The item can only be dispensed within artificial tasks that, although useful in the lab, do not reflect the real world. There, the attended item is the goal of search. Hulleman & Olivers' (H&O's) model can ignore the item only by reducing search to the question of whether a patch of 0s (distractors) contains a 1 (target).

We challenge the central idea proposed in Hulleman & Olivers (H&O) by arguing that the “item” is still useful for understanding visual search and for developing new theoretical frameworks. The “item” is a flexible unit that represents not only an individual object, but also a bundle of objects that are grouped based on prior knowledge. Uncovering how the “item” is represented based on prior knowledge is essential for advancing theories of visual search.

We proposed to abandon the item as conceptual unit in visual search and adopt a fixation-based framework instead. We treat various themes raised by our commentators, including the nature of the Functional Visual Field and existing similar ideas, alongside the importance of items, covert attention, and top-down/contextual influences. We reflect on the current state of, and future directions for, visual search.

In this review, we are pitting two theories against each other: the more accepted theory, the number sense theory, suggesting that a sense of number is innate and non-symbolic numerosity is being processed independently of continuous magnitudes (e.g., size, area, and density); and the newly emerging theory suggesting that (1) both numerosities and continuous magnitudes are processed holistically when comparing numerosities and (2) a sense of number might not be innate. In the first part of this review, we discuss the number sense theory. Against this background, we demonstrate how the natural correlation between numerosities and continuous magnitudes makes it nearly impossible to study non-symbolic numerosity processing in isolation from continuous magnitudes, and therefore, the results of behavioral and imaging studies with infants, adults, and animals can be explained, at least in part, by relying on continuous magnitudes. In the second part, we explain the sense of magnitude theory and review studies that directly demonstrate that continuous magnitudes are more automatic and basic than numerosities. Finally, we present outstanding questions. Our conclusion is that there is not enough convincing evidence to support the number sense theory anymore. Therefore, we encourage researchers not to assume that number sense is simply innate, but to put this hypothesis to the test and consider whether such an assumption is even testable in the light of the correlation of numerosity and continuous magnitudes.

Leibovich et al. propose that number sense is not innate but gradually emergent during ontogeny following experience. We argue that this hypothesis cannot be reasonably tested in humans, in which the contribution of neural maturation and experience cannot be experimentally manipulated. Studies on animals, especially fish, can more effectively provide critical insights into the innate nature of numerical abilities.

Leibovich et al. argue that evidence for an innate sense of number in children and animals may instead reflect the processing of continuous magnitude properties. However, some comparative research highlights responding on the basis of numerosity when non-numerical confounds are controlled. Future comparative tests might evaluate how early experience with continuous magnitudes affects the development of a sense of number.

Numerosity is inherently confounded by related stimulus attributes such as density and area, and many studies have reported interactions of various strengths between area, density, and numerosity. However, direct measurements of sensitivity within the area-density-numerosity space show that numerosity emerges as the most spontaneous and sensitive dimension, strongly supporting the existence of a dedicated number sense.

Leibovich et al. argue that the evidence in favor of a perceptual mechanism devoted to the extraction of numerosity from visual collections is unsatisfactory and propose to replace it with an unspecific mechanism capturing approximate magnitudes from continuous dimensions. We argue that their representation of the evidence is incomplete and that their theoretical proposal is too vague to be useful.

Leibovich et al. opened up an important discussion on the nature and origins of numerosity perception. The authors rightly point out that non-numerical features of stimuli influence this ability. Despite these biases, there is evidence that from birth, humans perceive and represent numerosities, and not just non-numerical quantitative features such as item size, density, and convex hull.

Visual number comparison does not require participants to choose a unit, whereas units are fundamental to the definition of number. Studies using magnitude estimation rather than comparison show that number perception is compressed dramatically past about 20 units. Even estimates of 5–20 items are increasingly susceptible to effects of visual adaptation, suggesting a rather narrow range in which subitizing-like categorization processes blend into greater reliance on adaptable magnitude information.

Leibovich et al. propose that continuous magnitudes and a number sense are used holistically to judge numerosity. We point out that their proposal is incomplete and implausible: incomplete, as it does not explain how continuous magnitudes are calculated; implausible, as it cannot explain performance in estimation tasks. We propose that we do not possess a number sense. Instead, we assume that numerosity judgments are accomplished by weighing the different continuous magnitudes constituting numerosity.

Leibovich et al. propose that non-symbolic numerosity abilities develop from the processing of more basic, continuous magnitudes such as size, area, and density. Here I review similar arguments arising in the visual perception field and further propose that the evolvement of discrete representations from continuous stimulus properties may be a fundamental characteristic of cognitive development.

Leibovich et al. do not present enough empirical support to overturn decades of work supporting a number sense nor to convince the reader that a magnitude sense provides a better explanation of the literature. Here we highlight what we feel are the main points of weakness and the types of evidence that could be provided to be more convincing.

Leibovich et al. argue persuasively that researchers should not assume that approximate number system (ANS) tasks harness an innate sense of number. However, some studies have reported a causal link between ANS tasks and mathematics performance, implicating the ANS in the development of numerical skills. Here we report a p-curve analysis, which indicates that these experimental studies do not contain evidential value.

Leibovich et al. challenge the prevailing view that non-symbolic number sense (e.g., sensing number the same way one might sense color) is innate, that detection of numerosity is distinct from detection of continuous magnitude. In the present commentary, the authors' viewpoint is discussed in light of the integrative theory of numerical development along with implications for understanding mathematics disabilities.

Leibovich et al. hypothesize that the absence of visual object individuation limits infants' numerical skills and necessitates a reliance on continuous magnitudes. We argue that parallels between infants' numerical discrimination in the visual and auditory modalities, their abilities to match numerosities across modalities, and their greater ability to discriminate changes in number compared with continuous magnitudes contradict the authors' assumptions.

Leibovich et al. claim that number representations are non-existent early in life and that the associations between number and continuous magnitudes reside in stimulus confounds. We challenge both claims – positing, instead, that number is represented independently of continuous magnitudes already in infancy, but is nonetheless more deeply connected to other magnitudes through adulthood than acknowledged by the “sense of magnitude” theory.

Where do human numerical abilities come from? Leibovich et al. argue against nativist views of numerical development noting limitations in newborns' vision and limitations regarding newborns' ability to individuate objects. I argue that these considerations do not undermine competing nativist views and that Leibovich et al.'s model itself presupposes that infant learners have numerical representations.

The theory put forward by Leibovich et al. of how children acquire a sense of number does not specify the mechanisms through which cognitive control plays a role in this process. We argue that visual attention and number word knowledge influence each other over development and contribute to the development of the concept of number.

The conclusions reached by Leibovich et al. urge the field to regroup and consider new ways of conceptualizing quantitative development. We suggest three potential directions for new research that follow from the authors' extensive review, as well as building on the common ground we can take from decades of research in this area.