In the wake of traditional rationalists versus empiricists debates, innateness and learning issues will never get old. In the field of numerical cognition, this takes the form of arguments regarding an allegedly innate, evolutionarily conserved and phylogenetically widespread “number sense” (Dehaene, 1999; Nieder, 2017), which is thought to underlie numerosity discriminations in both humans and other animals (Nieder, 2018). While this hypothesis has gained considerable popularity over the years, it continues to be animatedly debated.
Surprisingly, however, what is disputed is not if something about numerical discriminations is innate, but precisely what constitutes this natural mechanism. The evolutionary advantages of discriminating quantities are evident for many animal species and in numerous ecological niches (Rugani, 2018), but this does not directly endow animals with mathematical proficiency. What is the difference between discriminating quantities and working with arithmetic, counting, numbering? Can studying the brain clarify these issues?
In the light of recent evidence, hypotheses regarding an innate number system in the brain will be critically evaluated, with particular attention to avoid the loose terminology sometimes used in the field (Núñez, 2017). By showing what is innate and what depends on enculturation, better assistance can be provided to individuals who struggle to reach full mathematical proficiency, reducing the impact that low numeracy has both on individual life achievements (Parsons & Bynner, 2005) and nations’ economies (Butterworth et al., 2015).
The notion of an innate number system in the brain, first systematised by Dehaene (1999), has gained supporting evidence from various fields.
Firstly, there is evidence coming from prelinguistic infants (Cordes & Brannon, 2008), which shows that they can discriminate between small numerosities (1-4) of item sets in habituation (Strauss & Curtis, 1981) or violation of expectation paradigms (Kobayashi et al., 2005).
Secondly, studies on both trained and naive animals, from insects (Dacke & Srinivasan, 2008) to fish (Agrillo & Bisazza, 2018), birds (Rugani, 2018), and mammals (McComb et al., 1994) show that the ability to discriminate between quantities and small numerosities is not only prelinguistic but also phylogenetically widespread and advantageous in diverse ecological contexts.
Thirdly, electrophysiological work on monkeys trained to discriminate quantities revealed the presence of numerosity-selective neurons in prefrontal (Nieder et al., 2002) and parietal areas (Nieder & Miller, 2004; Figure 1d). These neurons have since been demonstrated to be active in a wide variety of tasks and across sensory modalities (Nieder, 2012; Figure 1a, 1b). Interestingly, numerosity-selective neurons show a gaussian-like activation profile across quantities, higher for their preferred numerosity and gradually lower for neighbouring ones (Figure 1c). This type of population response in overlapping tuning curves is thought to underlie the observation that numerosity judgments are subject to the psychophysical Weber-Fechner law, determining the well-established size, distance, and ratio effects (Dehaene, 2003).
Figure 1. Response and location of number neurons in the primate brain. a-b. A sequential enumeration protocol in two different sensory modalities. In this task, items are presented over time and need to be enumerated by the subject by matching them with the right numerosity of items in the test display. The task is often used to assess how neurons encode numerosities presented over time. c. Supra-modal number neurons’ tuning functions detected with the previous tasks. Note their preferential and decaying response, and nearly identical tuning across modalities. d. Frontolateral view of a monkey brain highlighting areas of the number network with high density of number neurons. e. Frontolateral view of the human brain showing areas consistently activated by number processing in fMRI studies. Part a-c based on Nieder, 2012. Original figure and partially adapted caption from Nieder, 2016. lPFC: lateral Prefrontal Cortex; VIP: ventral intraparietal area; IPS: intraparietal sulcus; mPFC: medial Prefrontal Cortex.
Lastly, studies on adult humans also seem to support a “number sense”. In particular, functional MRI studies show consistent activation in the prefrontal cortex (PFC) and intraparietal sulcus (IPS) during numerosity discriminations (Figure 1e), areas homologous to regions in the primate brain where “number neurons” have been found (Nieder & Dehaene, 2009). Additionally, psychophysics evidence indicates that numerosity is a stimulus characteristic subject to adaptation (Burr et al., 2018), which is taken as evidence of a dedicated perceptual mechanism.
It is immediately important to note that the evidence just surveyed can be un-controversially used to argue only for the innateness of an approximate quantity discriminating mechanism, which only rarely (i.e., for numerosities less than five) and often upon training results in exact quantification (Núñez, 2017). Elsewhere called an Approximate Number System (ANS; Feigenson et al., 2004), here this terminology is explicitly avoided because it implicitly uses the concept of number as a pre-existing primitive – existing where? – to which inexact mental representations are approximated (Núñez, 2017). In fact, though in the field the terms number, numerosity, and quantity are often used interchangeably, the notion of number more precisely entitles exact symbolic quantifiability, abstractness and cardinality, all properties that far from being innate and elementary depend on human cultural elaborations (Núñez, 2017).
Accordingly, additional proposals using loose terminology regarding the innateness of more elaborate mechanisms beyond quantity discrimination will also be omitted from discussion (examples are the Mental Number Line, Dehaene, 1999, and Spatial Numerical Associations more broadly, Dehaene et al., 1993), for such mechanisms are far less supported by the evidence and object of extensive criticism (Núñez et al., 2011; Shaki & Fischer, 2018).
Again, the only point to which various strains of evidence and perspectives converge is a phylogenetically widespread neural system for approximate quantities' representation and estimation.
Interestingly, small numerosities (up to four objects) are more accurately estimated than larger ones, a feature traditionally used to argue for the presence of two qualitatively different systems in representing quantities (Burr et al., 2018; Feigenson et al., 2004). However, recent modelling evidence shows that this behaviour can clearly arise from a single system processing numerosities optimally under the resource constraint of visual memory capacity (Cheyette & Piantadosi, 2020). This also clarifies why infants and other animals (with the possible exception of chimpanzees; Tomonaga & Matsuzawa, 2002) have a lower subitising range and are usually worse in numerosity discriminations than human adults.
Furthermore, it has been shown how numerosity discrimination and numerosity-selective units can emerge in deep neural networks designed for visuospatial processing (Zorzi & Testolin, 2018), reframing the innateness of the “number sense” in terms of cognitive architectural constraints during perceptual learning. This way, numerosity is seen as a “statistical invariant in highly variable input” (Zorzi & Testolin, 2018) that gets readily picked up by various perceptual systems, possibly justifying why quantity discriminations can be carried out even by insects.