How do we approximate large numerosities? The dominant view is that this is accomplished via the “approximate number system” (ANS), an innate number system that is able to process pure numerosity (e.g., Dehaene Reference Dehaene2003). Recently, others and we highlighted the role of continuous magnitudes (e.g., density, size, total surface of the dots, etc.) in numerosity judgments (e.g., Allik & Tuulmets Reference Allik and Tuulmets1991; Gebuis et al. 2016; Gevers et al. Reference Gevers, Cohen Kadosh, Gebuis and Henik2016). Leibovich et al. therefore challenge, in the current article, the idea that number sense is innate. They note that a natural correlation exists between continuous magnitudes and numerosity and argue that both types of information are used “holistically”Footnote
1
to judge numerosity. Their ideas are presented in a developmental framework within which processing of continuous magnitudes is innate but a number sense develops over time.
We fully agree with the authors that something is wrong with the ANS theory. We also agree that continuous magnitudes play an important role. The proposed idea of holistic processing of numerosity using both a number sense and continuous magnitudes is appealing at first sight. However, a closer inspection of the proposal makes us conclude it is both incomplete and implausible.
First, the account is incomplete, as it does not explain exactly how continuous magnitudes are judged and how they would bias the numerosity estimate. One possibility is that subjects can estimate all continuous magnitudes (size, density, diameter, contour length, etc.) simultaneously. This would allow the subjects to make an exact calculation of the numerosity, and hence, the numerosity estimate should be free from any bias. This result is clearly inconsistent with the literature, and this possibility can therefore be rejected. Another possibility is that the subject only decides which of the two stimuli contains “larger” or “more” continuous magnitudes. This could indeed cause the observed bias, but how would this work in cases in which only a few continuous magnitudes are larger in one stimulus compared with the other? Take for example two stimuli with the convex hull being larger but the surface smaller in one stimulus compared to the other. Do the subjects now rely only on the most prominent continuous magnitude? This is unlikely, as results have shown that the bias increases with the number of continuous magnitudes being manipulated (Gebuis & Reynvoet Reference Gebuis and Reynvoet2012b). Another alternative is that each continuous magnitude contributes to the final response relative to its size. Unless the authors have another suggestion on how continuous magnitudes could help in making numerosity judgments, the previously discussed reasoning leaves us with the latter solution. We already proposed this solution and implemented it in a model that relies only on continuous magnitudes (Gebuis et al. 2016; Gevers et al. Reference Gevers, Cohen Kadosh, Gebuis and Henik2016).
Second, the model provides an explanation only for comparisons tasks, but not for estimations (e.g., How many objects are presented?). We argue that performance in estimation tasks uncovers why the model is implausible. The model cannot explain how the biases or congruency effects as observed in a comparison task occur in estimation tasks. The reason is that in an estimation task, only a single numerosity is presented and the continuous magnitudes of this numerosity are not informative about the number presented. For example, knowledge about the diameter of the individual objects does not provide information about the numerosity presented. Related to the previous discussion, more detail is needed about how the continuous magnitudes influence a numerosity judgement and, more specifically, numerosity estimation. As outlined previously, an exact calculation/estimation of the exact size of each sensory cue would enable us to calculate the exact value presented. This means that our number sense and our continuous magnitudes would derive the same result and thus would not induce a bias. Furthermore, when a single set of objects has to be estimated, it is impossible to make a small/large judgment given that there is no other stimulus with which to compare it. The continuous magnitudes on their own are simply not informative for estimation and therefore should, according to the current model, not bias numerosity estimation. However, this is not right, as multiple studies have shown a bias induced by the continuous magnitudes when estimates are performed (e.g., Gebuis & Reynvoet Reference Gebuis and Reynvoet2012c; Izard & Dehaene Reference Izard and Dehaene2008). The authors could argue that subjects calculate a running average based on the set of stimuli used in the experiment and compare the continuous magnitudes with this running average. However, even in a study where only a single stimulus was presented (and hence no running average could be calculated), a bias was induced by the continuous magnitudes (Krueger Reference Krueger1982).
The proposed review makes it clear that the influence of continuous magnitudes on numerosity processing challenges the ANS theory. However, the alternative proposal made by Leibovich et al. can be rejected based on both logical and empirical grounds. We therefore propose a different solution, which is more parsimonious, stepping out of the comfort zone where researchers try to adapt the idea of an ANS to preserve it. We put forward the simple suggestion that we do not possess nor develop an ANS. Instead, we argue that it is much more straightforward to assume that numerosity estimations or comparisons are accomplished by weighing the different sensory cues constituting numerosity, whereas language is used to describe this numerosity (for an extensive review on this matter, see Gebuis et al. [2016] and Gevers et al. [2016]). Contrary to the current model, it does explain how numerosity can be derived from the sensory cues. This proposal will therefore spark the debate on the ANS, not when it comes into existence, but if it exists at all.
How do we approximate large numerosities? The dominant view is that this is accomplished via the “approximate number system” (ANS), an innate number system that is able to process pure numerosity (e.g., Dehaene Reference Dehaene2003). Recently, others and we highlighted the role of continuous magnitudes (e.g., density, size, total surface of the dots, etc.) in numerosity judgments (e.g., Allik & Tuulmets Reference Allik and Tuulmets1991; Gebuis et al. 2016; Gevers et al. Reference Gevers, Cohen Kadosh, Gebuis and Henik2016). Leibovich et al. therefore challenge, in the current article, the idea that number sense is innate. They note that a natural correlation exists between continuous magnitudes and numerosity and argue that both types of information are used “holistically”Footnote 1 to judge numerosity. Their ideas are presented in a developmental framework within which processing of continuous magnitudes is innate but a number sense develops over time.
We fully agree with the authors that something is wrong with the ANS theory. We also agree that continuous magnitudes play an important role. The proposed idea of holistic processing of numerosity using both a number sense and continuous magnitudes is appealing at first sight. However, a closer inspection of the proposal makes us conclude it is both incomplete and implausible.
First, the account is incomplete, as it does not explain exactly how continuous magnitudes are judged and how they would bias the numerosity estimate. One possibility is that subjects can estimate all continuous magnitudes (size, density, diameter, contour length, etc.) simultaneously. This would allow the subjects to make an exact calculation of the numerosity, and hence, the numerosity estimate should be free from any bias. This result is clearly inconsistent with the literature, and this possibility can therefore be rejected. Another possibility is that the subject only decides which of the two stimuli contains “larger” or “more” continuous magnitudes. This could indeed cause the observed bias, but how would this work in cases in which only a few continuous magnitudes are larger in one stimulus compared with the other? Take for example two stimuli with the convex hull being larger but the surface smaller in one stimulus compared to the other. Do the subjects now rely only on the most prominent continuous magnitude? This is unlikely, as results have shown that the bias increases with the number of continuous magnitudes being manipulated (Gebuis & Reynvoet Reference Gebuis and Reynvoet2012b). Another alternative is that each continuous magnitude contributes to the final response relative to its size. Unless the authors have another suggestion on how continuous magnitudes could help in making numerosity judgments, the previously discussed reasoning leaves us with the latter solution. We already proposed this solution and implemented it in a model that relies only on continuous magnitudes (Gebuis et al. 2016; Gevers et al. Reference Gevers, Cohen Kadosh, Gebuis and Henik2016).
Second, the model provides an explanation only for comparisons tasks, but not for estimations (e.g., How many objects are presented?). We argue that performance in estimation tasks uncovers why the model is implausible. The model cannot explain how the biases or congruency effects as observed in a comparison task occur in estimation tasks. The reason is that in an estimation task, only a single numerosity is presented and the continuous magnitudes of this numerosity are not informative about the number presented. For example, knowledge about the diameter of the individual objects does not provide information about the numerosity presented. Related to the previous discussion, more detail is needed about how the continuous magnitudes influence a numerosity judgement and, more specifically, numerosity estimation. As outlined previously, an exact calculation/estimation of the exact size of each sensory cue would enable us to calculate the exact value presented. This means that our number sense and our continuous magnitudes would derive the same result and thus would not induce a bias. Furthermore, when a single set of objects has to be estimated, it is impossible to make a small/large judgment given that there is no other stimulus with which to compare it. The continuous magnitudes on their own are simply not informative for estimation and therefore should, according to the current model, not bias numerosity estimation. However, this is not right, as multiple studies have shown a bias induced by the continuous magnitudes when estimates are performed (e.g., Gebuis & Reynvoet Reference Gebuis and Reynvoet2012c; Izard & Dehaene Reference Izard and Dehaene2008). The authors could argue that subjects calculate a running average based on the set of stimuli used in the experiment and compare the continuous magnitudes with this running average. However, even in a study where only a single stimulus was presented (and hence no running average could be calculated), a bias was induced by the continuous magnitudes (Krueger Reference Krueger1982).
The proposed review makes it clear that the influence of continuous magnitudes on numerosity processing challenges the ANS theory. However, the alternative proposal made by Leibovich et al. can be rejected based on both logical and empirical grounds. We therefore propose a different solution, which is more parsimonious, stepping out of the comfort zone where researchers try to adapt the idea of an ANS to preserve it. We put forward the simple suggestion that we do not possess nor develop an ANS. Instead, we argue that it is much more straightforward to assume that numerosity estimations or comparisons are accomplished by weighing the different sensory cues constituting numerosity, whereas language is used to describe this numerosity (for an extensive review on this matter, see Gebuis et al. [2016] and Gevers et al. [2016]). Contrary to the current model, it does explain how numerosity can be derived from the sensory cues. This proposal will therefore spark the debate on the ANS, not when it comes into existence, but if it exists at all.