Do We Have Innate Numerical Abilities? Number Concepts Represented through the Approximate Number System (ANS) in Cognition
Introduction
Concepts, the omnipresent element that constitutes our understanding of the world, is one of the primary building blocks of our cognition. A concept is a “dynamical distributed system” that mentally classifies and represents our perceptual experience to influence our interplays with the instances under those classifications or categories (Barsalou 2016). Even though many concepts, such as bicycles, are concocted after our interactions with the outside environment and learned a posteriori, some are argued to be a priori and innately present at birth, among which many concepts fall under. Many philosophers, such as Immanuel Kant, have argued the apriority of numbers which are independent of any experience even if they can be derived from experience (CPR 1998, B15), meaning that the concept of number 5, for example, is primitively and pre-linguistically present in our cognition before we ever experience counting of objects in the set of 5. In the modern field of psychological and cognitive research, this nativist tradition of “number sense” has grown to be postulated as an approximate number system (ANS) supported by many empirical studies on infants, adults, and even animals (Xu & Spelke 2000; Barth, Kanwisher, and Spelke 2003; Lipton & Spelke 2015; Corliss et al. 2020). In this view, ANS affords an inherent capacity to represent numbers such that infants and animals can discriminate between perceived magnitudes.
In this essay, I first overview some empirical evidence for ANS that accounts for this increasingly orthodox view of the number concept representation. Then, I address some critiques against this view, specifically the one concerning its inability to capture the numerical essence of precision on the grounds of the diminishing capability of discrimination. This line of philosophical objection argues that it represents “quantical” instead of “numerical” concepts, and I argue in response that the numerical notion rests on the foundation of quantical information. Lastly, I look at philosophical arguments that clarify the scope of ANS and postulate specific kinds of numerical contents of ANS representation, such as rational numbers, due to our ratio processing system. Overall, my essay supports that ANS is an innate faculty of our cognition and can genuinely allow number concepts to be represented pre-linguistically.
Number Sense through Discriminability
ANS is known through its primal discriminatory capabilities that differentiate different magnitudes, and it is evidenced through empirical research in infants, adults, and some animals. Nevertheless, this discriminability depreciates as the ratio between the two numerical quantities approaches 1:1, as subjected to Weber’s Law. For example, 4 is easier to be differentiated from 2 than 6 from 4. In this section, I survey some principal research that supports ANS’s discriminability to illustrate its overwhelming popularity.
To start with, ANS’s discriminability would be demonstrated most effectively if evidence can be discovered in infants as they have not been habituated with number concepts in language and mathematics. As such, Xu and Spelke (2000) visually exposed 6-month-old infants to arrays of dots that represent different numbers. After being accustomed to seeing 8-dot arrays, the infants exhibited more interest when presented with 16 or 4-dot arrays that comprise the 1:2 ratio with an 8-dot array. However, the 12-dot array did not have this effect. Such that, this study concludes that infants could discriminate approximate magnitudes that are reasonably and visibly different from each other by a ratio of 1:2. Later on, Lipton and Spelke (2003) further demonstrated the improved accuracy and enhanced capability of ANS’s discriminability when 9-month-olds were found to differentiate magnitudes of ratio 2:3. Moreover, instead of visual habituation, this 2003 study was conducted with auditory sequences of controlled duration, interval, and intensity. In this case, six-month-olds discriminated hearing sets of 16 sounds and 8 sounds (2:1) as opposed to 12 sounds and 8 sounds (3:2). By contrast, nine-month-olds discriminated sets with the ratio of 3:2 but not 5:4. Therefore, Lipton and Spelke conclude on an ANS in infants that refines with age and is cross-modal, such that infants could isolate low-level confounds in vision or sound.
However, studies in adults are harder to be organized as there are many confounds that need to be controlled, such as the cultivated ability to count with language, which supersedes the capability of ANS with great accuracy and precision. Nevertheless, studies with flashed numerical stimuli had been conducted so that counting could not be exercised as a tool given the rushed exposure of stimuli (Barth, Kanwisher, and Spelke 2003). There have also been studies that required participants to repeat a word while pressing a button for a specified number of times so that counting can be prevented in the multi-tasking set-up (Cordes, Gelman, Gallistel, & Whalen, 2001). Now, without counting, adults are left with ANS like infants to perceive magnitudes. Unsurprisingly, discriminability is still present and in accordance with Weber’s law, which again suggests an ANS in adults.
Interestingly, animals exhibit similar discriminatory capabilities when it comes to number sense. Corliss, Brown, Hurly, Healy, and Tello-Ramos (2020) discovered that hummingbirds use their number sense to ensure resources can be maximized in foraging. After a series of training for the birds to learn the higher rewards in terms of quantities of flowers, the birds showed discrimination between arrays of 5 and 7 flowers. In this case, ANS is not merely a conceptual tool but one that is vital to the survival and evolution of species.
In general, these studies all indicated the existence of an ANS that bypasses our linguistic abilities to represent numbers in terms of a discriminability that grows with our development and is shared among humans and animals.
The Imprecision of ANS
Even though ANS had been highly popular with the stacks of empirical studies that sought to prove its validity, it is not without criticisms. For instance, some question the physical evidence of ANS by arguing that the discriminability origins from irrelevant factors that influence the perception of magnitudes. As an example, the visual arrays of dots do not just differ in terms of the quantities of dots contained but also differ by their geometrical size with different visual densities or diameters of the dots. And therefore, the need for an ANS that can “extract pure numerosity” (Gebuis et al. 2016) would seem to be undermined by our tendencies also to perceive non-numerical factors that also indicate magnitudes. However, this argument seems relatively weak as these congruency effects are present in all kinds of perceptual content, and given the imprecision of ANS, magnitudes of the numerical and non-numerical kinds are naturally helpful for subjects to form discriminability. With that being said, ANS is still subjected to other challenges. And most prominently, on the imprecision aspect itself.
Because the discriminability of ANS continuously diminishes when the magnitudes of different sets of entities approach the same quantity level, it means that ANS cannot represent numerosities in their exactness. Then, a line of critique naturally arises by suggesting that ANS fails to capture the numerical essence of precision and therefore is not able to represent numbers at all. This argument relies heavily on a strong sensitivity principle, which states that a concept can only be represented when all its properties need to be attended to sensitively. In the case of numbers, they carry the property of being quintessentially precise. Then, given the strong sensitivity principle, any true representation of numbers should reflect this property sensitively, which is not granted by the postulation of an ANS that is only approximate. One may object to having this principle modified slightly from representing all properties to some, as our representations of many concepts are not sensitive to all their properties. For example, our mental representation of fire does not usually give due attention to the emitted gas of CO2 and CO1. Now, the weak sensitivity principle only requires the representation of concepts to be sensitive to some properties instead of all properties. And this principle, when applied to number concepts, could include precision property or not. Nevertheless, one can still object by emphasizing the essentiality of the precision property, and we now need to define how our understanding of this essential property of precision came to be in numbers.
Núñez (2017) investigated our understanding of numbers and arithmetic from perspectives of biological evolution, pre-industrialized linguistic cultures, and trained animal experiments. From a biological perspective, our brain seems to be hard-wired with a mathematical ability since it affords selective advantages to those who can discriminate resources better in terms of quantities. And these “biologically evolved preconditions” of number sense are also argued by Núñez to be teleological as they are effective in reaching the goal of modern industrialization with cultural practices that involves arithmetic heavily. It is a view that is tracing number concepts backward by situating biological evolution in terms of a linear progression toward contemporary society today. Particularly, Núñez argues that our understanding of numbers is only investigated from the perspectives of human beings in the industrialized world. The fact that we associate exact numerosity as the essence of numbers is due to our education system, measuring conventions, and linguistic capability that completes the symbolizations of numbers. Surveys of 193 hunter-gatherer societies across the globe found that most of their languages do not distinguish numbers above the number 5 (Epps et al. 2012). Núñez, therefore, argues that our ideas of number sense are an industrial-centric view that neglects the cultural varieties that subsisted without numerical notions and any writing practices for them. And therefore, the ANS, as an innate capacity for exact numbers, does not hold as a necessary component of our cognition as endowed at birth. Thereby, Núñez objects to ANS as its imprecise and non-symbolic “quantical” discrimination is not definitive enough to infer any “numerical” representation.
In Núñez’s argument, the implication of a weak sensitivity principle is assumed so that the hunter-gatherer societies which live without the accuracies afforded by numerical concepts are evidence enough to show that an approximate number system can be inexistent for humans to function. Núñez is relying on the imprecision of ANS to attack its numerical representation capability. However, numerical and quantical properties are intricately linked together. Numbers, as defined by Gottlob Frege’s logicism, are abstract objects of language used in statements to infer the numerosity of other concepts (1980 FA). For example, number 4 can be defined as the number of apples on my table as they are being counted by me through a sortal. However, this numerical exactness depends upon a critical quantical differentiation that generates an immediate number sense to indicate how the pile of apples on my table is in a different level of magnitudes to that pile of apples in the supermarket. Counting is not necessary to activate that number sense for us to know that, at least, they are in different quantities. And this is, arguably, the basis for numbers to exist at all. Even though numbers carry the precision property with them, they have to be invented because of our innate number sense that appeals to natural discriminations of magnitudes. Therefore, the “numerical” and “quantical” differentiation that Núñez proposed to contest ANS does not hold because the precise numerical notions necessarily depend upon an approximate intuition of quantical notions.
Overall, ANS represents numbers in terms of their approximate quantities, but this capability does not violate or contradict the precision property of numbers. Instead, the rough discriminability afforded by ANS is the core foundation for any number of concepts to be formulated in languages at all.
The Scope and Content of ANS Representation
It is vital to distinguish the scope of ANS for number representations now, given that it does not precisely represent accurate numbers. Clarke and Beck (2021) clarified the scope of ANS with two main categorizations. The first one is that ANS represents numerosity instead of precise numbers, and the second one is that ANS represents numbers but has a different mode of representation. Clarke and Beck argue that the second interpretation is more worthy of support as ANS obeys Weber’s law, making its mode of representation to be analog, like a mercury thermometer, instead of distinct and concrete, like a digital thermometer. Just as literal number concepts, ANS also represents numbers, but in a different format which is imprecise but still indicative of number concepts.
Furthermore, Clarke and Beck postulated the kinds of numbers represented by ANS, further clarifying the content of its representations. To begin with, ANS is usually, by default, assumed to be representing natural numbers such as 8 or 16. Because in the empirical studies surveyed in the first section, ANS demonstrated representations of whole items in a set expressed as positive integers, whether it was flowers or dots in an array, Clarke and Beck accepted the baseline where ANS represents positive integers. Moreover, the authors argue that ANS represents non-natural numbers that include rational numbers such as 1.5 in addition to positive integers. This postulation is supported by developmental and psychology research that suggests a “ratio processing system” (RPS) that enables our understanding of fractional numbers. Additionally, the authors posit that our need to draw inferences with probabilistic reasonings due to uncertainties of future events makes representations of rational numbers such as fractions and probability percentages advantageous to survivals. McCrink and Wynn (2007) discovered that 6-month infants, after being exposed to instances of numerical ratios, would exhibit longer observant behaviors when new ratios are represented. And there have also been studies in nonhuman apes that supported discriminability when it comes to numerical ratios. The empirical evidence suggests that humans and animals can process and differentiate rational numbers in terms of ratios as part of ANS’s discriminability.
Lastly, I support Clarke and Beck’s conjecture of a ratio representing the ANS system for it the studies in humans and animals with whole entities evince a ratio processing system as discriminability strengthens when the ratios between the quantities of two sets increase. This seems to suggest our ability to capture the comparison of quantities through ratio processing, such that we have immediate intuitions when given sets of two collections that contain varied resources and have higher success rates of choosing the voluminous collection of objects when they are in sharper contrast to the other collection quantitively. This ratio-processing ability of ANS can also be seen as crucial to enhance our survival chances in biological evolutions.
In general, ANS does represent numbers, abide that it is in an analog mode instead of a concrete mode like literal numerical concepts. Moreover, I argued that ANS represents rational numbers such as 1.2 and 0.5, aided by a ratio-processing system that is evolutionally advantageous.
Conclusion
Overall, this essay overviewed many empirical studies that suggested an innate capability of number representation in the name of an Approximate Number System (ANS). By refuting objections that question ANS’s imprecision by emphasizing how the approximation of quantities is the foundation of representing numerical notions, I defended ANS against the argument that it does not capture the numerical essence of precision. Lastly, I elucidated and defended the content of ANS representations by relating it to a ratio-processing system in our evolutionary history. Therefore, this essay concludes that number concepts can be innately and imprecisely represented through ANS with discriminatory capabilities and that this representation can range from natural numbers to non-natural ratios.
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