Leibovich et al. review a series of experiments supposedly demonstrating the existence of a “number sense” in human adults, infants, or nonhuman animals, and argue that none of them is really conclusive. Although we tend to agree with this general conclusion, we find that the authors's alternative theory is on equally uncertain ground. In the following, we first critically discuss some of the arguments the authors put forward, before concluding that data analysis based on precise, computational models does offer a more promising way to make progress.
The authors go into considerable detail in explaining why previous methods for detecting the number sense are flawed. Their critique on the stimuli centers around the inability to uncouple numerosity from various continuous variables, like surface area, density, object size, and convex hull size. Their critique on the measured variables, moreover, is that what is measured could often be something else than pure “number perception” itself, like, for example, the “summation of object area.”
However, the authors' casual use of the term “continuous variables” deserves some scrutiny. In their description, they make no distinction between for example object size, convex hull size, cumulative surface area, outline, and density. Distinctions between these quantities are, although often subtle, very relevant. For example, surface density (How many pixels are black vs. white per surface area?) differs from object density (How many objects are there per surface area?), as their formulas show: To get surface density, one simply divides the total surface area of all objects (black) by the total area of white space. To arrive at the object density, one divides the number of objects by the total surface. Note that to do this, one needs to know both the total surface, or convex hull, and the number of objects. That means that to perform the calculation that gives you object density, numerosity is required! Therefore, in studies in which object density is the “continuous variable” that participants could have used to estimate numerosity, they may very well have been using a derivative of numerosity.
An even bigger issue is the authors' implicit assumption that the processing of continuous variables contradicts a number sense. But even if the authors are right in claiming that continuous variables play a role in many of the cited experiments, that is not grounds to conclude a “number sense” does not exist. The existence of one mechanism does not exclude the existence of another.
Does that mean that no conclusions about the discrete or continuous, or innate or learned nature of number cognition are possible? No, by developing precise models of both, we can say much more about these issues than we currently can, as demonstrated in other domains of cognitive science that once seemed stuck in unproductive debates about nurture versus nature (e.g., Borensztajn et al. Reference Borensztajn, Zuidema and Bod2009) or continuous versus discrete variable accounts (e.g., Westermann Reference Westermann, Gleitman and Joshi2000; Zuidema & de Boer Reference Zuidema, Bart de Boer, Podesva and Sharma2014).
Model-based data analysis would allow us to evaluate whether the continuous factors can play a consistent explanatory role, or that only ad hoc combinations of these factors, different for each experiment, fit the data as well as the numerosity theory popular in the literature. This is particularly important in the case of animal experiments, which the target article claims are all inconclusive, but which play no role in the final part of the article where an alternative theory is laid out, despite the fact that the influence of language in that theory would lead to strong predictions about animal behavior. We might not be able to exclude all alternative hypotheses, but we can make statements about which model (when specified in sufficient detail) provides the better fit to the data.
Fortunately, some prior work exists (e.g., Stoianov & Zorzi Reference Stoianov and Zorzi2012; van Woerkom Reference van Woerkom2016) providing ideas on how to build precise computational models for the different theoretical positions outlined in the target article. Stoianov and Zorzi's (Reference Stoianov and Zorzi2012) model is a type of neural network (a restricted Boltzman machine), trained on reproducing a binary input image. The model shows signs of encoding numerosity as well as cumulative surface area, and has been interpreted as evidence that a number sense might not be innate per se, but that it is an emergent property of (visual) perception. By looking at the mechanisms behind how such a model performs a numerosity detection task, we can form and test hypotheses about the nature of human and other animals' numerosity detection.
For example, Stoianov and Zorzi (Reference Stoianov and Zorzi2012) analyzed the connection weights in models that had been trained on numerosity estimation. In doing so, they found both “cumulative surface detectors” (responding to the continuous variable of surface area) and a different set of numerosity detectors (whose activity was correlated only with numerosity, not with surface area), showing that detectors for both quantities can emerge, work in tandem, and complement, rather than compete, with each other. These types of analyses are critical in forming our understanding of estimating numerosities and the use of the “number sense” concept.
But more modeling work in this area is required to explore the broader area of models similar to Stoianov and Zorzi's and embodying alternative positions. For example, in our replication of the model (Van van Woerkom Reference van Woerkom2016), we end up with networks that are equally good at the training task (reproducing images), but do so without any apparent numerosity detectors. That raises questions about the robustness of the “number sense as side effect” result and its explanatory power for empirical results in cognitive science (e.g., those relating to the influence of language). That said, this domain is a perfect example of a domain where we should report how well each model fits the data, rather than making binary choices between rejecting and accepting alternative theories. Modelers should get to work!
Leibovich et al. review a series of experiments supposedly demonstrating the existence of a “number sense” in human adults, infants, or nonhuman animals, and argue that none of them is really conclusive. Although we tend to agree with this general conclusion, we find that the authors's alternative theory is on equally uncertain ground. In the following, we first critically discuss some of the arguments the authors put forward, before concluding that data analysis based on precise, computational models does offer a more promising way to make progress.
The authors go into considerable detail in explaining why previous methods for detecting the number sense are flawed. Their critique on the stimuli centers around the inability to uncouple numerosity from various continuous variables, like surface area, density, object size, and convex hull size. Their critique on the measured variables, moreover, is that what is measured could often be something else than pure “number perception” itself, like, for example, the “summation of object area.”
However, the authors' casual use of the term “continuous variables” deserves some scrutiny. In their description, they make no distinction between for example object size, convex hull size, cumulative surface area, outline, and density. Distinctions between these quantities are, although often subtle, very relevant. For example, surface density (How many pixels are black vs. white per surface area?) differs from object density (How many objects are there per surface area?), as their formulas show: To get surface density, one simply divides the total surface area of all objects (black) by the total area of white space. To arrive at the object density, one divides the number of objects by the total surface. Note that to do this, one needs to know both the total surface, or convex hull, and the number of objects. That means that to perform the calculation that gives you object density, numerosity is required! Therefore, in studies in which object density is the “continuous variable” that participants could have used to estimate numerosity, they may very well have been using a derivative of numerosity.
An even bigger issue is the authors' implicit assumption that the processing of continuous variables contradicts a number sense. But even if the authors are right in claiming that continuous variables play a role in many of the cited experiments, that is not grounds to conclude a “number sense” does not exist. The existence of one mechanism does not exclude the existence of another.
Does that mean that no conclusions about the discrete or continuous, or innate or learned nature of number cognition are possible? No, by developing precise models of both, we can say much more about these issues than we currently can, as demonstrated in other domains of cognitive science that once seemed stuck in unproductive debates about nurture versus nature (e.g., Borensztajn et al. Reference Borensztajn, Zuidema and Bod2009) or continuous versus discrete variable accounts (e.g., Westermann Reference Westermann, Gleitman and Joshi2000; Zuidema & de Boer Reference Zuidema, Bart de Boer, Podesva and Sharma2014).
Model-based data analysis would allow us to evaluate whether the continuous factors can play a consistent explanatory role, or that only ad hoc combinations of these factors, different for each experiment, fit the data as well as the numerosity theory popular in the literature. This is particularly important in the case of animal experiments, which the target article claims are all inconclusive, but which play no role in the final part of the article where an alternative theory is laid out, despite the fact that the influence of language in that theory would lead to strong predictions about animal behavior. We might not be able to exclude all alternative hypotheses, but we can make statements about which model (when specified in sufficient detail) provides the better fit to the data.
Fortunately, some prior work exists (e.g., Stoianov & Zorzi Reference Stoianov and Zorzi2012; van Woerkom Reference van Woerkom2016) providing ideas on how to build precise computational models for the different theoretical positions outlined in the target article. Stoianov and Zorzi's (Reference Stoianov and Zorzi2012) model is a type of neural network (a restricted Boltzman machine), trained on reproducing a binary input image. The model shows signs of encoding numerosity as well as cumulative surface area, and has been interpreted as evidence that a number sense might not be innate per se, but that it is an emergent property of (visual) perception. By looking at the mechanisms behind how such a model performs a numerosity detection task, we can form and test hypotheses about the nature of human and other animals' numerosity detection.
For example, Stoianov and Zorzi (Reference Stoianov and Zorzi2012) analyzed the connection weights in models that had been trained on numerosity estimation. In doing so, they found both “cumulative surface detectors” (responding to the continuous variable of surface area) and a different set of numerosity detectors (whose activity was correlated only with numerosity, not with surface area), showing that detectors for both quantities can emerge, work in tandem, and complement, rather than compete, with each other. These types of analyses are critical in forming our understanding of estimating numerosities and the use of the “number sense” concept.
But more modeling work in this area is required to explore the broader area of models similar to Stoianov and Zorzi's and embodying alternative positions. For example, in our replication of the model (Van van Woerkom Reference van Woerkom2016), we end up with networks that are equally good at the training task (reproducing images), but do so without any apparent numerosity detectors. That raises questions about the robustness of the “number sense as side effect” result and its explanatory power for empirical results in cognitive science (e.g., those relating to the influence of language). That said, this domain is a perfect example of a domain where we should report how well each model fits the data, rather than making binary choices between rejecting and accepting alternative theories. Modelers should get to work!