Leibovich et al. criticize the results of different studies employing non-symbolic numerical tasks, because the effect of continuous magnitudes would have not been adequately controlled. By definition, a non-symbolic number is the numerosity extrapolated from an array of elements (Feigenson et al. Reference Feigenson, Dehaene and Spelke2004). We agree that it is impossible to equate simultaneously both the overall area and the perimeter of two different arrays of elements, and that considering only the overall area is only a partial control. Nevertheless, Leibovich et al. did not consider that when the overall perimeter of two arrays of two-dimensional squares is equated, a negative correlation with the area occurs.

From a theoretical viewpoint, given a first array depicting a number $n_i \ge 1$ of squares of side $l_{n_i}$ , and a second array depicting a number $n_j \gt n_i$ of squares of side $l_{n_j}$ , it is impossible to simultaneously keep constant both areas $\left[ {\left( {A_{n_i} = n_il_{n_i}^2 } \right) = \left( {A_{n_j} = n_jl_{n_j}^2 } \right)} \right]$ and perimeters $\left[ {\left( {P_{n_i} = n_i4l_{n_i}} \right) = \left( {P_{n_j} = n_j4l_{n_j}} \right)} \right]$ of the arrays. In fact, to do so, the following system has to be solved for $l_{n_j}$ :

$$\left\{{\matrix{ {l_{n_j} = \displaystyle{{n_{\rm i}l_{n_i}} \over {n_{\rm j}}}} & {\hbox{when}\, P_{n_{\rm i}} = P_{n_{\rm j}}} \cr {l_{n_j} = \displaystyle{{\sqrt {n_{\rm i}} l_{n_i}} \over {\sqrt {n_{\rm j}} }}} & {\hbox{when}\, A_{n_{\rm i}} = A_{n_{\rm j}}} \cr } } \right.$$

This system can be solved if and only if $n_i=n_j$ , thus violating the hypothesis $n_j\gt n_i$ .

We can now evaluate the relation between $P_{n_i}$ and $P_{n_j}$ when the overall area is kept constant (i.e., $A_{n_i}=A_{n_j}$ ).

$$\hskip 8.8pc P_{n_j}{\rm \; }=\displaystyle{{n_j} \over {\sqrt {n_j} }}4\sqrt {n_i} l_{n_i}\; \;$$

For any natural number $n_j \gt 0\comma \; $ it holds true that $n_j \gt \sqrt {n_j}$ . Moreover, if $n_i \lt n_j$ , then $\displaystyle{{n_i} \over {\sqrt {n_i} }} \lt \displaystyle{{n_j} \over {\sqrt {n_j} }}$ . Therefore, the size of $P_{n_j}$ increases as a direct function of the increase of $n_j$ whenever $A_{n_i}=A_{n_j}$ .

We can finally evaluate the relation between $A_{n_i}$ and $A_{n_j}$ _, whenever the overall perimeter is kept constant.

$$\hskip 9pc A_{n_j}=\displaystyle{{n_j} \over {n_j^2 }}\lpar {n_il_{n_i}} \rpar ^2\; \;$$

For any natural number $$ n_j \gt 0 $ , it holds true that $ n_j \lt n_j^2 $ . Moreover, if $ n_i \lt n_j$ , then $ \displaystyle{{n_i} \over {n_i^2 }} \gt \displaystyle{{n_j} \over {n_j^2 }}\;$ . Therefore, it is evident that $ A_{n_j}$ will decrease as a function of the increase of $n_j$ whenever $P_{n_i} = P_{n_j}$$ .

To sum up, number and perimeter are positively correlated when the area is kept constant, whereas number and area are inversely correlated when the perimeter is kept constant.

For example, we can draw two different scenarios:

1. 1. A square of side 3 cm has an equivalent area (9 cm²) with respect to three squares of side 1,732 cm (3×3 cm²=9 cm²). However, the perimeter increases proportionally, shifting from 12 cm in the first case to 20,785 cm in the second case.

2. 2. A square of side 3 cm has an equivalent perimeter (12 cm) with respect to three squares of side 1 cm (3×4 cm=12 cm). However, the area decreases proportionally, shifting from 9 cm² in the first case to 3 cm² in the second case.

This evidence prompts us to consider a hypothetical case. If you are in the first scenario and you choose the set of three squares rather than the single square, the choice can be ambiguously based on numerosity (3 vs. 1, respectively) or on perimeter (20,785 vs. 12 cm, respectively). However, if you are in the second scenario and you make the same choice, this is likely based on numerosity (3 vs. 1, respectively) rather than on area (3 vs. 9 cm², respectively). This is indeed a crucial test, enabling us to exclude the role played by continuous magnitudes (i.e., area and perimeter) in numerical tasks. However, we cannot exclude the influence of other crucial variables such as occupancy and density.

We contend that the simultaneous control for an entire set of continuous variables is not only possible, but also should be guaranteed, as recently demonstrated (Rugani et al. Reference Rugani, Vallortigara, Priftis and Regolin2015). In this study, implying numerical discriminations in 3-day-old chicks, a simultaneous control for multiple variables was obtained: (1) perimeter (summation of the perimeters of all elements in each array was identical), (2) area (summation of the area of all elements was negatively correlated with numbers), (3) occupancy (the overall space occupied by each set of elements was, on average, the same for small and large numbers), and (4) density (the mean distance among the elements was not statistically different). Subjects' behavior revealed that only number was driving their actions, not perimeter, density, occupancy, or area.

Nonverbal subjects, thus, can master numerical tasks, purely on the basis of numerical information. But this is not the only evidence in the scientific literature. In some cases, human infants and young chicks were shown to be more sensitive to continuous magnitudes, rather than numbers (Clearfield & Mix Reference Clearfield and Mix1999; Reference Clearfield and Mix2001; Feigenson et al. Reference Feigenson, Carey and Spelke2002; Rugani et al. Reference Rugani, Regolin and Vallortigara2010).

Moreover, when both continuous magnitudes and numerical cues are available and consistent, human infants (Suanda et al. Reference Suanda, Tompson and Brannon2008) and nonhuman animals (Rugani et al. Reference Rugani, Regolin and Vallortigara2011; Stancher et al. Reference Stancher, Rugani, Regolin and Vallortigara2015) can solve increasingly complex numerical tasks. This suggests that subjects can keep on hold and integrate multiple sources of information, so that redundancy can be exploited to increase mathematical abilities.

Why is scientific evidence on non-symbolic numerical cognition so puzzling? A possibility is that it reflects what a non-symbolic number is. By definition, a non-symbolic number is the numerosity extrapolated from an array of perceived stimuli. Leibovich et al. conceive non-symbolic numbers as abstract representations of discrete quantity (number). We believe, instead, that non-symbolic numbers are part of a more general system for representing quantity, discrete or continuous (see also Gallistel Reference Gallistel, Dehaene and Brannon2011).

To sum up, here we propose a new and more fruitful approach that can be helpful for controlling all of the continuous variables, to create tasks that can be solved solely on the basis of numerical (discrete) information. Controlling occupancy and density, and applying our algorithm to equate overall perimeter (obtaining, thus, a negative correlation between number of elements and overall area) would allow us to assess non-symbolic numerical competences, in human and nonhuman animals.

A future challenge in numerical cognition would be, then, to disentangle the relative role and weight of different cues (number and continuous magnitudes) and how they interact to influence estimation.