Possible or unclear is not enough

The main argument the authors lay out is that it is impossible to control for all non-numerical magnitude properties that co-vary with numerosity at once, and, as such, all previous work attempting to control for non-numerical magnitude properties to study numerical processing cannot be straightforwardly interpreted as having to do with “number.” Unfortunately, simply saying that all magnitude dimensions were not controlled in a given study does not provide any empirical support for the position that the observed responses were non-numerical. To be taken seriously, the authors need to say not only which particular magnitudes confounded number in each example, but also why the empirical evidence favors that particular non-numerical magnitude interpretation over a numerical one in each of those cases. More broadly, the authors need to explain how alternatively explaining the literature by various and different non-numerical confounds in each experimental context provides a more comprehensive and consistent explanation of the literature than the number sense.

The authors also seem to misunderstand the controls used in neural adaptation (e.g., Piazza et al. Reference Piazza, Izard, Pinel, Le Bihan and Dehaene2004), where the impossibility of controlling for all non-numerical magnitude properties in a single stimulus (or pair of stimuli) is avoided by requiring a more complex pattern of responses across many stimuli (see Kourtzi & Grill-Spector Reference Kourtzi, Grill-Spector, Rhodes and Clifford2005). In adaptation experiments, participants see a sequence of novel images that vary on many non-numerical magnitude dimensions. A majority are “adaptation” images with the same number of items. Occasionally, “deviant” test images are presented that deviate on numerical or non-numerical parameters (e.g., shape). Critically, adaptation and test images are drawn from different control distributions where number and non-numerical magnitude properties share different relationships to one another. Although researchers focus on equating different magnitude dimensions across adaptation and test image sets (e.g., item size and item spacing vs. total area and luminance), many magnitude dimensions are interrelated, and systematically changing relationships between some dimensions necessarily changes the others. For example, changing the relationship between item size and number also changes the relationship between number and item spacing, item contour length, total area, total contour length, brightness, and so forth. Even if a particular undefined non-numerical magnitude dimension was inadvertently confounded with number and led to attenuation in the brain response during the adaptation images, the relationship between that particular dimension and number would change in test images. Thus, one would not observe differential release from adaptation between numerical and non-numerical deviants (e.g., shape) because the relationship between the particular (undefined) non-numerical magnitude and number that was confounded during adaptation necessarily changes in all test types, as test images are drawn from a different set of controls. In contrast, if the brain response attenuates to repeated presentation of the same number of items and selectively recovers with a change in number despite these sorts of controls, it can be concluded that the response was numerical (see Kourtzi & Grill-Spector Reference Kourtzi, Grill-Spector, Rhodes and Clifford2005).

An influence on number does not equal non-numerical

The main positive empirical evidence drawn upon by the authors to support the hypothesis that previously observed responses are non-numerical comes from studies showing influences of non-numerical properties on numerical judgments or neural processing (e.g., Gebuis & Reynvoet Reference Gebuis and Reynvoet2014; Leibovich & Henik Reference Leibovich and Henik2014; Leibovich et al. Reference Leibovich, Henik and Salti2015). Although this point highlights the fact that the field currently lacks a model of how exactly number is extracted from visual arrays and provides evidence that non-numerical properties may be used to do so, evidence of an influence of non-numerical magnitudes on number is not evidence that the response is not numerical in nature. In a similar vein, the existence of a brain region that processes both numerical and non-numerical magnitudes (e.g., Walsh Reference Walsh2003) is not evidence that number is not represented distinctly at other levels or regions. To be taken seriously, the authors would need to provide evidence that responses thought to be numerical are not only influenced, but also better explained by non-numerical magnitudes.

No actual developmental evidence provided

The authors fail to provide any positive empirical evidence in support of their main developmental claim: Continuous magnitude representation is innate and number representation is not. Moreover, the authors disregard some of the strongest evidence to date for an innate number sense, cross-modal numerical processing in infants (e.g., Feigenson Reference Feigenson2011; Izard et al. Reference Izard, Sann, Spelke and Steri2009; Jordan & Brannon Reference Jordan and Brannon2006). Cross-modal number studies with infants are the smoking gun here, as they provide evidence of an innate number sense while avoiding the challenge of controlling for non-numerical magnitudes within a single sense modality. Leibovich et al. assert that cross-modal number studies with neonates (e.g., Izard et al. Reference Izard, Sann, Spelke and Steri2009) cannot be trusted because neonatal vision is too “blurry,” not allowing object individuation until about 5 months. Furthermore, they claim that results from cross-modal numerical matching studies with infants are highly inconsistent and, therefore, should not be trusted. The claim that infants cannot perceptually individuate objects until 5 months is simply false. Moreover, the authors inaccurately estimate study reliability (not considering the different predictions of preferential matching vs. violation of expectation paradigms). In both cases, the cited references (Cantrell & Smith Reference Cantrell and Smith2013; Carey Reference Carey2001) do not support the authors' claims. The authors are right that neonatal vision is much worse than child or adult vision. However, to convince the reader to disregard the cross-modal number studies with neonates, the authors would need to provide actual evidence that typical newborn visual acuity is below the threshold needed to see the stimuli used in these particular studies (e.g., estimates from Fantz et al. [Reference Fantz and Ordy1962] and Kellman & Arterberry [Reference Kellman, Arterberry, Siegler and Kuhn2007] suggest otherwise).

In the words of David Hume (Reference Hume and Millican1748/2007), “A wise man [sic], therefore, proportions his belief to the evidence” (p. 80). Although the magnitude sense proposal is intriguing and highlights important gaps of knowledge in the field, the authors would need to present stronger empirical evidence to actually overturn the number sense literature and garner widespread support for their own view.