The idea that discrete-numerical quantity and basic continuous magnitudes share some characteristics is uncontested. However, in their recent article, Leibovich et al. propose that a separate faculty for number does not exist in the first place. The article is, however, rather selective, as the authors reach this conclusion based on a survey that is idiosyncratic not only in what it attacks, but also in what it omits.

First, they adopt an unjustifiably narrow definition of “number sense.” The term “number sense” was first coined by Tobias Dantzig (Reference Dantzig1930). However, he is nowhere mentioned in the target article. Dantzig (Reference Dantzig1930, p. 1) writes, “Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense.” Although Dantzig only talks about a faculty to assess numerical quantity, the target article conveys a rather simplistic view of an “innate” and fully hardwired system that extracts numerical information just like a reflex. However, such a narrow definition does not hold for any cognitive capability and is not maintained by protagonists of the “number sense” (Anobile et al. Reference Anobile, Cicchini and Burr2016c; Burr & Ross Reference Burr and Ross2008; Viswanathan & Nieder Reference Viswanathan and Nieder2013). Physiological faculties are plastic (subject to maturation and/or learning processes); they are embedded in – and interact with – other faculties. The finding that the number faculty interacts with general magnitude representations can therefore not refute its existence.

Second, a key argument of the article is that varying the number of items in a set inevitably changes physical stimulus parameters. Although this is undisputed, it is far too premature to conclude that investigations of numerical representations are therefore a priori useless. The two main reasons are as follows:

1. 1. Potential sensitivity to simple sensory parameters is not specific to number investigations but pervasive to all investigations targeting abstract representations. Semantic groups can only be tested with specific stimulus representatives. Continuous magnitudes are, of course, no different in that respect. Resorting to continuous magnitude therefore does not solve the problem.

2. 2. In contrast to the impression caused by omissions in the target article, many researchers painstakingly selected their stimuli and went to great lengths to demonstrate number representations. Because it is not physically possible to equate all possible stimulus parameters at the same time, the best way is to control – unbeknown to the subject – one parameter after the other in separate stimulus configurations. If the subject responds equally to systematically varied numerosity stimuli, it is safe to conclude that the subject responds to number (Nieder Reference Nieder2016). One of the main research agendas over the last two decades therefore was to test numerosity representations over a broad range of stimuli and formats. For example, humans have recently been shown to be far more sensitive to numerosity than to continuous magnitudes in dot displays (Cicchini et al. Reference Cicchini, Anobile and Burr2016). Greater sensitivity to changes in numerosity was present both spontaneously and in tasks where participants were explicitly instructed to judge continuous parameters of the dot displays. Therefore, humans extract number information based on dedicated mechanisms. In addition, studies using controlled stimuli with conditioned animals demonstrated clear numerosity judgments. For example, the seminal monkey study by Brannon and Terrace (Reference Brannon and Terrace1998) controlled for item location, overall surface area, item size, and item type. Later, Nieder et al. (Reference Nieder, Freedman and Miller2002) controlled for item position, overall item area, overall item circumference, high and low density, item type, shape-like item configurations, and linear item arrangements, the latter one also abolishing convex hull. Monkeys also extracted the number of elements that appeared sequentially one-by-one and matched it to the number in spatial dot arrays (Nieder et al. Reference Nieder, Diester and Tudusciuc2006). In this sequential presentation format, temporal parameters such as duration, rhythm, and accumulated intensity have been controlled for and were neglected by the monkeys. Moreover, monkeys assessed number also independently of the sensory modality and discriminated both the number of sequential visual dots and auditory sounds within the same session (Jordan et al. Reference Jordan, Maclean and Brannon2008a; Nieder Reference Nieder2012). The animals did not care about non-numerical magnitude changes and responded to number information. Similar results have been obtained in preschool children (Barth et al. Reference Barth, La Mont, Lipton and Spelke2005). In sum, evidence for the capability of nonverbal subjects to represent numerical quantity is stronger than ever.

Third, another unfortunate omission of Leibovich et al. concerns abstract number representations in the brain. The single-neuron code underlying number representations has been addressed over the past years with a broad range of controlled stimuli. These studies in animals showed surprisingly abstract number representations (“number neurons”). As reviewed in Nieder (Reference Nieder2016), number neurons recorded in monkeys performing the aforementioned numerical tasks were tuned to preferred numerosities while being largely insensitive to changing sensory features. Number neuron responses in prefrontal cortex (PFC), and to some extent in the intraparietal sulcus (IPS), generalized across spatial features in visual item arrays (Nieder et al. Reference Nieder, Freedman and Miller2002), spatio-temporal visual presentation formats (Nieder et al. Reference Nieder, Diester and Tudusciuc2006), and also visuo-auditory presentation formats to signal numerosity supramodally (Nieder Reference Nieder2012). Moreover, in monkeys trained to associate shapes with numerosities, neurons signaled the numerical meaning of signs (Diester & Nieder Reference Diester and Nieder2007). Number neurons were present even if monkeys were not trained on number (Viswanathan & Nieder Reference Viswanathan and Nieder2013). After training, PFC showed improved responses to numerosity during active discrimination, whereas ventral intraparietal area (VIP) neurons remained stable (Viswanathan & Nieder Reference Viswanathan and Nieder2015). Of course, such highly generalized responses of number neurons cannot (and should not) be expected to be the only code for numerical quantity. Abstract number information can also be extracted from population activity (Ramirez-Cardenas et al. Reference Ramirez-Cardenas, Moskaleva and Nieder2016; Tudusciuc & Nieder 2007). Collectively, these single-neuron recordings strongly support the idea of a dedicated number faculty residing in a parieto-frontal network, with striking similarities between numerical representations in nonhuman and human primates (Nieder Reference Nieder2016).

Leibovich et al. also err when claiming that only one study (Castelli et al. Reference Castelli, Glaser and Butterworth2006) had directly compared brain areas during number and continuous magnitude comparison tasks. For example, Pinel et al. (Reference Pinel, Piazza, Le Bihan and Dehaene2004) found that number and size, but not luminance, activated overlapping parietal regions during functional imaging. More directly, single-cell recordings in monkeys that discriminated numerical, spatial, and sensory magnitudes in one session showed that coding was largely dissociated at the single-neuron level (Eiselt & Nieder Reference Eiselt and Nieder2016; Tudusciuc & Nieder Reference Tudusciuc and Nieder2009). Therefore, numerical representations are based on distributed coding by single neurons that are anatomically intermingled within the same cortical area.

Contrary to the claim of the target article, overwhelming evidence supports a dedicated number faculty that operates independent from continuous magnitude. The target article's attempt to reduce number judgments to simple magnitude representations is a lost case. Far from being put to rest, the number faculty is alive and kicking.