate to continuous magnitudes as an attribute and a feature and not as a confounding element.

Importantly, the use of the word “control” is misleading and meaningless. What does it mean to control continuous magnitudes? Does it mean to abolish the influence of continuous magnitudes? This option is impossible. Hence, “controlling” continuous magnitudes means different things in different studies. For Rugani et al., for example, it means dissociating different continuous magnitudes in different trials. For others, it means equating different continuous magnitudes in different trials. The ambiguity of the word “control” makes it difficult to interpret the results of studies involving number and continuous magnitudes and to understand the limitations of such studies. Therefore, it is important to provide details of the measures taken in order to deal with the confound of number and continuous magnitudes (e.g., equating, dissociating, etc.).

R4.1. Prominent continuous magnitudes in numerosity comparison tasks

Continuous magnitudes differ in the type of information they convey; for example, intrinsic magnitudes (i.e., diameter, circumference, and area) are indicators of item size. Intrinsic magnitudes can be calculated for a single item or a set of items. For example, one can calculate the area of a dot or the total area of three dots. In contrast, extrinsic magnitudes (i.e., density and convex hull – the smallest polygon containing all items) provide information on the size and spatial location of the items (Salti et al. Reference Salti, Katzin, Katzin, Leibovich and Henik2017). Accordingly, extrinsic magnitudes that contain information about both individual items' size and the spatial relationship between the items have been suggested to have greater influence on numerical judgments.

One example for an extrinsic magnitude that has been demonstrated to directly affect number perception is density. More specifically, Durgin (Reference Durgin1995) adapted participants to a large number of dots on one side of the visual field and to a small number of dots on the other side of the visual field. In the test stage, a patch of dots was presented to either the visual field adapted to high numerosities or that adapted to low numerosities. Participants were then asked to estimate the number of dots. The results revealed that estimates were affected by the type of adaptation: Adaptation to high numerosities yielded higher estimates of numerosities. This was evident particularly for high numerosities (more than 40). These findings led the authors to conclude that density affected the perception of numerosity.

Convex hull was also recently suggested as one of the most influential continuous magnitudes in numerosity comparison tasks. Gilmore et al. (Reference Gilmore, Cragg, Hogan and Inglis2016) showed that the ratio of convex hull areas in a dot comparison task consistently influenced responses for all ages (5–20 years old) and at all stimuli display times (16, 300, and 2,400 ms), whereas the influence of total surface area was stronger in childhood but diminished with age. Accordingly, the authors highlighted the importance of controlling convex hull in numerosity comparison tasks. The mechanism by which convex hull might influence number perception is still unclear. Interestingly, convex hull has played a major role in the subitizing literature. One of the prominent theories about subitizing involves pattern recognition. Involvement of pattern recognition suggests that non-symbolic numerosities that are arranged canonically form a pattern that is automatically translated to a numerosity (Mandler & Shebo Reference Mandler and Shebo1982). For example, three dots in the form of a triangle are automatically perceived as three. Katzin et al. (Reference Katzin, Katzin, Salti and Henik2016) recently suggested that the shape of the convex hull could account for the different ranges of enumeration (subitizing, counting, and estimation).

R4.2. Is the relationship between continuous magnitudes and number static?

Even studies that acknowledged the tight relationship between numerosity and continuous magnitudes, and tried to account for it, assumed that this relationship is constant and not dynamic. However, we have already shown that this relationship could change because of context (Leibovich et al. Reference Leibovich, Henik and Salti2015) or because of saliency (Salti et al. Reference Salti, Katzin, Katzin, Leibovich and Henik2017)

Although Gilmore et al. (Reference Gilmore, Cragg, Hogan and Inglis2016) found that convex hull was the most influential continuous magnitude, some studies have shown other magnitudes to be more influential. For example, in a numerosity comparison task, Leibovich and Henik (Reference Leibovich and Henik2014) used the ratio between five continuous magnitudes and numerosity magnitudes as predictors of response time in two different settings. In the first setting, the groups of dots appeared to the left and right of the center of the screen, at the same latitude. Under this condition, stepwise regression revealed that after numerosity ratio, total circumference was the most influential magnitude. In the second setting, however, the same dot stimuli and procedure were used, but the groups of dots appeared at different latitudes, so one appeared “higher” on the screen than the other did. Under this condition, density was the most influential among the continuous magnitudes. Importantly, unlike Gilmore et al.'s design, the design of Leibovich and Henik contained different levels of congruity that were not taken into account. This demonstrates that the influence of different continuous magnitudes may change because of difference in stimuli, setting, or context.

Task context can also influence the dynamics between number and continuous magnitudes. Leibovich et al. (Reference Leibovich, Henik and Salti2015) asked participants to compare either the number of dots (in the subitizing range, i.e., the numerical task) or their area (i.e., the continuous task). Half of the participants started with the numerical task and half with the continuous task. The order of the tasks affected performance: Participants who started with the continuous task did not show any congruity effect; namely, their comparative judgments of area were not affected by the number of dots. In contrast, comparative judgments of area, by participants who started with the number task, were affected by the irrelevant number of dots, as demonstrated by the significant congruity effect in the area task.

Another factor that can influence the relationship between continuous magnitudes and numerosity is the way in which the stimuli are constructed. This has recently been demonstrated by Salti et al. (Reference Salti, Katzin, Katzin, Leibovich and Henik2017). In this work, the authors used three different sets of stimuli. In one set, the average diameter ratio was equal to the numerical ratio. In the second set, the total surface area ratio was equal to the numerical ratio. In the third set, the convex hull area ratio was equal to the numerical ratio. In all sets, all five continuous magnitudes (average diameter, total circumference, total surface area, density, and convex hull) were either congruent with numerosity (in half of the trials) or incongruent with numerosity (in the other half). Participants were divided into three groups. Each group saw only one set of stimuli and performed either a numerical task or a continuous task (as described in Leibovich et al. Reference Leibovich, Henik and Salti2015). The results revealed that the way in which stimuli were constructed (e.g., equating average diameter ratio and numerical ratio) affected performance in both tasks. For example, in the set where average diameter ratio was equal to the numerical ratio, the congruity effect in the continuous task was larger than the congruity effect in the numerical task. In the set where convex hull ratio was equal to the numerical ratio, there was a very small congruity effect in both tasks. Importantly, the tasks included numerosities between 2 and 4 (i.e., in the subitizing range); hence, more studies using different ranges of number are needed to generalize this result to larger numerosities.

The study of Salti et al. (Reference Salti, Katzin, Katzin, Leibovich and Henik2017) demonstrates how different ratios between magnitudes may affect performance. More specifically, if, for example, the ratio between two numerosities is close to 0 (i.e., a very large difference) and the ratio between total surface areas is closer to 1 (i.e., a very small difference), then it is more likely that numerosity would be a more salient cue than total surface area and would be used by the participant to compare magnitudes. This could be the case that Burr describes: Burr cites the work of Cicchini et al. (Reference Cicchini, Anobile and Burr2016), who presented participants with three dot patches and asked them to pick the odd patch. The authors reported that regardless of task instructions (to choose by a specific magnitude or freely), participants tended to rely on numerosity when choosing the odd patch. However, a closer examination of Cicchini et al.'s stimuli reveals that the numerical ratio was smaller (closer to 0) than the ratios of the continuous magnitudes, suggesting that numerosity was more salient. Because the most salient cue was numerosity, it cannot be generalized to conclude that number is always more salient than continuous magnitudes.

Another example of the dynamic relationship between number and continuous magnitudes is the task itself. Although in a comparison task we can see that more area, larger size, and so forth are indicators of larger numerosity, the opposite occurs in number estimation tasks (Gebuis et al.). When participants are asked to estimate the number of presented items, continuous magnitudes have the opposite influence: The quantity of smaller items is overestimated, whereas the quantity of larger items is underestimated. Although this finding has been reported many times (Cleland & Bull Reference Cleland and Bull2015; Gebuis & van der Smagt Reference Gebuis and van der Smagt2011; Ginsburg & Nicholls Reference Ginsburg and Nicholls1988), there is still no explanation for this phenomenon. One possible explanation is that we compare the relative area (Sidney, Thompson, Matthews, & Hubbard [Sidney et al.]) of the items to the total area (of the screen, for example). From our experience, we know that we can fit more small objects into the same area compared with larger objects, explaining why the quantity of large items is underestimated and that of small items is overestimated. This, however, is just a suggestion that should be empirically confirmed.

To conclude, the influence of continuous magnitudes is dynamic and depends on task, saliency, the stimuli themselves, and so forth. This complexity highlights the importance of studying these factors and understanding their role in non-symbolic number and size perception. Progressing in this direction of research can deepen our knowledge of numerical cognition.

R5.1. What can we learn from nonhuman animal studies?

As mentioned by Beran & Parrish, nonhuman animals have demonstrated a wide variety of magnitude-related behaviors. For example, parrots and chimpanzees were able to learn to associate a quantity with a specific label. This often required lengthy training (years), but nevertheless, it was possible. Chimpanzees that were trained to choose the larger number of items were able to do so using a variety of different continuous magnitudes. We concur with Beran & Parrish that there is a lot to be learned from such training studies in animals. The actual training process can be insightful: Would the learning curve be steeper if at first the stimuli were composed so that the correlation between number and continuous magnitudes would be high? Which incongruent continuous magnitude would affect the rate of learning the most? These are only some of the questions we could test. In the case of the chimpanzee study (Cantlon et al. Reference Cantlon, Platt and Brannon2009b), it would be interesting to analyze the congruity effect throughout the training because it is possible that during training the chimpanzees learned to inhibit the irrelevant continuous magnitudes. It has been demonstrated that nonhuman animals have some cognitive control abilities (Deaner et al. Reference Deaner, Isler, Burkart and Van Schaik2007), and it would be interesting to study the interaction of cognitive control abilities and magnitude processing abilities in animals.

R5.2. What can we learn from computational models?

Another line of studying the possible role of continuous magnitudes in non-symbolic number processing is by using computational models. These are mathematical models that do not require human or nonhuman participants. Instead, a computational model aims to “imitate” brain processes of a computation (like when comparing two non-symbolic numerosities). The model is validated if the computational results are similar to behavioral results.

The advantage of computational models is that they are independent of strategy and prior knowledge. However, in the context of non-symbolic number processing, the problem of the correlation between number and continuous magnitudes still exists. Accordingly, computational models have produced mixed results regarding the independence of number and continuous magnitudes. Stoianov & Zorzi found, in their computational model, both number and area detectors operating in cooperation but independently of one another. In contrast, van Woerkom & Zuidema attempted to replicate Stoianov & Zorzi's model but failed to find number detectors, casting doubt on the robustness of the number sense.

Because of the great potential of computational models, it is important to keep using this tool to study the approximate magnitude system (AMS) and our suggested model under different scenarios. For example, Stoianov & Zorzi suggested in their model that non-symbolic images undergo a normalization process before being enumerated by a numerosity detector. This normalization process allows the system to ignore the different continuous magnitudes of the to-be-counted items. This normalization process could be equivalent to the suggested role of cognitive control in our model. Namely, it could be that instead of being completely inhibited, continuous magnitudes are normalized. This possibility should be further examined.

R5.3. The role of cognitive control in numerical cognition

The role of cognitive control in our suggested model should be further tested. As suggested by Merkley, Scerif, & Ansari (Merkley et al.), cognitive control does not work in isolation. Instead, it works in conjunction with domain-specific knowledge. Merkley et al. raise the intriguing possibility that both top-down and bottom-up attention processes can divert attention toward the discrete aspects of non-symbolic stimuli (i.e., numerosity). For example, knowledge of number words may direct attention in a top-down manner toward the numerosity of a set; physical features of the stimuli (e.g., the range of numerosity) can divert attention in a bottom-up manner toward numerosity (like in small quantities in the subitizing range) or to continuous magnitudes (like in large quantities). In other words, top-down and bottom-up control processes may play a role in diverting attention toward different features of the stimuli.

R5.4. Levels of representation

An important question raised by Odic concerns the different levels of representation of number and continuous magnitudes: Do they share a common abstract representation? Or, do they have separate representations that share similar encoding or decision-making components? A recent study by Sokolowski et al. (in press) shed some light on this question. In a quantitative meta-analysis of more than 90 functional magnetic resonance imaging studies, Sokolowski et al. demonstrated that the representation of symbolic and non-symbolic numbers in the brain was distinct. However, the representation of continuous magnitudes (brightness, line length, area, etc.) was not distinguishable from that of non-symbolic numbers. These results suggest that non-symbolic numbers and continuous magnitudes share a common representation. Of course, one should always keep in mind that even if both non-symbolic numbers and continuous magnitudes activate the same brain regions, the pattern of activation might differ. Therefore, it is important to directly study activation patterns of continuous magnitudes and non-symbolic numbers in the regions found by Sokolowski et al.

R5.5. Acquisition of the symbolic number system

The current model is limited to processing of non-symbolic numerosities. The literature discusses two ways in which symbolic numbers are acquired, that is, the symbol grounding problem. The first is that symbolic numbers are acquired by mapping them into an existing approximate number system (ANS). The second possibility is that symbolic and non-symbolic numbers are learned independently of each other and influence each other reciprocally (Leibovich & Ansari Reference Leibovich and Ansari2016). Sasanguie & Reynvoet suggest that if number is not processed automatically, the former theory seems unlikely. They suggest that the initial stage of acquiring symbolic numbers is the mapping of small symbolic numbers (up to four) to the object tracking system, and that large numbers are not mapped onto the ANS. We agree and highlight that a prerequisite for this initial stage is an understanding that numerosity is independent from continuous magnitudes, for example, an understanding that the quantities of two ants and two firetrucks are equal, despite their vast differences in size. Combining our model with Merkley et al.'s suggestion, we argue that acquiring the first number words enhances attention toward numerosity, thereby allowing a child to map the first symbolic numbers. Importantly, learning the independency of numerosity does not mean that continuous magnitudes will not bias performance as we know they bias even adults (Leibovich et al. Reference Leibovich, Henik and Salti2015).

R5.6. Acquisition of fractions and other types of numbers

Sidney et al. mentioned that our model can also account for acquisition of fractions and proportions, not only whole numbers. They suggest that infants learn about ratios and proportions from a single stimulus also. For example, in a pizza box with eight slices, when two are missing, you can assess how much pizza is left by comparing the area covered by pizza and the area that is not covered. The assessment may not be exact. Importantly, unlike Sidney et al., we argue that this assessment does not involve knowing the exact number of pizza slices (i.e., numerosity). We do agree with Sidney et al. that understanding ratios and continuous magnitudes are interconnected, and that because continuous magnitudes are processed relatively, they can be used to represent all real numbers, not only whole numbers (see also Leibovich et al. Reference Leibovich, Kallai, Itamar and Henik2016).