Leibovich et al. advocate for a single “sense of magnitude” to which a dedicated faculty for number could allegedly be reduced. This conclusion is unjustified as the authors adopt an unnecessarily narrow definition of “number sense,” neglect studies that demonstrate non-symbolic numerosity representation, and furthermore ignore abstract number representations in the brain.

Leibovich et al. suggest that congruency effects in number perception (biases towards smaller, denser, etc., dots) are evidence for the number's dependence on these dimensions. I argue that they fail to differentiate between effects at three distinct levels of number perception – encoding, representations, and decision making – and that differentiating between these allows the number to be independent from, but correlated with, non-numeric dimensions.

The evaluation of magnitudes serves as a foundation not only for numerical and mathematical cognition, but also for decision making. Recent theoretical developments and empirical studies have linked numerical magnitude evaluation to a wide variety of core phenomena in decision making and challenge the idea that preferences are driven by an innate, universal, and stable sense of number or value.

The authors rightly point to the theoretical importance of interactions of space and number through the life span, yet propose a theory with several weaknesses. In addition to proclaiming itself unfalsifiable, its stage-like format and emphasis on the role of selective attention are at odds with what is known about the development of spatial-numerical associations in infancy.

The target article dismisses all prior work purporting to demonstrate that number is a conceptual primitive. Here, we take issue with their misrepresentation of our recent line of work on numerosity perception, which demonstrates rapid and direct encoding of numerosity and undermines the thesis of the target article that “continuous magnitudes are more automatic and basic than numerosities” (sect. 1, para. 2).

We agree with Leibovich et al.'s argument that the number sense theory should be re-evaluated. However, we argue that highly efficient skills (i.e., fluent and highly accurate, “automatic,” performance) can be acquired and mastered at all ages. Hence, evidence for primacy or fluency in perceiving continuous magnitudes is insufficient for supporting strong conclusions about the innateness of this aptitude.

Leibovich et al. argue that it is impossible to control for all continuous magnitudes in a numerical task. We contend that continuous magnitudes (i.e., perimeter, area, density) can be simultaneously controlled. Furthermore, we argue that shedding light on the interplay between number and continuous magnitudes – rather than considering them independently – will provide a much more fruitful approach to understanding mathematical abilities.

Leibovich et al. proposed that the processing of numerosities is based primarily on a “sense of magnitude.” The consequences of this proposal for how numerical symbols acquire their meaning are, however, neglected. We argue that symbols cannot be learned by associating them with a system that is not yet able to derive discrete numbers accurately because of immature cognitive control.

Leibovich et al. overlook numerous human infant studies pointing to an early emerging number sense. These studies have carefully manipulated continuous magnitudes in the context of a numerical task revealing that infants can discriminate number when extent is controlled, that infants fail to track extent cues with precision, and that infants find changes in extent less salient than numerical changes.

Leibovich et al.'s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

We provide an emergentist perspective on the computational mechanism underlying numerosity perception, its development, and the role of inhibition, based on our deep neural network model. We argue that the influence of continuous visual properties does not challenge the notion of number sense, but reveals limit conditions for the computation that yields invariance in numerosity perception. Alternative accounts should be formalized in a computational model.

Leibovich et al. argue that that none of the experiments they review really establishes that human adults, infants, or nonhuman animals are sensitive to numerosity independent of a range of continuous quantities. We do not dispute their claim that the empirical record is inconclusive but argue that model-based data analysis does offer a way to make progress.

Leibovich et al. argue that because it is impossible to isolate numerosity in a stimulus set, attempts to show that number is processed independently of continuous magnitudes are necessarily in vain. I propose that through clever design and manipulation of confounding variables, we can gain deep insight into number representation and its relationship to the representation of other magnitudes.

In response to the commentaries, we have refined our suggested model and discussed ways in which the model could be further expanded. In this context, we have elaborated on the role of specific continuous magnitudes. We have also found it important to devote a section to evidence considered the “smoking gun” of the approximate number system theory, including cross-modal studies, animal studies, and so forth. Lastly, we suggested some ways in which the scientific community can promote more transparent and collaborative research by using an open science approach, sharing both raw data and stimuli. We thank the contributors for their enlightening comments and look forward to future developments in the field.

The presence of general intelligence poses a major evolutionary puzzle, which has led to increased interest in its presence in nonhuman animals. The aim of this review is to critically evaluate this question and to explore the implications for current theories about the evolution of cognition. We first review domain-general and domain-specific accounts of human cognition in order to situate attempts to identify general intelligence in nonhuman animals. Recent studies are consistent with the presence of general intelligence in mammals (rodents and primates). However, the interpretation of a psychometric g factor as general intelligence needs to be validated, in particular in primates, and we propose a range of such tests. We then evaluate the implications of general intelligence in nonhuman animals for current theories about its evolution and find support for the cultural intelligence approach, which stresses the critical importance of social inputs during the ontogenetic construction of survival-relevant skills. The presence of general intelligence in nonhumans implies that modular abilities can arise in two ways, primarily through automatic development with fixed content and secondarily through learning and automatization with more variable content. The currently best-supported model, for humans and nonhuman vertebrates alike, thus construes the mind as a mix of skills based on primary and secondary modules. The relative importance of these two components is expected to vary widely among species, and we formulate tests to quantify their strength.

Here, we specifically discuss why and to what extent we agree with Burkart et al. about the coexistence of general intelligence and modular cognitive adaptations, and why we believe that the distinction between primary and secondary modules they propose is indeed essential.

We welcome the cross-disciplinary approach taken by Burkart et al. to probe the evolution of intelligence. We note several concerns: the uses of g and G, rank-ordering species on cognitive ability, and the meaning of general intelligence. This subject demands insights from several fields, and we look forward to cross-disciplinary collaborations.

Burkart et al. suggest that social learning can explain the cognitive positive manifold for social animals, including humans. We caution that simpler explanations of positive trait intercorrelations exist, such as genetic load. To test the suggested explanation's specificity, we also need to examine non-social species and traits, such as health, that are distal to cognitive abilities.

Conceptualizing intelligence in its biological context, as the expression of manifold adaptations, compels a rethinking of measuring this characteristic in humans, relying also on animal studies of analogous skills. Mental manipulation, as an extension of object manipulation, provides a continuous, biologically based concept for studying G as it pertains to individual differences in humans and other species.

We argue that general intelligence, as presented in the target article, generates multiple distinct and non-equivalent characterisations. Clarifying this central concept is necessary for assessing Burkart et al.'s proposal that the cultural intelligence hypothesis is the best explanation for the evolution of general intelligence. We assess this claim by considering two characterisations of general intelligence presented in the article.