Although it is impossible to simultaneously control for all continuous quantities in a single numerosity display, some studies have developed ingenious designs controlling these variables across all of the experiment's displays, as for example in Xu and Spelke's (Reference Xu and Spelke2000) seminal study. Six-month-old infants saw first several arrays of a fixed numerosity (either 8 or 16, in different groups), varying in dot size and position. Once habituated, all infants were tested with two numerosities in alternation (8 and 16). Crucially, different aspects of stimuli were controlled in the habituation and test phases: the summed area of all dots (as well as brightness and contour length) and the array area were matched on average between the 8 and 16 habituation groups, while the density and dot size were matched between the two tested numerosities. Therefore, if infants attend to dot size or density, they will respond in the same way to test numerosities 8 and 16; whereas if infants attend to summed area or array area, the two groups will respond similarly to the test stimuli. Sensitivity to non-numerical parameters, either a single parameter or a combination of them, thus cannot explain the interaction pattern observed: In both groups, infants looked longer at the novel numerosity. This finding has been replicated by a different group (Brannon et al. Reference Brannon, Abbott and Lutz2004), using different numerical values (Xu Reference Xu2003), in the auditory modality (Lipton & Spelke Reference Lipton and Spelke2003), and the same parameter control strategy was employed to demonstrate sensitivity to numerosity at the brain level (Izard et al. Reference Izard, Dehaene-Lambertz and Dehaene2008, Piazza et al. Reference Piazza, Izard, Pinel, Le Bihan and Dehaene2004).

Similar controls for non-numerical features were used to demonstrate newborns' sensitivity to number (Izard et al. Reference Izard, Sann, Spelke and Steri2009). While hearing a fixed value of numerosity (e.g., 12), newborns looked longer to arrays matched in numerosity than to non-matching arrays (e.g., 4). Because the stimuli were presented across two different modalities (auditory and visual), the newborns' response was necessarily based on an abstract property of the stimuli. Following the logic of Xu and Spelke (Reference Xu and Spelke2000), extensive parameters were controlled in the auditory stimuli across the two groups by equating the duration, and intensive parameters across the two test numerosities in the visual modality by equating density and item size. Therefore, infants' preference for the matching stimuli could be explained only by numerosity, not by sensitivity to an abstract notion of amount, or rate. Moreover, as infants received only one numerosity in the auditory modality, they could not be responding to relative quantity (“more” or “less”). In that respect, the numerosity paradigm departed crucially from another paradigm used later (de Hevia et al. Reference de Hevia, Izard, Coubart, Spelke and Streri2014), in which newborns matched two values, one small and one large, across the two dimensions of numerosity and spatial extent. Newborns are able to relate increases versus decreases of quantities at a generic level, but also to perceive numerosities, calibrated across senses.

In line with these findings, studies investigating newborns' visual perception have demonstrated that they are able to represent individual objects, at the same age as in the numerosity study. In particular, human newborns can perceive complete shapes over partial occlusion (Valenza et al. Reference Valenza, Leo, Gava and Simion2006), and they can both distinguish individual elements of a stimulus or group them into a holistic percept (Antell and Caron Reference Antell and Caron1985, Farroni et al. Reference Farroni, Valenza, Simion and Umiltà2000, Turati et al. Reference Turati, Gava, Valenza and Ghirardi2013). Moreover, newborns respond differently to faces displaying direct versus averted gaze (Guellai & Streri Reference Guellai and Streri2011), a much finer cue than the shapes used in the numerosity experiment. Perceptual abilities to individuate items from the background and from one another likely fed into the numerosity percept evidenced by Izard et al.'s (2009) study.

Despite the common belief that numerosity perception must be more complex, and therefore a later developmental achievement, than the perception of continuous quantity, developmental studies have provided evidence that numerosity discrimination is easier and more automatic. In particular, infants show higher sensitivity to, and prefer to look at, changes in numerosity over changes in item or total surface area, when difference ratios are equated across dimensions (Brannon et al. Reference Brannon, Abbott and Lutz2004; Cordes & Brannon Reference Cordes and Brannon2008; Reference Cordes and Brannon2011), and even when variations in number are smaller (Libertus et al. Reference Libertus, Starr and Brannon2014). Similarly, children show higher sensitivity to number than to density (Anobile et al. Reference Anobile, Castaldi, Turi, Tinelli and Burr2016b). That perception of numerosity is more automatic than other continuous quantities is true in adults too: Even without an explicit task, numerosity of visual arrays is processed faster than other continuous features of those arrays (Park et al. Reference Park, DeWind, Woldorff and Brannon2016b). In this context, it is important to note that although Stroop studies on adults indicate that continuous quantities interfere with number perception, much of the behavioral and neuroscientific evidence cited by Leibovich et al. is based on interference paradigms in which non-numerical quantities varied by considerably larger ratios (and, thus, likely had higher perceptual discriminability and salience) than numerosity.

At the brain level, areas in the intraparietal sulcus respond to numerosity, and not simply to non-numerical cues. In particular, Eger et al. (Reference Eger, Michel, Thirion, Amadon, Dehaene and Kleinschmidt2009) used intraparietal sulcus activations to train a classifier to discriminate between patterns evoked by different numerosities across which item size was equated and found that this classifier generalized without accuracy loss to patterns evoked by numerosities across which total surface area was equated (and vice versa). Numerosity was also decodable from the intraparietal sulcus when low-level factors such as contrast energy were equated (Castaldi et al. Reference Castaldi, Aagten-Murphy, Tosetti, Burr and Morrone2016). Finally, in the right superior parietal lobe Harvey et al. (Reference Harvey, Klein, Petridou and Dumoulin2013) observed an orderly topographical structure of numerosity responses, correlated across stimulus sets implementing different controls. Although the same region also responds to object size (Harvey et al. Reference Harvey, Fracasso, Petridou and Dumoulin2015), the tuning curves and map organization differ, thus highlighting the specificity of the numerosity response.

In summary, the literature brings uncontroversial evidence that humans perceive and represent numerosity from birth on. As pointed out by Leibovich et al., the literature also brings uncontroversial evidence that numerosity perception is imperfect, often subject to the influence of non-numerical aspects of stimuli. These phenomena are fascinating, as they open up a new research agenda – if perception of numerosity relies on an imperfect algorithm, we now need to crack up its functioning.