© 2016 Canadian Psychological Association 1196-1961/16/$12.00 http://dx.doi.org/10.1037/cep0000070

The Symbol-Grounding Problem in Numerical Cognition: A Review of Theory, Evidence, and Outstanding Questions Tali Leibovich and Daniel Ansari The University of Western Ontario How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the “symbol-grounding problem,” is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem. Keywords: numerical cognition, symbol-grounding problem, approximate number system, numerical magnitudes, nonnumerical magnitudes

imate Number System” or ANS). Against this background, the present literature review will examine (a) the extent to which the available behavioural and neuroscientific evidence supports the theory that number symbols are grounded in the ANS and (b) the extent to which the purported ANS is numerical in nature. The review will close by outlining questions for future theory and research.

Whether it is to estimate the distance to the nearest bus stop, to choose the shortest line in the supermarket or to calculate the change given to you, we use numbers constantly in our everyday lives. How we represent and process numbers has been studied extensively in humans at different developmental stages, as well as in a variety of animal species. One of the key questions in the literature concerns how humans learn the meaning of numerical symbols (sometimes referred to as the “symbol-grounding problem”), such as Arabic numerals, which are uniquely human cultural representation of numerosity that children need to learn over the course of development. The present article begins by reviewing the most prominent proposal for the resolution of the symbolgrounding problem in numerical cognition, which postulates that symbols are grounded in a system for the approximate representation of number that we share with other species (the so-called “Approx-

The Number Sense/ANS Account of the Symbol-Grounding Problem One of the most prominent theories in the field of numerical cognition was put forward by several researchers (e.g., Dehaene, 1997; Gallistel & Gelman, 1992). According to this theory, the ability to process numerical quantities (i.e., the total number of items in a set or its numerosity) is a basic, automatic, and innate ability that can be found across species. In other words, humans are born with a capacity to process nonsymbolic numerosities (e.g., dot arrays, groups of objects, number of sounds, etc.). Furthermore, according to this theory, symbolic representations of number, such as Arabic numerals and number words, which children learn over the course of development, are thought to acquire their meaning by being mapped onto the preexisting, nonsymbolic representations of number. This theory is supported by studies demonstrating that preverbal babies and animals are able to spontaneously discriminate between numerosities (e.g., Libertus & Brannon, 2009; Nieder, 2005). For example, in a habituation study (repeated presentation of a particular stimuli followed by the presentation of a stimuli that differs in the variable of interest) by Xu and Spelke (2000), infants were

Tali Leibovich and Daniel Ansari, Numerical Cognition Laboratory, Department of Psychology & Brain and Mind Institute, The University of Western Ontario. This work was supported by operating grants from the Natural Sciences and Engineering Council of Canada (NSERC), the Canadian Institutes of Health Research (CIHR), the Canada Research Chairs Program, an E.W.R Steacie Memorial Fellowship from the Natural Sciences and Engineering Council of Canada (NSERC) to Daniel Ansari, as well as a Brain & Mind Institute (Western University) Postdoctoral Scholarship to Tali Leibovich. Correspondence concerning this article should be addressed to Daniel Ansari, Department of Psychology & Brain and Mind Institute, The University of Western Ontario, Westminster Hall, Room 325, London, ON N6A 3K7, Canada. E-mail: [email protected] 12

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habituated to groups of dots in different sizes but that contained the same numerosity. In the test stage, the infants saw two displays— one display with the same numerosity as in the habituation stimuli and another display containing a novel number of dots. The results of this experiment revealed that infants looked longer toward the novel numerosity compared with the numerosity to which they had been habituated. Furthermore, it was found that 6-month-old infants only exhibited the ability to discriminate between the habituated and novel numerosity when the ratio between the two stimuli was 0.5 (e.g., 8 vs. 16 dots), but not when the ratio was higher (e.g., 8 vs. 12 dots). These findings have since been replicated in a number of different labouratories (Coubart, Izard, Spelke, Marie, & Streri, 2014; Izard, Sann, Spelke, & Streri, 2009). In addition to work on infants, it has also been demonstrated that a variety of animals are able to discriminate between different nonsymbolic numerosities (birds, Bogale, Kamata, Mioko, & Sugita, 2011; Watanabe, 1998; fish, Agrillo, Dadda, Serena, & Bisazza, 2008, 2009; rodents, Meck & Church, 1983; lions, McComb, Packer, & Pusey, 1994; and primates, Cantlon & Brannon, 2006; Nieder, Freedman, & Miller, 2002). Discrimination between different numerosities shows a similar response pattern to that of discriminating between different magnitudes, such as brightness, pitch of sound, physical size, and weight (Cantlon, Platt, & Brannon, 2009; but see also Leibovich, Ashkenazi, Rubinsten, & Henik, 2013; Leibovich, Diesendruck, Rubinsten, & Henik, 2013). In all these cases, the ability to detect change (or to understand that two presented magnitudes are different from each other) depends on the ratio between the to-becompared magnitudes. For example, it is faster and easier to decide that 10 dots are more numerous than 3 dots (ratio of 0.3), than deciding that 10 dots are more numerous than 8 dots (ratio of 0.8). The ratio effect is thought to result from noisy representation of numerosities (Figure 1A). It is thought that numerosities are represented in a logarithmic analog format where numerosities that have a larger ratio share more representational overlap and are thus more easily confused than numerosities that have a smaller numerical ratio and thus share less representational overlap (Figure 1B). This ratio dependency is also known as “Weber’s law,” namely, that the difference in intensity needed to detect a difference between two stimuli (e.g., two different numerosities) is proportional to the objective intensities of the stimuli (Cantlon, Platt, et al., 2009). An individual differences measure derived from this law is the so-called Weber fraction—the minimal differences of the intensity between stimuli that can still be discriminated. This measure represents an individuals’ acuity of numerosity representation: Individuals with low Weber fraction scores are able to discriminate much closer numerosities than are individuals with high Weber fraction scores. Weber’s law is a characteristic of both human and nonhuman performance and explains the noisy representation of numerosities in the ANS (Cantlon & Brannon, 2006). The processing of features that were found to obey Weber’s law in classic psychophysical experiments (e.g., loudness, brightness, line-length, etc.) is considered very fast and automatic. Because numerosity processing was found to obey the same law, it has been suggested that numerosity processing is as basic, fast, and innate as processing of brightness, weight, pitch of sound, and so forth (Cantlon et al., 2009; Feigenson, Dehaene, & Spelke, 2004).

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Figure 1. Basic effect in numerical cognition. (A) The numerical ratio effect. The plot illustrates the relationship between numerical ratio and response time, in a number comparison task. The x-axis describes the numerical ratio: smaller/larger numerosity. The y-axis represents the time it takes to respond to the larger number. Task difficulty increases when the numerical ratio is closer to 1 (i.e., with an increase in numerical ratio). Inside the plot are examples for symbolic and nonsymbolic stimuli: The numerical ratio of 3 and 8 is ⬃0.37, and the numerical distance is 4; the numerical ratio of 7 and 9 is ⬃0.77, and the numerical distance is 2. It is also true, then, that task difficulty increases with the decrease in numerical distance. (B) Approximate representation of numerosities. Representation of numbers is thought to be represented on a logarithmic scale. This representation is assumed to be approximate and noisy: Larger numbers are represented more approximately, and the representation of adjacent numbers overlap. See the online article for the color version of this figure.

The ANS and the Symbol-Grounding Problem To this point, the discussion of number representations has focused on nonsymbolic representations of number. This raises questions regarding the relationship between nonsymbolic representation of numbers and their symbolic representation (e.g., Arabic numerals, number words, etc.): How do symbols acquire their meaning? Once that meaning is acquired, is the representation of symbolic and nonsymbolic numbers similar? When children first encounter symbolic representations of number, they are meaningless words or visual symbols. Therefore, children need to connect symbolic representations to their semantic meaning. How symbols, such as numerical symbols, become representations of semantic referents (such as numerosity) has been referred to as the “symbol-grounding problem” (Harnad, 1990, 2003). In the field of numerical cognition, the most widely accepted theoretical account to resolve this problem is the notion that symbols are mapped on the preexisting ANS (Dehaene, 2007; Piazza, 2010; Stoianov, 2014). In support of this account, it has been demonstrated that, just like nonsymbolic number discrimination, symbolic number discrimination is also ratio-dependent, namely, symbolic number comparison has been found to obey Weber’s law. This was first demonstrated by Moyer and Landauer (1967). In this study, the authors presented two single-digit symbolic numbers and asked adult participants to decide which number is numerically larger. Manipulating the numerical distance (or difference) between the two symbolic numbers, it was found that both response times (RT) and accuracy were affected by the numerical distance between the two numbers; responses were faster and more accurate for large numerical distances (e.g., 2 and 9) than for smaller distances (e.g., 7 and 8). The numerical distance effect is highly correlated with the ratio effect, because when the difference between two numbers is larger, the ratio between them is smaller (i.e., closer to 0). The

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numerical ratio, however, takes into account not only the difference between two numbers, but also their absolute size, so, in accordance with Weber’s law, it can explain more variability in performance than the distance effect. The numerical distance (and ratio) effect is a robust finding that has been replicated in numerous studies since it was first published (Ansari, Garcia, Lucas, Hamon, & Dhital, 2005; Dehaene, Dupoux, & Mehler, 1990; Leibovich, Ashkenazi, et al., 2013; Leibovich, Diesendruck, et al., 2013; Tzelgov, Meyer, & Henik, 1992). The similarity in distance and ratio effects across both symbolic and nonsymbolic number processing has been one of the central pieces of evidence behind the suggestion that humans come to learn symbolic numbers by mapping them onto their more primitive and possibility innate nonsymbolic representations of number. In other words, the similarity of the representational signatures (ratio and distance effects) has led to the suggestion that numerical symbols are grounded in an innate, approximate representation of number that exhibits both ontogenetic (developmental) and phylogenetic (evolutionary) continuity. The idea that symbolic and nonsymbolic representations of number are tightly connected to one another has also been substantiated by brain imaging research. Such research has shown that the intraparietal sulcus (IPS) is engaged during both symbolic and nonsymbolic number processing (Cohen Kadosh, Lammertyn, & Izard, 2008; Dehaene, Piazza, Pinel, & Cohen, 2003; Nieder, 2005). Furthermore, brain imaging studies have shown that the activity in the IPS is modulated by the numerical ratio of both symbolic and nonsymbolic numerosities, demonstrating ratiodependent processing of symbolic and nonsymbolic number at both behavioural and brain levels of analysis (Fias, Lammertyn, Reynvoet, Dupont, & Orban, 2003; Holloway, Price, & Ansari, 2010; Kaufmann et al., 2005).

Symbols Grounded in the ANS? While there is much consensus in the literature for the notion that numerical symbols, such as Arabic numerals and number words, acquire their meaning through being mapped onto the ANS, there is a surprising dearth of studies that have directly tested this account of the “symbol-grounding problem” in numerical cognition. There are several ways of examining whether numerical symbols are grounded in the ANS. One prediction one can make is that, if numerical symbols acquire their meaning by being mapped onto the ANS, there should be a correlation between individual differences in the ANS and individual differences in symbolic number processing, such as arithmetic. Indeed, in recent years there has been an explosion of studies examining such relationships by investigating correlations between, on the one hand, individual differences in children’s and adults’ nonsymbolic numerosity processing (most typically measured by means of comparison tasks) and, on the other, symbolic number processing abilities, which have been most commonly assessed by means of standardized tests of symbolic calculation abilities. Already at the outset of such investigations, the results of studies across different labouratories were conflicting. While some of the earliest correlational studies reported a positive association between individual differences in the ANS and symbolic math (Halberda, Mazzocco, & Feigenson, 2008), others were unable to uncover such relationships (Holloway & Ansari, 2009). A recent comprehensive review of the literature suggests that the initial contradictory evidence

(with some reporting an association between ANS and symbolic math and others unable to obtain such positive evidence) characterizes the extant literature on the correlations between the ANS and symbolic math (De Smedt, Noël, Gilmore, & Ansari, 2013). Indeed, the review by De Smedt et al. suggests that symbolic number processing (most typically measured by symbolic number comparison tasks) is more consistently associated with symbolic math (typically measured using standardized tests of mental arithmetic) than are measures of nonsymbolic numerical processing. Taken together, current evidence from studies probing the dominant account of the “Symbol-Grounding Problem” in numerical cognition via studies that investigate the correlation between the ANS and symbolic math reveal a picture that is far from straightforward. This evidence suggests that associations between the ANS and symbolic math are inconsistent across different investigations, thereby not supporting the notion that individual differences in the nonsymbolic ANS are strongly predictive of betweensubjects variability in symbolic math. Now, one might reasonably argue that correlational studies assessing the relationship between the ANS and symbolic math are a rather indirect, nonexperimental way to assess the prediction that the “symbol-grounding problem” in numerical cognition can be resolved by postulating that numerical symbols become semantic representations of numerosity by being mapped onto the ANS. In other words, correlations with individual differences cannot be used to measure the similarity in the processing mechanisms of two systems. This is a general problem because correlating individual processes cannot separate a correlation of state from a correlation of process (Cantlon, 2015; Garner & Morton, 1969). A decisively more experimental approach to testing the predictions of the symbol-grounding problem was taken by Lyons et al. (2012). More specifically, the authors tested whether numerical symbols are tightly associated with nonsymbolic representations (underpinned by the ANS) by examining the efficiency with which adults can compare symbolic with nonsymbolic representations of numerosity (henceforth compared with mixed comparisons). In their study (Lyons, Ansari, & Beilock, 2012), adults were asked to make comparisons both within and across symbolic and nonsymbolic representations of numerosity. In other words, participants were asked to compare (a) which of two Arabic numerals was larger; (b) which of two dot arrays was larger and, in the critical, mixed condition; (c) which of a simultaneously or sequentially presented Arabic numeral and dot array was numerically larger. In view of the dominant account of the above discussed resolution to the “symbol-grounding problem,” the authors argued that if symbolic and nonsymbolic representations of numerosity are strongly connected to one another, then mixed (symbolic–nonsymbolic) comparison should be at least as efficient as within format (symbolic or nonsymbolic) comparison. Contrary to this prediction, the findings revealed that the performance in the mixed condition (comparison of symbolic with nonsymbolic) numerosities was by far the most difficult condition. In addition, the authors demonstrated that this mixing cost was specific to the comparison of symbolic and nonsymbolic representations of numerical magnitude, because the comparison of number words to Arabic numerals did not show a similar cost in performance relative to within format comparisons (symbolic and nonsymbolic). Furthermore, in a recent study, Lyons, Nuerk, and Ansari (2015) demonstrated that only a minority (30%) of kindergarten through

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sixth-grade children showed a significant ratio effect in a symbolic comparison task, whereas the majority (75%) of children that age showed a significant ratio effect in a nonsymbolic comparison task. This result is difficult to explain if one assumes that symbols are grounded in the ANS. Indeed, these findings suggest that symbolic and nonsymbolic representations of number may be qualitatively different from one another. These findings, therefore, cast significant doubt on the notion that symbolic and nonsymbolic representations of numerosity are strongly associated with one another and undermines the widely suggested resolution to the symbol-grounding problem. However, it is important to point out that the data discussed above were obtained from adults and older children, hence it is possible that such an association may exist earlier in developmental time and that, as the authors of the study argue, symbols become “estranged” from nonsymbolic, ANS-supported representations of numerosity over the course of developmental time. Indeed, developmental studies do exist that have addressed exactly this question. More specifically, there exists a growing body of studies that have investigated whether children’s development of counting abilities, and specifically children’s understanding that counting serves to determine the numerosity of sets (the “Cardinality Principle”), is scaffolded by the ANS. If children’s developing understanding of the meaning of symbolic number is grounded in the ANS, then there should exist a strong connection between the ANS and children’s understanding of the meaning of counting, because arguably the acquisition of the cardinality principle represents the starting point of children’s semantic representation of symbolic number. To date, as the case for the previously discussed investigations of a correlation between the ANS and symbolic math, the data on the association between the ANS and children’s understanding of the meaning of counting are contradictory, with some suggesting that children acquire an understanding of the meaning of counting without the involvement of the ANS (Le Corre & Carey, 2007) and others suggesting a correlation between children’s understanding of the cardinality principle and the ANS (Mussolin, Nys, Leybaert, & Content, 2012; Wagner & Johnson, 2011). These contradicting results can also be explained by lack of power in some of the studies. Recent data, however, suggest that children only map nonsymbolic to symbolic numbers successfully after they have acquired the cardinality principle (Odic, Le Corre, & Halberda, 2015), while at the same time they can map symbolic numbers to nonsymbolic numerosities more successfully. This finding suggests that children’s ANS-symbolic number mappings are not fully bidirectional after acquiring the cardinality principle. This finding also suggests, at the very least, that the ANS cannot be considered the sole contributor to children’s development of an understanding of the meaning of counting, which is, arguably, their first step toward an understanding of how symbols (in this case number words) represent numerosity. Thus, contrary to the notion of “symbolic estrangement” (Lyons et al., 2012), these developmental studies suggest the early acquisition of a semantic meaning of numerical symbols (number words) is not strongly supported by the ANS. Therefore, even when looking at the very outset of symbolic number development (children’s understanding of the meaning of count words), there does not appear to be strong support for the notion that symbols are grounded in the ANS.

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In addition to data investigating the mapping between the ANS and children’s early understanding of the meaning of counting, there is also a growing body of evidence that speaks against a strong mapping between the ANS and symbolic number in older children. To investigate whether symbolic and nonsymbolic representations of numerosity are related to one another during the transition from kindergarten to first grade, Sasanguie, Defever, Maertens, and Reynvoet (2014) tested children’s nonsymbolic numerosity comparison abilities in kindergarten and their nonsymbolic and symbolic numerosity comparison abilities 6 months later. While children’s performance on the nonsymbolic comparison task was found to be correlated across times, the authors failed to find a correlation between nonsymbolic numerosity processing in kindergarten and symbolic number comparison abilities 6 months later. Moreover, the authors were unable to find a significant correlation between symbolic and nonsymbolic number processing during the second testing time point. These findings suggest that nonsymbolic number processing abilities, thought to be underpinned by the ANS, do not predict future symbolic number processing competencies. Furthermore, contrary to the prediction that the ANS predicts growth in individual differences in symbolic number processing (because symbolic numbers are grounded in the ANS), a recent longitudinal study (Mussolin, Nys, Content, & Leybaert, 2014) with 3- to 4-year-old children has revealed that while early symbolic number processing abilities predict later nonsymbolic numerosity processing, the reverse (early nonsymbolic predicts later symbolic) is not true. Therefore, contrary to the suggestion that symbols acquire their meaning by being grounded in the ANS, these data, consistent with those reported by Odic et al. (2015) discussed earlier, show that symbolic scaffolds nonsymbolic, while the reverse is not true. While these results speak against the notion that individual differences in the nonsymbolic ANS scaffold symbolic number processing, there does exist some evidence from training studies to suggest that training nonsymbolic numerosity processing has an impact on symbolic number processing in both adults and children (Hyde, Khanum, & Spelke, 2014; Park & Brannon, 2013). However, the precise mechanisms underlying such training-induced transfer effects from nonsymbolic to symbolic number processing remain unclear; data suggest that it is the training-related manipulation of nonsymbolic numerosities rather than training-induced enhancements of ANS representations that leads to improvements in symbolic number processing following nonsymbolic ANS training (Park & Brannon, 2014). Another line of studies relevant to the symbol-grounding problem is semantic priming. In semantic priming studies, the target is preceded by a prime stimulus that is presented briefly to the participant. The effect of the prime on the target stimulus is then measured. For example, Koechlin, Naccache, Block, and Dehaene (1999) presented a number prime in different notations (verbal, digit, or nonsymbolic group of dots) and tested the effect on the target number. Participates had to indicate whether the target number was smaller or larger than 5. The results of this study revealed that performance improved only when both the prime and the target stimulus were of the same notation. When the prime and the target had different notations, response was not faster even when both stimuli were smaller or larger than 5. In contrast to these results, in a different experimental condition, when participants had to respond to both the prime and the target, performance

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always benefited from the prime and the target leading to the same response, even across notations. Accordingly, the authors suggested that the internal representation of numerosities dissociate into multiple notation-specific subsystems. These results are not in line with the notion of a shared representation of symbolic and nonsymbolic numerosities. It is interesting that in a more recent priming study, Gabay, Leibovich, Henik, and Gronau (2013) used pictures of small (cat, dog) and large (elephant, horse) animals as prime and asked participants to judge whether a target digit represented an odd or an even number. The authors found that prime influenced performance: performance was slower when the prime was a large animal and the target number was smaller than 5, and performance was faster when both the prime and the target were either big (e.g., 8 and elephant) or small (e.g., 2 and dog). Accordingly, the authors suggested that conceptual size and numerical value share an underlying representation, and that symbolic representation of numbers has evolved from, or is rooted in, a more evolutionary ancient system for the representation of magnitudes. Beyond behavioural data, the symbol-grounding problem can also be assessed by investigating the similarities between symbolic and nonsymbolic representations of numerosity in the brain. To test whether symbolic and nonsymbolic representations of numerical quantity are correlated with activation in similar brain networks, several researchers have used a method referred to as conjunction analysis. This statistical tool enables researchers to examine which brain regions are activated by both symbolic and nonsymbolic representations of numerical quantity. The results from these studies have revealed that areas of the IPS are activated by both symbolic and nonsymbolic numerical quantity (Holloway & Ansari, 2010; Holloway, Price, & Ansari, 2010). Another method to test whether there is an association between nonsymbolic and symbolic representations of numerical quantity in the brain is to examine whether the presentation of one representational format also leads to the activation of the other (i.e., cross-format activation). Using a method commonly referred to as functional MRI (fMRI) adaptation, Piazza, Pinel, Le Bihan, and Dehaene (2007) tested whether adaptation to symbolic number quantities would also activate nonsymbolic representations and vice versa. More specifically, participants were first adapted (through the repeated presentation of the stimulus of interest) to either a symbolic (Arabic numerals) or nonsymbolic (dot arrays) numerical quantity. Following adaptation (repeated presentation), they were presented with numerical deviants in the other format (with the hypothesis that deviants will elicit a recovery of the adapted response in neural regions sensitive to the change in the stimulus between adaptation events and the deviant). Responses to these deviants were found in a bilateral network of frontal parietal regions. It is important that the degree of deviant-related response in the IPS was related to the numerical distance between the number to which the brain was adapted and the deviant. In other words, the amount of deviant-related activation increased as the numerical distance between the adaptation and deviant numbers increased. Critically, this distance effect occurred both when the change from notation was from symbolic to nonsymbolic and vice versa. This cross-format numerical distance effect suggests that symbolic and nonsymbolic representations of numerical quantity tap into a common approximate, neural code of quantity representation. As the authors put it, the findings: “. . . support the idea that symbols acquire meaning by linking neural populations coding

symbol shapes to those holding nonsymbolic representations of quantities” (Piazza et al., 2007, p. 293). While the findings reported by Piazza et al. (2007) support the notion of a format-independent representation of numerical quantity in the parietal cortex, more recent findings have been unable to reveal similar evidence. Also using fMRI adaptation to study cross-format processing in the brain, Cohen-Kadosh et al. (2011) found that a change of format in the absence of a change in numerical quantity (e.g., a change from 12 dots to the symbolic representation “12”) led to activation in the same IPS regions that responded to a change in quantity. Furthermore, format changes led to greater activation in the IPS than did quantity changes, which is at odds with the existence of a common, amodal quantity code. The use of both conjunction analysis and fMRI adaptation to examine commonalities in the neuronal correlates of symbolic and nonsymbolic numerical quantity processing suffers from a number of limitations regarding the inferences that can be drawn about the presence or lack of a strong link between symbolic and nonsymbolic quantity processing in the brain. Because fMRI has a relatively coarse spatial resolution and involves averaging the activation across points of observation (called voxels; three-dimensional pixels), it is entirely possible that the common activation of a brain region for symbolic and nonsymbolic numerical quantity processing does not imply equivalence of processes engaged by each condition. In other words, it is possible that the activation of a brain region by two experimental conditions is driven by two different underlying mechanisms. The use of multivariate analytic techniques can help to overcome some of the aforementioned limitations. In particular, rather than averaging activation across voxels, these new tools analyse the pattern of activation of multiple voxels and compare and correlate the pattern of activation between conditions. This method is often referred to as Multi-Voxel Pattern Analysis (MVPA). Using this methodology, Bulthé, De Smedt, and Op de Beeck (2014) revealed that while it is possible to distinguish the patterns of activation elicited by different quantities within formats (e.g., distinguishing 4 dots from 6 dots or the numeral 5 from the numeral 9), there was no evidence for cross-format classification. This evidence points to a lack of overlap in the distributed (across voxels) representations of symbolic and nonsymbolic numerical quantities. Similar evidence was obtained in a study by Damarla and Just (2013), who also showed good within-format classification of numerical quantities, but poor cross-format classification (i.e., the multivoxel patterns of different numbers could be distinguished within but not across formats of numerical quantity representation). Another study by Eger et al. (2009), using MVPA to classify the patterns of symbolic and nonsymbolic numerical quantities, revealed modest cross-format classification (just over 50% correct) but only in one direction: The multivoxel patterns for dot arrays could be used to predict the activation of corresponding Arabic numerals, but the patterns of activation elicited during the processing of the numerals could not be used to predict the patterns correlated with processing dot arrays. Moreover, through examining the correlation between multivoxel pattern activity (an approach referred to as Representational Similarity Analysis), Lyons, Ansari, and Beilock (2014) found no correlation between the patterns of individual symbolic and nonsymbolic numbers.

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Taken together, both behavioural and neuroimaging data do not provide strong evidence in support of the notion that symbolic representations of number are grounded in the nonsymbolic ANS. In view of this, it is clear that the symbol-grounding problem in numerical cognition is far from resolved. Clearly, in order for numerical symbols to become semantic representations of numerosity, they need to be grounded in some nonsymbolic representation of number. Some have suggested that instead of being grounded in the ANS, number symbols become linked to small quantities (⬃1– 4) that can be enumerated exactly and without counting (a process referred to as subitizing; Trick & Pylyshyn, 1994). In other words, early in development, the number words 1– 4 become associated with nonsymbolic numbers in the subitizing range, but larger numbers are not linked to nonsymbolic representations of numerical quantity (Carey, 2001, 2004). According to this hypothesis, children recognize the link between number words 1–3/4 and sets of objects in this numerosity range. From this association, they can bootstrap critical principles of the symbolic number sequences such as ordinality (that each number is part of a sequence) and the successor function (that the next symbol in the sequence is exactly 1 larger). Another, not mutually exclusive, explanation for the lack of a strong association between numerical symbols and the ANS is that the way in which the ANS is typically measured in tasks using nonsymbolic stimuli may tap into processes other than the representation of approximate numerosities. It is this possibility that is discussed in the next section of this review article.

How Numerical is the ANS? When studying the mapping of symbolic numbers onto the ANS, a central assumption is that the ANS is numerical; namely, that what we extract from a visually presented set of items is their numerosity and that this nonsymbolic numerosity taps into a fundamental numerical representation. An example for this view is demonstrated in a model of Dehaene and Changeux (1993). This computational model describes a three-step process for exact representation of numerosities. First, an input records the location of each item; then, the locations of the items are being mapped topographically. This mapping is not influenced by the continuous variables of the items or the array (such as the density of the dots, their total area, etc.). In the third step, a “numerosity detector” sums all the locations, producing a number. Verguts and Fias (2004) provided computational modelling data to suggest that the same network modules that subserve the nonsymbolic numerosities are also used in the developmental construction of symbolic representations of numerosities. Theoretical considerations and recent empirical evidence, however, reveal a more complicated picture. The stimuli that are used to study nonsymbolic numerosity processing are sets of items (e.g., dot arrays). Such sets are characterised, in addition to their numerosity, by different nonnumerical, continuous variables: the average size of each item, the total surface area of all the items, their density, the total occupied space of the items and the area between them (i.e., convex-hull), and so forth. In the real world, numerosity and continuous variables are often correlated or anticorrelated. Usually, for example, more items will occupy more area and will be denser than fewer items. In an experimental design, it is possible to present the same numerosity while changing the continuous variables of a set (Figure 2A). It is

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Figure 2. Nonnumerical variables change with numerosity. (A) Each array contains six dots. The continuous variables of these arrays, however, are different from one another. (B) Different dot arrays will always have different continuous variables. The lefthand side of each rectangle contains five dots, and the right-hand side contains 10 dots. As can be seen in the different versions, manipulating one variable results in differences in other variables as well, making it impossible to control for all the continuous variables at the same time. See the online article for the color version of this figure.

impossible, however, to change the numerosity without changing the continuous variables of the set (Figure 2B). In other words, changing the numerosity of a set will always lead to change in the continuous variables of this set. In view of this, it is impossible to be sure that participants do not process the continuous variables of a set when performing estimation or comparison tasks of nonsymbolic numerosities. It is unclear, then, to what extent the processing of nonsymbolic numerosities is purely numerical. This questions the degree to which the processing of any nonsymbolic array of items is purely numerical, as is stipulated by the ANS theory. With this theoretical constraint in mind, different studies have manipulated continuous variables in an effort to ensure that such variables do not drive the response of participants in nonsymbolic number processing studies. Such approaches have used various strategies, such as (a) manipulating one continuous property at a time (Mussolin, Mejias, & Noël, 2010), (b) assigning a random dot size to each array (Piazza et al., 2010), (c) using a single array containing two different colours of dots where participants must indicate the colour of the more numerous dots (Mazzocco, Feigenson, & Halberda, 2011), and so forth. The assumption of such studies is that if a continuous variable is not a reliable cue of numerosity, it will be ignored, will not be processed, and thus will not affect performance. For example, equating the average size of two dot arrays creates a correlation between total surface area and numerosity: The total area of all the dots will be greater in the array containing more dots. Equating the total surface area of the dots will result in a negative correlation between numerosity and average size: The average size of a dot in the array containing more dots will be smaller than the array containing fewer dots. In such a design, some continuous variables are correlated with numerosity and can be used to identify the array containing more dots. If, however, in half the trials the average size of the dots will be equated (across different numerosities) and on the other half the total surface area will be equated, then none of these continuous variables will be a reliable cue of numerosity and will not affect performance (for a review see Leibovich & Henik, 2013). This assumption, however,

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is not supported by recent studies, demonstrating that even when continuous variables are task-irrelevant and not correlated with numerosities, they still affect performance in both estimation and comparison tasks (Leibovich & Henik, 2014). For example, in comparison tasks, both adults and children in various developmental stages were found to be susceptible to the influence of continuous variables. In one such study, Gebuis and Reynvoet (2012) asked adults to compare the numerosity of two successively presented dot arrays. Continuous variables in this set of stimuli were not correlated with numerosity; namely, different continuous variables (total surface area, density, average dot-size, total circumference, etc.) were manipulated all at once (in the same dot arrays and not in separate dot arrays as done before), so none of these variables could consistently predict numerosity. It is important that the authors manipulated the number of continuous variables that were congruent with numerosity. In one condition, only density was congruent with numerosity; namely, the more numerous dots were denser than the less numerous dots. In another condition, both density and total surface area were congruent with numerosity, and so forth. Critically, consistent with the notion that continuous variables play a role in nonsymbolic numerosity discrimination, it was found that accuracy increased with the number of continuous variables that were congruent with numerosity. Hence, continuous variables affected performance even when they were task-irrelevant and could not reliably predict numerosity. These results are important because in many studies, the congruity between numerosities and continuous variables was manipulated as a way of attempting to prevent participants from using continuous variables as cues. Because in such studies congruent and incongruent trials are not analysed separately (e.g., Halberda, Mazzocco, & Feigenson, 2008; Odic, Libertus, Feigenson, & Halberda, 2013), it is impossible to evaluate the potential influence continuous variables had on performance. In addition to the correlation between numerosity and continuous magnitudes, these magnitudes have been found to be processed in similar time-windows in event-related potential (ERP) studies; in such studies, brain activity is being recorded while participants are engaged in an active or passive task. This technique has better temporal resolution than that of fMRI and is being used to answer questions regarding the time in which different cognitive processes occur. Some ERP studies have revealed that continuous magnitudes are processed automatically in a passive viewing task, in the same time window that was previously attributed to processing numerosities; that is, N2, which occurs around 200 ms following the presentation of the stimulus of interest (Gebuis & Reynvoet, 2013). A more recent ERP study also found that processing of numerosities occurs later than processing of basic visual cues, such as shape, and suggested that numerosity is a “higher-level property assembled from naturally correlating perceptual cues, and hence, it is identified later in the cognitive processing stream” (Soltész & Szu˝cs, 2014, p. 203). In contrast, Park, DeWind, Woldorff, and Brannon (2015) reported that numerosities are being processed extremely early (starting at 75 ms from display onset) in the visual stream. In this study too, however, continuous magnitudes (and specifically convex hull) could have contributed to these early activations. There is, however, the possibility that what is evident in this time point (N2) is the integration of numerosity and continuous magnitudes. This option should be empirically tested. The problem with the correlation between numerosity and continuous variables is also critical in neuroimaging studies aiming to reveal

the neural underpinnings of the ANS. To reveal the brain regions that subserve the ANS, many studies have employed nonsymbolic comparison tasks. In such a task, one theoretically needs to use pairs of dot arrays that are different only in their numerosities. Because this is impossible, neuroimaging studies tried to either manipulate continuous variables in the way described previously (so they are not correlated with numerosity), or to create a control task in which only continuous variables are compared; for example, comparing the area of two squares. That way, contrasting the numerosity comparison task and the area comparison task will yield brain areas that are dedicated to numerosity comparison (Chassy & Grodd, 2012). This idea is problematic for several reasons. First, the two sets of stimuli for the different tasks are visually different from one another; stimuli used in numerosity comparison tasks (e.g., two sets of dots) are composed of more items than stimuli used in an area comparison task (e.g., one small and one large square). Second, while the area comparison task can be solved only by relying on continuous variables, participants can use both numerosity and continuous variables in the numerosity comparison task. Third, one thing that is required only in the numerosity task is summation or integration of all the visual properties of the items in the array. When comparing the numerosity of two sets of items, one first needs to extract the numerosity of items in each array and then compare these numerosities. Such a process requires one to first sum the dots and then compare them. Such summation is not required when comparing the size of two squares (i.e., area comparison task). Therefore, when contrasting a numerosity comparison task with, for example, an area comparison task, the brain areas revealed by such a contrast might reflect the stage of summing-up the dots, rather than a direct access to the representation of numerosity (see also Roggeman, Santens, Fias, & Verguts, 2011). Moreover, there exists direct evidence demonstrating that continuous variables affect ANS acuity—a measure that is supposed to rely solely on numerosity. Specifically, Tokita and Ishiguchi (2013) measured precision (Weber fraction) and accuracy (the point of subjective equality; when two arrays become indistinguishable to the participants) of adults and 5- to 6-year-old children in a numerosity comparison task. The total area and convex hull of the standard and the comparison arrays were identical in one condition and different in the other. It was found that both adults and children overestimated the number of items when their total surface area and convex hull was larger. In other words, Weber fraction was directly affected by different manipulations of continuous variables. Accordingly, the authors suggest that numerosity judgments were not solely based on the numerical information available, but that continuous variables were part of the decision making process. These findings, therefore, show that measures of nonsymbolic numerical acuity are not pure measures of numerosity processing but critically depend on the correlation between numerical and continuous variables (Leibovich, Henik & Salti, in press; Leibovich, Vogel, Henik & Ansari, in press). Not only the way in which continuous variables are manipulated, but other factors, such as stimuli presentation time (Inglis & Gilmore, 2013) task context, the range of the tested numerosity, task difficulty (i.e., the numerical ratio of the to-be-compared numerosities), and even culture, were shown to modulate how much participants might rely on nonnumerical variables (Leibovich, Henik, & Salti, in press; Leibovich, Vogel, Henik, & Ansari, in press). Related to this, a recent study by Cantrell, Kuwabara, and Smith (2015) compared performances of Japanese and U.S preschoolers in a nonsymbolic comparison task. Children first saw a target stimuli

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION

(array of items) and then were presented with two different arrays and were asked to choose the array that was most similar to the target. The array matched the target in either the total area or the numerosity of the dots. Children were found to choose the array similar in numerosity to the target array when the number of items was small and when there was a noticeable difference between the numerosity of the target and the test array (i.e., in small numerical ratios). Moreover, children chose the array that was similar in area when the number of dots was large and the difference between the dots was very small (i.e., in higher numerical ratios). Japanese children relied on total area at relatively smaller set sizes than did U.S. children. Accordingly, it has been suggested that numerosity range, numerical ratio, and cultural differences influenced the level of attention (or the weight) given to numerosity as opposed to continuous variables. These findings seriously question the notion that there exists a nonsymbolic representation of number that is numberspecific. Instead, these findings point to a system of magnitude representation in which continuous and numerical variables interact with one another during numerosity processing. In the context of the symbol-grounding problem in numerical cognition, such findings raise the distinct possibility that previous attempts to unveil associations between the ANS and symbolic number did not measure pure nonsymbolic numerosity processing; instead, the process of mapping symbolic numbers to both numerical and continuous variables is inherent in nonsymbolic numerosity stimuli. A recent study by Merkley and Scerif (2015) directly tested this notion by means of a training study. In particular, adult participants were trained to associate abstract symbols with nonsymbolic numerosities (i.e., dot arrays). The symbols were associated to dot arrays in which the numerosity and continuous variables were either congruent or incongruent with one another. During the test phase, participants performed a comparison task of the abstract symbols. Consistent with the evidence discussed previously, the authors found that comparison was modulated by the ratio between the numerosities associated with the symbols. Critically, performance was also affected by congruity, because comparisons were also slower and less accurate for symbols that had been associated (during training) with numerosities that were incongruent in terms of their continuous variables. This result supports the claim that not only numerosity, but also continuous variables may be associated with the symbolic representation of number. The congruity effect found in the symbol comparison task suggests that continuous variables continue to affect performance even after the symbols are learned and associated with numerosity. Overall, the results of this training study emphasize the important role continuous variables play in the process of associating symbolic and nonsymbolic representations of number.

Future Directions While the most dominant theories in the field of numerical cognition postulate that the ANS is a system that is number specific (e.g., Burr & Ross, 2008; Dehaene & Changeux, 1993; Dehaene, 1997), recent evidence suggests that this view should be reconsidered. As discussed previously, it has been shown that continuous variables play a pivotal role in nonsymbolic numerosity processing (Cantrell & Smith, 2013; Leibovich & Henik, 2013; Mix, Huttenlocher, & Levine, 2002). This theoretical shift profoundly affects how the symbol-grounding problem can be conceptualised. In other words, how can the correlation between exact, symbolic number knowledge

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and nonsymbolic numerosity processing be studied when numerosity processing cannot be studied in isolation of continuous variables? Next, we raise outstanding questions and suggest future research directions that will hopefully move the field closer to a resolution of the symbol-grounding problem in numerical cognition.

Inhibitory Control Plays a Critical Role in Nonsymbolic Numerosity Processing To prevent participants from relying on continuous variables during numerosity comparisons, researchers have varied the degree to which numerosity and continuous variables are congruent with one another. For example, it might be the case that in half the trials, the more numerous dots are denser (i.e., congruent). Conversely, in the other half, the less numerous dots are denser than the more numerous dots (i.e., incongruent). Such a design will mean that density will not be a reliable cue of numerosity, and thus should not influence performance. There is, however, evidence suggesting that this is not the case; under these conditions it has been found that participants inhibit the processing of the incongruent continuous variables, resulting in a congruity effect—faster and more accurate performance when numerosity and continuous variables are congruent than when incongruent (Clayton & Gilmore, 2014; Gebuis & Reynvoet, 2012; Hurewitz, Gelman, & Schnitzer, 2006; Leibovich, Henik, & Salti, in press; Leibovich et al., in press; Nys & Content, 2012). These congruity effects suggest that nonsymbolic numerosity comparison tasks require inhibitory control abilities. This problem leads to an impasse: When numerical and continuous variables correlate, there is no conflict and need for inhibition, but then one cannot exclude the possibility that processes other than pure representation of approximate numerical quantity are being recruited. In other words, it is impossible to be sure that what was measured during nonsymbolic numerosity comparison solely reflects ANS. Against this background, it becomes apparent how ANS acuity and inhibition abilities can be confounded with one another. To illustrate this point, consider a recent study by Szu˝cs, Nobes, Devine, Gabriel, and Gebuis (2013). These authors asked 7-year-old children and adults to engage in a typical nonsymbolic numerosity comparison task. Weber fractions were calculated separately for congruent and incongruent trials. This measure, commonly thought to reflect ANS acuity, was found to be modulated by congruity. Specifically, acuity was worse in incongruent compared with congruent trials. In addition, the difference between Weber fractions in congruent and incongruent trials was greater in children than in adults. Accordingly, the authors suggested that Weber fraction is not a pure measure of ANS acuity but may also reflect individual differences in inhibitory control abilities. In a similar vein, Gilmore et al. (2013) found that that only incongruent trials during nonsymbolic numerosity comparisons were correlated with math abilities. Furthermore, Bugden and Ansari (2015) found that children with DD differed from their typically developing peers on a nonsymbolic number discrimination task only when the area and numerosity of the stimuli were incongruent but not when they were congruent. In the context of such an impasse, that is, the inability to evaluate pure numerosity representation without confounding it with either reliance on continuous variables or inhibition abilities, the existence of a correlation between ANS acuity and exact number knowledge should be reevaluated. For example, Mussolin et al. (2012) reported results supporting the association between exact number knowledge

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and the ANS; in this study, 3– 6 year-old children’s performance in a numerosity comparison task was positively correlated with exact number knowledge. In view of the strong association between numerical and continuous variables, however, Negen and Sarnecka (2015) repeated the experiment carried out by Mussolin et al. (2012) but systematically manipulated continuous variables, to ensure that these variables were no longer a reliable cue for numerosity. Contrary to Mussolin et al. (2012), Negen and Sarnecka (2015) found that there was no longer a correlation between ANS acuity and exact number knowledge when continuous variables were no longer a reliable cue to numerosity. Against the background of these data, the authors concluded that the association between ANS acuity and exact number abilities may be the result of an “artefact of the procedure used to assess ANS acuity in children” (Negen & Sarnecka, 2015, p. 92). Taken together, it is clear that nonsymbolic numerosity processing cannot be studied in isolation of continuous variables, because numerosity processing is confounded by continuous variables and cognitive control abilities.

A New Approach to the Symbol-Grounding Problem: The Relationship Between ANS Acuity, Cognitive Control, and Exact Number Knowledge The pattern emerging from the literature discussed previously suggests that the symbol-grounding problem in numerical cognition cannot be resolved by assuming that symbols are mapped onto the ANS. Specifically, if numerosity is only one of many visual variables processed when encountering nonsymbolic numerosity, how can nonsymbolic representations of numerosity be directly linked to symbolic numbers? We contend that cognitive control may be an important component in this mapping process (if indeed it does occur). In other words, cognitive control serves to disentangle numerical and continuous cues in nonsymbolic numerosity tasks and therefore may play a critical role in allowing for any connection between nonsymbolic and symbolic representations of number to occur (whether this be to the ANS or the representation of small set sizes). It allows for attention to numerosity while inhibiting the influence of anti-correlated, nonnumerical variables. Inhibition abilities (or more generally cognitive control) mature with development (e.g., Morton, 2010). Thus, in very young ages, the inability to inhibit irrelevant information results in the inability to properly compare numerosities when numerosities and continuous magnitudes are incongruent. This point was already demonstrated by Jean Piaget nearly 70 years ago in his famous experiments on children’s conservation abilities. In the conservation task, children are first presented with the two rows of objects in which the objects are equally spaced (i.e., same convex-hull) and asked whether the rows of objects contain the same number of items. Then, in a second stage of the experiment, the experimenter increased the spacing between items in one of the rows of items, thereby manipulating convex-hull, and then repeated the question. Only children who answered that both rows contained the same number of items in both stages of the task are said, according to Piaget, to be able to conserve number (Piaget, 1952), or in other words, to ignore, or inhibit, the changes in continuous variables while maintaining a representation of numerical equivalence. Recent work has demonstrated that not only children but also adults are sensitive to continuous variables in conservation tasks (Leroux et al., 2009). In light of the profound connection between numerosity, continuous variables, and cognitive control, we argue that

understanding the interplay between these factors will lead to a more complete picture of the development of the ability to represent numerosities, and connect them with exact symbolic representations. We reviewed evidence that Weber fraction, a measure of ANS acuity, is affected by cognitive control abilities that are required to resolve the conflict between numerical and continuous variables, as well as the ratio of the numerosities that are being compared. Hence, the current definition of ANS acuity can be said to confound numerosity processing and cognitive control. This confound should not be viewed simply as a side effect of experimental design that should be avoided and overcome. Instead, it is imperative to explore the nature of this confound further and to investigate the extent to which the resources required to disentangle numerosity from continuous variables in nonsymbolic number tasks is key to the way in which children learn about nonsymbolic number and how they form associations between symbolic and nonsymbolic representations of number. In other words, to understand the meaning of a number and to understand that the same numerosity is preserved even if continuous variables change, it is necessary to have the ability to inhibit nonnumerical magnitudes when they do not correlate with numerosity (i.e., cognitive control) and to use continuous magnitudes as a cue of numerosity when numerical and continuous variables do correlate. We suggest that similar set of abilities might also be required when a child first learns to connect approximate nonsymbolic numerosities with exact symbolic representations. To understand that 6 apples and 6 watermelons correspond to the same symbolic representation, it is necessary to inhibit other properties of these items that usually correlate with numerosity, such as their size.

Summary and Conclusions A fundamental issue in the study of numerical cognition concerns the way in which numerical symbols, such as number words and Arabic numerals, become semantic representations of numerical quantity. At present, a dominant theoretical framework postulates that humans share with other species an approximate system for the nonsymbolic representation of number (ANS) and that symbols acquire their meaning by being mapped onto/associated with this evolutionarily ancient and possibly innate system. We examined available evidence from both behavioural and neuroimaging studies to evaluate the extent to which this hypothesis can be supported. The overall conclusion we draw from this review of evidence is that the symbol-grounding problem in numerical cognition cannot be simply resolved by postulating that numerical symbols become representations of numerosity by being linked to the nonsymbolic ANS. We contend this because our review clearly demonstrates that investigations which have directly tested this hypothesis, have failed to find strong links between symbolic and nonsymbolic number processing in both children and adults. Furthermore, it is becoming increasingly clear that any examination of the nonsymbolic ANS involves the processing both numerical and continuous variables. Thus, whenever links between symbolic and nonsymbolic number processing have been investigated, it is likely that both the processing of the numerosity of nonsymbolic sets as well as continuous variables correlating with numerosity were measured. Put differently, given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION

significant cognitive control resources are required in order to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there does exist any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, the symbol-grounding problem in numerical cognition has not been resolved, and more questions than answers surround this fundamental theoretical issue in numerical cognition. A future research agenda that embraces the role of cognitive control and continuous variables in nonsymbolic number processing promises to shed further light on how (and indeed whether) number symbols are associated with approximate, nonsymbolic numerical magnitudes. Alternatively, it remains possible that a resolution to the symbol-grounding problem of numerical cognition may not require the involvement of the ANS.

Résumé Comment les symboles numériques, tels que des mots exprimant un nombre, acquièrent-ils leur sens? Cette question, ou « problème du fondement des symboles », est au cœur du domaine de la cognition numérique. Les théories actuelles suggèrent que les symboles acquièrent leur signification lorsqu’ils sont intégrés dans un système approximatif pour la représentation non symbolique du nombre (système du nombre approximatif; SNA). Dans la présente revue de littérature, on s’est demandé, premièrement, dans quelle mesure les données actuelles sur le comportement et les données de neuroimagerie appuient cette théorie, et deuxièmement, dans quelle mesure le SNA, dans le cadre duquel il est assumé que les nombres symboliques sont ancrés, est de nature numérique. Nous concluons que a) les preuves actuelles du rapport entre le SNA et les symboles numériques n’appuient pas la notion que ceux-ci sont ancrés dans le SNA et, 2) que vu la forte corrélation entre la numérosité et les variables continues dans la tâche de traitement de nombres non symboliques, il est a` peu près impossible de mesurer l’association nette entre la numérosité symbolique et non symbolique. Toutefois, il est clair que d’importantes ressources cognitives sont requises pour distinguer les variables numériques des variables continues durant le traitement de nombres non symboliques. Ainsi, s’il existe une mise en correspondance entre le SNA et le nombre symbolique, ce processus doit être géré par un contrôle cognitif. Nous suggérons que l’étude a` la fois du rôle du contrôle cognitif et des variables continues dans les tâches de comparaison de numérosité permettra de dresser un portrait plus complet du problème du fondement des symboles. Mots-clés : cognition des nombres, problème du fondement des symboles, système du nombre approximatif, grandeurs numériques, grandeurs non numériques.

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Received May 21, 2015 Accepted August 6, 2015 䡲