Arithmetic, Culture, and Attention

Arithmetic, Culture, and Attention Abstract: The study of numerical cognition has undergone tremendous progress in recent years, accumulating scores of data on cognitive systems that could be involved in the uniquely human ability to practice formal arithmetic. Among the important questions tackled by this burgeoning domain of research is what happens to limited cognitive systems that we share with many animal species to allow us to develop arithmetically-viable numerical content. While answers to this questions have varied, most have appealed to the presence of culturally-inherited extracranial cognitive support to explain how numerical content emerges from our innate cognitive machinery. In this paper I challenge this externalist approach and argue that we should favor explanations that focus on cognitive processing inside the head. To support my claim, I take a look at research into Spontaneous Focusing on Numerosity (SFON) to show that internalism can explain some aspects of the development of numerical cognition without needing to factor in the cultural setting in which this development takes place. 1 Introduction: the main problem for numerical cognition studies The study of the cognitive and perceptual systems underlying our numerical abilities has progressed tremendously in the past few decades, yielding scores of data on the potential role played by the so-called Approximate Number System (ANS) and the Object-File System (OFS) in the development of natural number concepts (Carey 2009; Dehaene 1997/2011; Feigenson et al. 2004; Cohen Kadosh & Dowker 2015). While there is still disagreement on the relationship between these systems and on the extent to which they produce representations with numerical content, there is overwhelming consensus that, on their own, neither of them produces representations with sufficient precision and numerical range to account for the development of natural number concepts. For example, not only is there compelling evidence that the OFS does not produce representations with explicit numerical content (Carey 2009), it can only keep track of a very small number of items (up to 4). On the other hand, the ANS’s representations are, as its 1 name suggests, increasingly approximate as stimulus numerosity1 increases, so that it is unable to accommodate distinct numerical magnitudes that are too close together (e.g. 21 and 22). Given these obvious limitations to our innate cognitive systems, the question that interests me in this paper is: what is the best way to find out how we come to understand what natural numbers are? Most answers to this question have adopted a form of externalism about cognition in which culturally-inherited symbols and extracranial artefacts allow us to bridge the gap between natural numbers and the output of our innate cognitive machinery. In most of these accounts, the role played by such artefacts in the development of numerical content is explained by appealing to features of 4E cognition (Menary 2010b), where things outside our head extend and augment our cognitive capacities. There is certainly good reason to appeal to cultural factors in explaining how we bridge this gap: not only do we learn and manipulate numbers by relying on culturally-inherited symbols for these, but the practice of arithmetic as we know it is the result of a gradual, cumulative process of cultural evolution that took place over many generations, involving the transmission of cultural practices and numeration systems from one person to another, and from one generation to the next. And yet, as I argue in this paper, if we want to bridge the gap between natural number concepts and the content produced by systems like the ANS and the OFS, extending the mind beyond the skull and relying on cumulative culture leads, at best, to an incomplete picture of the origins of numerical cognition. This is because the insight that leads to an understanding of what numbers are occurs in individuals’ heads, without any accompanying change in their environment. Instead, I claim that any answer to this developmental question must appeal to internalist, psychological processes that do not depend on attributing a constitutive status to culturally-inherited extracranial numerical artefacts. To make my case, I summarize the main lines of a research program that seems to illustrate well the limitations of cultural factors in explaining how we bridge the gap, namely, the study of Spontaneous Attention to Number. 1 While this term is rightfully criticized for being imprecise (Beck 2014; Schlimm 2018), I use it here to describe the number of items of a perceived collection of objects. In other words, when we focus on the number of objects in a perceptual scene, numerosity describes the number of things we perceive. 2 The paper is divided as follows. In section 2 I sketch the reasoning behind externalism about numerical cognition and then discuss why this approach may not apply to how we bridge the gap in section 3 by taking a look at the learning trajectory followed by children learning what numbers are. I present the main lines of the study of Spontaneous Focusing on Numerosity in sections 4 and 5, to illustrate how explanations of how we bridge the gap may not need to factor in the presence of culturally-inherited extracranial cognitive aids. In section 6 I discuss the potential input of cultural factors in SFON studies and highlight their limitations. I then show that focusing on attention with respect to how we bridge the gap is already present in many externalist proposals and discuss potential explanatory benefits of attention to numerosity in section 7. Finally, in section 8 I explain why the fact that externalism is true does not mean it is always explanatorily useful. 2 Numerical cognition and externalism To explain the ontogenetic emergence of numerical cognition, the standard approach is to rely on a form of external symbols or artefacts with numerical content. For example, Stanislas Dehaene (1997/2011) holds that when children learn to count, numerals such as those in a count list come to be associated with representations of the ANS. Learning the meaning of the number words is then a matter of mapping these to a precise location on what has come to be called a “mental number line” (MNL).2,3 For Dehaene, the answer to how we bridged the gap from the content of our evolved cognitive systems to the content of arithmetically-viable number concepts lies in culture: “Cultural inventions, such as the abacus or Arabic numerals…transformed [the intuition of number] into our 2 Galton (1880) was the first to describe human numerical representations as being ordered on a left- right line, while Restle (1970) was the first to explicitly tie in Moyer & Landauer's (1967) work to an analog line. See also Gallistel & Gelman 1992, 2000; Dehaene 2003; Cantlon et al. 2009; van Dijk et al. 2015) 3 Carey (2009) offers two main reasons to doubt such mapping-based accounts succeed: first, they do not explain why learning the meaning of number words proceeds in a stepwise manner (Wynn 1990, 1992). Second, while the evidence that such a mapping occurs is strong, there is also evidence that children have an understanding of what number words mean before this mapping occurs. See Carey 2009. 3 fully- fledged capacity for symbolic mathematics.” (Dehaene 1997/2011:x) If Dehaene is right, without precise symbols, there would be no advanced number concepts: Certain structures of the human brain that are still far from understood enable us to use any arbitrary symbol, be it a spoken word, a gesture, or a shape on paper, as a vehicle for a mental representation. Linguistic symbols parse the world into discrete categories. Hence, they allow us to refer to precise numbers and to separate them categorically from their closest neighbours. Without symbols, we might not discriminate 8 from 9. (Dehaene 1997/2011:79) Similarly, offering a philosophically-oriented version of Dehaene’s cultural Darwinism, Helen De Cruz (2008) explains the development of arithmetical abilities by appealing to the effect of cultural selection, variation, and inheritance on the output of innate cognitive systems like the ANS. Unlike Dehaene, however, De Cruz’s Darwinism explicitly adopts Clark and Chalmers’ (1998) extended approach to the mind, which considers the use of extra-cranial artefacts and symbols as essential, constitutive components of some cognitive processes. This externalist framework focuses on the dynamic interaction between individual brains, bodies, and the objects in their environment to understand how objects outside our heads, including culturally accumulated artefacts, can be constitutive parts of cognitive systems. Given that theories of extended cognition challenge the traditional boundaries of cognition, their implications are far reaching and have potential impact on any discipline in which cognition makes extensive use of external supports. This, of course, includes numerical cognition. In fact, one of the first examples mentioned by Clark & Chalmers is that of using pen and paper to solve an arithmetical problem, and Clark’s (2008) monograph opens with an anecdote in which physicist Richard Feynmann describes the paper on which he had done his work as more than a mere record of his thought process, but as an integral part of his work. The fact that the practice of arithmetic seems to rely on mastery of culturally-inherited numeration systems like the Indo-Arabic numerals makes it look like an obvious case of cognition necessarily relying on things outside the head and of culture allowing us to go beyond our innate cognitive limitations by supplying us with material extensions of our mind. This may explain why virtually all of the explanations that have been put forward to bridge the gap between the approximate and limited output of evolutionarily-inherited systems for quantification and the precision and scope of natural numbers adopt an extended approach to cognition. Implicitly or 4 explicitly, answers to the gap problem have generally relied on attributing a constitutive, irreplaceable role to culturally-inherited extracranial objects, artefacts, or symbols in explaining how we move beyond the limits of systems like the ANS and the OFS.4 Here, the idea is that culturally-inherited external representations of numbers – whether in the form of knot systems, body parts, tally systems, number words, or numerals – are constitutive parts of extended cognition systems in which extracranial cognitive aids play an essential part. These external aids to cognition are typically held responsible for our ability to form precise number concepts using systems like the ANS and the OFS as our building blocks. 3 The Induction and the limits of externalism The role of external objects and symbols in the development of numerical cognition is illustrated by the importance of number words in Karen’s Wynn’s celebrated research on the development of numerical abilities (Wynn 1990, 1992),5 which shows that children reliably go through the same stages in learning what numbers are. These stages typically go as follows: the ‘No-Numeral-Knower’ stage describes when children are unable to give even one object when asked to, even though they have memorized a list of number words, in order. Then, typically between 24 and 30 months of age, they correctly give one object when asked to, but fail at any other number, at which point they are ‘One-knowers’. Six to nine months later, they become ‘Twoknowers’, where they can correctly give one object when asked to, or two objects when asked for two, but give random numbers of objects for any number word larger than two. Another, distinct three-knower stage follows.6 4 E.g. Hurford 1987; 1997/Dehaene 2011; Lakoff and Núñez 2000; Wiese 2004; De Cruz 2008; Carey 2011; Coolidge and Overmann 2012; Menary 2015; Malafouris 2010, 2013; Ansari 2008. 5 See also Sarnecka & Lee 2009. Children at these stages are often referred to as ‘subset-knowers’, to highlight the fact that, for 12 to 18 months, they only know how to correctly apply an initial segment of the count list. 6 5 The crucial knowledge stage almost invariably occurs once children have become threeknowers (in some rare cases, after being four-knowers), when what is sometimes called the Induction7 happens: around age 3 1/2 on average, English middle-class children become cardinal principle knowers—they work out the numerical meaning of the activity of counting and can now reliably produce sets with the cardinal value of any numeral in their count list. (Carey 2009: 298) In other words, once the Induction happens, children’s knowledge of the meaning of number words is no longer stuck at individual numbers, and extends to their entire count list. While the knower-stages are well documented, the mystery persists as to what changes occur inside children’s head when the Induction happens: what role do number words play in the Induction, and what effect do they have on systems like the ANS and the OFS, if any, to allow us to learn what numbers are? The claim I want to defend in the rest of this paper is that, to know what happens to the ANS and the OFS that allows us to go through with the Induction, we need to look inside the head, rather than focusing on how we interact with an enculturated environment populated by number words and other cognitive aids, as externalists are prone to do. In a nutshell, the reason I want to claim this is that the Induction happens in our heads despite there being no change in our external environment: when children learn the meaning of the first few number words, they have access to the same cognitive aids throughout their learning process. Given that that the Induction occurs without there being an accompanying change in the extracranial features of the individual’s environment, I claim that whatever happens during the Induction that allows us to understand what numbers are 8 can be described in internalist terms. To illustrate the worth of this internalist 7 E.g. Margolis & Lawrence 2008; Rips et al. 2008b. For criticism see Rips et al. 2008a. I should mention that the Induction does not lead directly to a full understanding of what numbers are. For example, there is evidence that it is possible to become a CP-Knower without understanding exact equality, and vice versa (JaraEttinger et al. 2016). This being said, one can question to which extent the tasks used to determine presence of an understanding of exact equality is one of exact numerical equality in this research. Also, the understanding that there is no largest number and that there are infinitely many of these comes much later than the Induction, if at all (Cheung et al. 2016). This being said, the Induction does allow children to generalize their knowledge of how the count list works to allow them to correctly identify the number of objects presented to them by enumeration for any number in their count list, which is not something they can do before the Induction happens. 8 6 approach, in the next section, I take a look at a research program that has studied the relationship between the development of numerical cognition and attention. 4 Spontaneous Focusing on Numerosity Admittedly, my comments about internalism being a better way to explain how we bridge the gap sound both speculative and vague at this point. Having said that, in this section I want to argue that, on top of the extensive research done by Dehaene and many others into how the brain processes numerical information, there is an active research program that has been studying the origins of numerical cognition for quite some time, whose approach falls in the internalist camp. Since 2000, Hannula-Sormunen and colleagues have been studying what they dubbed Spontaneous Focus on Numerosity, or SFON.9 In a nutshell, SFON describes the tendency to behave in reaction to numerical features of the environment without having been directed to do so (i.e. spontaneously). The aim of SFON studies is to determine if there is a distinct mental process that characterizes paying attention to numerosity versus other features of stimuli, and to which extent such a process is related to arithmetical development. One of the hypotheses motivating this research is that a child’s tendency to attend to numerosity instead of other features of the environment can be considered a stable behavioral trait that should manifest itself across many tasks and over developmental time. The main idea behind SFON studies is that, in order for someone to learn what numbers are, they first have to pay attention to quantities of discrete objects in their environment - the numerosity of collections. If paying attention to numerosity is related to the development of numerical cognition, then people who tend to pay more attention to numerosity than others have more chances of developing an understanding of what numbers are, other things being equal. This means that SFON research could shed some light on why some individuals have less of a hard time learning what numbers are by identifying the process responsible for a person’s attending to numerical aspects of their environment. It also means that if we find ways to help people pay more 9 For a review see Hannula-Sormunen 2015. See also Hannula, 2000; Hannula & Lehtinen, 2001, 2005; Hannula et al. 2010; Hannula-Sormunen et al. 2016. 7 attention to numerosity, we might help then learn what numbers are. SFON studies, as well as the related study of Spontaneous Attention to Number (SAN),10 have tried to hone in on potential effects of individual differences in attention to quantities of discrete objects by attempting to let test subjects choose which aspect of a stimulus they respond to, instead of explicitly telling them to respond to numerical aspects of stimuli. To do this, researchers in SFON studies give participants ambiguous instructions that can be interpreted in many ways. Unlike research that studies children’s numerical abilities via explicit linguistic instructions such as “tell me which pile has more”, or “how many in this card?”, the use of ambiguous directives in SFON studies does not force participants to attend to a particular (numerical) aspect of displays, thereby allowing them to freely attend to any feature, including numerosity, to complete the task. By putting the burden of figuring out which aspect of the stimulus to respond to on the participants, it is possible to single out to which extent some participants have a tendency to react to numerical features of stimuli without being explicitly told to do so (i.e. spontaneously),11 and thus, to get an idea of the level at which they spontaneously attend to numerical features of their environment. This way, researchers can determine which children have a tendency to focus on quantities of discrete objects in displays, versus any of the many other continuous dimensions that can – or, according to critics of the ANS,12 must – co-vary with numerosity. To test the existence of lasting individual differences in SFON, longitudinal studies involving almost 30 different experimental tasks were carried out with the same individuals over extended periods of time,13 to see to which extent individuals’ tendency to attend to numerosity changes over time. For example, researchers will scatter small numbers of objects on a mat in front of them and then instruct children sitting in front of similar mats to “make your mat like mine” without specifying which aspect of the collection of objects they are meant to copy. Participants can then 10 Baroody et al. 2008, 2015. See brief discussion on the relationship between SFON and SAN below. Chen & Mazzocco (2017) doubt that SFON is truly spontaneous given that contextual factors other than instructions can affect where attention is focused. For example, depending on what other features number is pitted against in SFON studies, attention to numerosity will fluctuate, which suggests that it is not entirely spontaneous, since researchers can still guide it by varying non-numerical perceptual dimensions. This being said, they do admit that SFON is only malleable to a certain extent, which suggests that SFON research is still focusing on self-initiated attention to numerosity. 12 See e.g. Leibovich et al. 2017; Gebuis et al. 2016. 13 E.g. 3 years: Hannula & Lehtinen: 2005. See also Hannula-Sormunen 2015. 11 8 match the collection in front of the researcher for numerosity, but also for total area, orientation, color, composition, or other features of the objects in the collection. Or researchers will feed a puppet a specific (small) number of morsels of food and instruct the child to ‘do the same’, monitoring whether the child copies the act of feeding or also copies the specific number of morsels fed. Using such ambiguous instructions, Hannula-Sormunen and colleagues set out to find evidence that SFON exists, that it is stable throughout a person’s development, and that it is related to the development of arithmetical abilities. The data collected strongly suggest that there are indeed individual differences in SFON (Hannula & Lehtinen 2005) and, importantly, that these are correlated with (2005) and even predictors of (Hannula-Sormunen et al. 2010, 2015; Batchelor et al.2015) arithmetical proficiency, though this latter claim has been questioned (Baroody & Li 2015). For example, children aged 37 with a more pronounced tendency to focus on numerosity tend to be better at counting and subitizing than those with less pronounced SFON (Hannula et. al 2005), while children aged 4-5 have been found to perform better on symbolic arithmetical tasks if they have stronger SFON tendencies (Batchelor et al. 2015). 5 An Aside: SFON vs SAN As noted by Baroody and colleagues (Baroody et al. 2008, 2015) one issue with SFON studies, however, is that they mainly focus on children who have already developed some number concepts. This might mean that in SFON tasks, children can match collections based on numerosity in part because they have learned to attend to numerosity when learning the meaning of number words. Since these children have already bridged the gap, they are not ideal candidates to study the processes that allow the child to complete the Induction. To study potential effects of attention to numerosity on how children learn the meaning of number words, Baroody and colleagues (2008, 2015) have offered a variant on SFON, which they call Spontaneous Attention to Number (SAN), that focuses more explicitly on unprompted attention to particular small numbers in children that are learning the meaning of number words (e.g. when children perceive ‘twoness’ but not 9 ‘threeness’).14 Because Baroody and colleagues focus on learning the meaning of the first few number words, their research would appear more adapted to shining light on what happens when children bridge the gap and to which extent culture plays a constitutive role in this development, while Hannula and colleagues’ focus seems to be aimed at children who have already bridged the gap. Regardless of which approach is best, as I discuss in the next section, SFON and SAN studies illustrate well how cultural factors are limited in explaining how we bridge the gap. 6 Culture and SFON Recall that the point of discussing SFON and SAN studies here is to support my claim that internalism is better suited than externalism to explain how we bridge the gap. So, how do SFON studies support internalism? Given that there is reason to believe that individual differences in SFON can predict individual differences in arithmetical development, research into SFON supports my claim that attention to numerosity can be a promising domain of research to explain how we bridge the gap, since the processes underlying SFON are attentional and seem positively related to the development of numerical cognition. Given that attention is an individual-level mental process (Allport 2011), this would seem to suggest that cultural factors are of limited use in this study. 14 It bears mentioning that there is an unusually public debate between Baroody and collaborators on one side and Hannula-Sormunen and colleagues on the other, regarding to which extent SAN is a form of SFON and whether Baroody and colleagues have properly acknowledged the influence of SFON on SAN (Hannula-Sormunen et al. 2015, 2016; Baroody et al. 2015). Though I will remain neutral on this matter here, some alleged differences between the two (according to Baroody and colleagues) could be of interest to some readers. First, while SFON aims to determine to which extent we can detect when individuals respond to numerosity instead of other perceptual dimensions of their experience, SAN focuses more on whether children respond to particular numerosities and not others. In relation to this, SAN uses different data analysis methods, in that it is number-sensitive data, and it also involves younger participants, since it focuses on when children are learning the meaning of number words. SAN studies use nonverbal matching tasks similar to those used in SFON, though since they are matching and not production tasks they require less memory load, since the stimulus is always visible to the participant. One apparently major difference between the two approaches is that SAN researchers claim that SFON ability does not predict numerical abilities. There is also the issue of to which extent SFON needs to be voluntary, or whether it is possible for this spontaneous focusing on numerosity to occur without a participant intending to do so. 10 However, the reader familiar with SFON and SAN will no doubt object that I am proposing an inaccurate characterization of these research programs. After all, both explicitly mention culture as an important factor in shaping an individual’s SFON. Indeed, when discussing the possibility of remedying children’s delayed arithmetical development by enhancing their SFON, Hannula & Lehtinen write that “It would be of a particular interest to broaden our knowledge about the role of cultural environment, as well as that of adults (see Saxe et al. 1987) in how children learn to focus on numerosity and formulate the goals of quantitative tasks in social interaction.” (2005:254).15 So it could appear inaccurate of me to say that SFON research supports my claim that we don’t need to look outside the head to see how we bridge the gap. It is true that there are many factors that can affect SFON and SAN, including the size, composition, and saliency of the features making up the stimuli, the nature of the task,16 as well as an individual’s age and cultural background. There is indeed compelling evidence of the importance of culture-specific factors on the development of numerical cognition, and that the cultural backdrop in which individuals grow up will affect the extent to which they manage to understand what numbers are, as well as the extent to which they pay attention to numerical features of their environment. 17 , 18 One well-known case of cultural influence on numerical cognition is Tang et al.’s (2006) study, which found cultural variation was reflected in how the brain processes numerical tasks, with increased activation in the left premotor cortex for Chinese speakers and increased activation of left perisylvian areas for English speakers. As for SFON, Cantrell and colleagues (2015) found evidence suggesting that cultural factors influence the degree to which children’s behavior in match-to-sample tasks of the sort used in SFON studies is based on numerical features of the stimuli – at least, for larger numerosities. Here, researchers found that Japanese speakers are more likely to attend to non-numerical cues than English speakers for 15 Similarly, Baroody and colleagues write that “children’s understanding and functional use of even the intuitive numbers may not unfold naturally (i.e., readily or spontaneously) but may require scaffolding by parents, early childhood teachers, and others” (2008: 266) and that “a conceptual understanding of number only gradually directs [children’s] attention to collections larger than two because a concept of such numbers must be socially constructed.” (2008:264) 16 E.g. matching vs producing, viewing vs manipulating. See Chen & Mazzaco 2017. 17 Assuming, of course, they do at all. Notwithstanding innovators and creative individuals, most people living in anumerate cultures will never develop an understanding of number. 18 See Menary 2015. 11 numerosities larger than sixteen.19 Data like these suggests culture can influence how we process numerical information, and that culture has an effect on SFON. This is not a surprise: clearly, the things to which people around us pay attention to will influence what we pay attention to. Indeed, it is almost platitudinous to claim that which perceptual dimension we pay attention to depends on cultural and perceptual context. So why am I claiming that SFON illustrates the need to look inside the head to bridge the gap, rather than at cultural factors? Three main reasons motivate my thinking that cultural factors are of limited use in explaining how we bridge the gap: i) behavioral and developmental data suggests many culture-independent processes are involved in bridging the gap; ii) individual differences in SFON are studied against a fixed cultural background; and iii) there is a distinction between knowing what directs attention and knowing how attention affects representational systems like the ANS and the OFS. First, it is important to consider that many of the recent discoveries in the study of numerical cognition involved cognitive systems whose functioning is culture-independent. I mentioned above that there are data showing culture being a factor in determining how much we pay attention to numerosity, but it there is also tremendous amounts of data showing that there are many cultureindependent aspects of numerical cognition. This includes what might be responsible for individual differences in SFON, and whether SFON can help explain how the Induction goes through. For example, while Cantrell and colleagues found some cultural effects on SFON for larger numerosities, it is also important to mention that well-known culture-independent effects were also found, including that children’s attention seems biased towards number for small collections containing 1-4 items, and that attention to numerosity decreases with set size – in both cases, irrespective of cultural background. Similarly, while cultural background can certainly have a large influence on the extent to which an individual pays attention to number and the age at which this happens, there are no culturally19 The explanation given is that speakers of languages like Japanese that do not mark the count-mass distinction as much as English are more likely to attend to continuous properties of stimuli than speakers of languages that emphasize this distinction. However, this link between marking the count-mass distinction and attending to continuous properties has been questioned (e.g. Barner et al. 2009). 12 directed effects on how the induction goes through. For example, as far as I know, there are no cultures where the induction occurs when children learn the meaning of ‘eight’, or ‘twelve’, or ‘two’. For example, while there is data showing that Japanese speakers are slower to become oneknowers,20 this does not change the fact that they go through the same knower stages as children speaking Russian and English (Sarnecka et al. 2007). The same incremental learning trajectory occurs even later for learners living in less industrialized cultures like the Tsiname, where early education is less of a priority than in more industrialized countries (Piantadosi et al 2014). Nevertheless, here too we find that learning the meaning of number words is divided into the same knower stages, and the Induction occurs at the same knower level.21 Similarly, studies of bilingual learners suggest that while there is considerable evidence of language-specific effects on onset times of learning, “the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.” (Wagner et al. 2015:2) Moreover, when we consider the study of number-related deficits like dyscalculia, for example, it is not clear what sort of input cultural factors can have in identifying what the problem is. While it could hypothetically turn out that dyscalculia is more prevalent in certain cultures than others due to culture-specific effects, say, certain eating habits that would have negative impact on the brain’s development –and this seems like quite a stretch – the reference to cultural practices in the explanation of why individuals with dyscalculia have problems processing numerical information would be secondary, in that reference to cultural factors is not required in order to explain what is not working well in these people’s brains. Rather, the active ingredient in the explanation would be relative to individual-level processes that can be described independently of cultural factors. Further, even accepting that culture has important consequences on what individuals pay attention to, as SFON studies show, there are considerable individual differences in SFON between 20 This could potentially be explained by the fact that classifier languages like Japanese do not emphasize the singularplural distinction as much as English does, which means Japanese speakers’ attention is not as solicited by quantityrelated information as English speakers. See Carey 2009 or Sarnecka et al. 2007 for more on this. 21 Such independence of cultural factors prompted Piantadosi and colleagues to conclude that “The presence of a similar developmental trajectory likely indicates that the incremental stages of numerical knowledge – but not their timing — reflect a fundamental property of number concept acquisition which is relatively independent of language, culture, age, and early education.” (2014:1, emphasis added) 13 individuals from the same cultural background, which suggests that individual-level differences in how these people’s heads work must be appealed to if we wish to explain why they pay more attention to number than other individuals. But if our attention is directed to certain aspects of our environment as a result of differences in how our brains work, then it would seem inaccurate to attribute constitutive status to culturally-inherited extracranial cognitive aids in explaining why these differences are there – let alone explanatorily useful – given that the differences are observed against a fixed cultural backdrop. Another reason to doubt that it is explanatorily useful to attribute constitutive status to cultural factors in bridging the gap is that the cultural factors often mentioned in SFON studies are limited to affecting what an individual pays attention to. Culture might give us an idea at where an individual will tend to direct their attention, but it does not explain how attention works nor what SFON can do to help map number words on representations of the ANS, for example. This to be expected, given that attention is something individuals do (e.g. Allport 2011), not cultures. Further, if culture can direct our attention towards numerosity, this does not mean it is the only factor that can do this. After all, the ‘spontaneous’ in SFON refers to the fact that individuals are not explicitly prompted to pay attention to numerosity by others, which suggests that there can be personal motivations for doing so. This, in turn, means that culture is not what determines what an individual pays attention to in specific instances of SFON, which means it would appear limited in its ability to explain how attention to numerosity can affect the development of numerical abilities. The claim being made here is that the relevance of culture in SFON studies is limited to its effects on the extent to which an individual pays attention to numerosity and number in their environment, and that this means that focusing on cultural factors will not allow us to identify the active (psychological) process that modifies systems like the ANS and the OFS due to the effects of attention, since whatever effect culture has on this mechanism stops at the interface between the individual and the outside world, whereas the attention-driven modification occurs inside the head. 14 7 Attention and externalism Despite their focus on extracranial aids to explain how we bridge the gap, externalists do not deny the importance of internal processes of attention and noticing. On the contrary, if we look at some of the externalist accounts considered above, we quickly see that they appear to depend on a form of noticing that leads to the discovery of novel numerical content. For example, Dehaene claims that “[a]ll children spontaneously discover that their fingers can be put into one-to-one correspondence with any set of items” (1997/2011:81). Similarly, Carey’s description of the crucial step where children learn that words in count lists refer to precise quantities of objects centers around attending and noticing: in the course of counting, children discover that when an attended set would be quantified with the dual marker “two,” the count goes “one, two,” and when an attended set would be quantified with the trial marker “three,” the count goes “one, two, three.” The child is thus in the position to notice that for these words at least, the last word reached in a count refers to the cardinal value of the whole set. At this point, the stage is set for the crucial induction. The child must notice an analogy between next in the numeral list and next in the series of models ({i}, {j k}, {m n o}, {w x y z}) related by adding an individual. (Carey 2009:326-7) Thus, on these externalist approaches, an important element in how individuals come to bridge the gap is the presence of a form of realization that follows noticing a correspondence between representations of quantities and objects in the environment. It is uncontroversial that attention is an important part of such noticing. For example, Wu (2014) offers a theory of attention as selection for task in which what we pay attention to determines what sorts of perception-based beliefs we end up forming, since what we pay attention to determines what we notice (e.g. Wu 2014:249).22 Indeed, the idea that attention is necessary for the development of numerical content is neither new nor particularly controversial, as seen by the fact that a number of authors have already appealed to general attentional mechanisms to explain how infants get numerical content from one-to-one correspondence on the output of the OFS. For example, Simon (1997) offers “a “nonnumerical” account that characterizes infants’ competencies with regard to numerosity as emerging primarily from some general characteristics of the human perception and attention system.” (Simon 1997:349), while Izard and colleagues similarly propose that “infants may also 22 Similarly, Hannula and colleagues write that “The numerosity of items depends on the way one carves up the set of items and, thus, on the goal of quantification.” (Hannula et al. 2005:238) 15 be able to use their attentional resources to extract numerical information from displays containing only a small number of objects.” (Izard et al. 2009: 492) Such explanations of infant behavior in terms of attentional mechanisms have been well received and lend credence to an internalist account where attentional learning can lead to the development of novel representational content. While the precise role of attention in the development of numerical content has yet to be identified, there is little disagreement that it is an essential component to the process. However, attention on its own would be far too general to be specifically responsible for novel numerical content. After all, it could be argued that attention is necessary for any action (e.g. Wu 2014, but see Jennings & Nanay 2014 who disagree), and that it is too general and poorly delineated a theoretical construct to warrant its ubiquity in psychological explanation (Walsh 2003). However, SFON is a much more specific construct than is attention, and, in that sense, is a more promising alternative that can explain individual and cultural differences in numerical abilities. For example, the internalist can appeal to cultural effects on SFON (or lack thereof) to explain why anumerate cultures like the Piraha (Frank et al. 2008) and the Mundurucu (Pica et al. 2004, 2008) do not develop number concepts: their lifestyle and social organization does not require nor encourage paying close attention to quantities of objects. This means that SFON is not encouraged in such cultures and that the potential modification of systems like the ANS that could be caused by SFON do not occur. Virtually every author that has speculated about the historical origins of numerical cognition has agreed that specific cultural conditions were required for this to happen. More specifically, concepts for numbers and precise quantities must have emerged in societies where keeping precise track of quantities of discrete objects was not only useful, but had lasting utility for those who managed to do it. For example, Gelman and Butterworth speculate that anumerate cultures do not develop words for precise quantities because “numbers are not culturally important and receive little attention in everyday life” (Gelman & Butterworth 2005: 9) in such cultures. The importance of cultural context can also be explained by reference to attention to quantity, since trade-oriented cultures would supply a context where paying attention to precise quantities of objects would be 16 valued, which differentiates numerate cultures from those where “numbers are not culturally important and receive little attention in everyday life” (Gelman & Butterworth 2005: 9). 8 The explanatory limits of cultural evolution and extended cognition These are, of course, speculative comments about the potential effects of SFON on other cognitive systems, aimed at illustrating how an internalist approach can help bridge the gap without attributing a constitutive role to external artefacts. Even if it were to turn out that SFON does not have such effects, the point is that individual differences and culture-independent processes involved in bridging the gap warrant paying closer attention to what goes on in our heads, rather than at what we inherit from our cultural background. It is important to emphasize here that I am not denying the importance of enculturation for the development of bodies of knowledge like mathematics, nor am I claiming that externalism cannot explain many important aspects of numerical cognition. Clearly, no single person could ever accomplish what we do as an enculturated species. And it is just as clear that – at least for large enough numbers – mastery of culturally-developed numeration systems is an integral part of numerical cognition (Schlimm 2018). But acknowledging the importance of enculturation for the development of mathematics and arithmetic does not necessarily always explain it. I have tried to argue that the origins of numerical cognition – in particular, how children manage to learn what numbers are via the Induction – is best left to internalist approaches. The fact that humans gradually accumulate innovations over generations is indeed a cultural process, one that is undoubtedly responsible for the incredible achievements of mathematics, and science as a whole. There is no question that numerical content did emerge in an enculturated context, perhaps due to the demands of increasingly complex commercial exchanges and practices which benefitted from keeping tallies on precise quantities. But the fact that an event or practice takes place within an cultural context does not mean that enculturation can meaningfully explain the origins of this practice. Also, this cultural evolution relies on individuals responding to their (cultural) environment and building on it. It is this building process, the generation of novel content in an individual’s head, that needs to be explained in our case. For this, appealing to the fact that 17 human cognition extends into the environment to include cultural processes is limited in its explanatory power. A good way to illustrate that it is not always explanatorily rewarding to appeal to culture or extended cognition, irrespective of whether or not they describe how our minds work, is to consider our digestive system. Citing Wrangham (2009), Sterelny (2010) points out that our jaws – and, relatedly, our brains – have evolved to their present shape because because we started cooking our food a very long time ago.23 In a sense, then, our digestive system could be described as being extended beyond our stomach, given that we essentially rely on external artefacts to process, select, ferment, and store food in ways that have profoundly affected our digestive system and jaw: “[t]he physiological demands on hominin jaws, teeth and guts have been transformed by cooking and more generally by food preparation and food targeting…[w]e are obligatorily cooks. Moreover, we supplement cooking by pre-engineering our food sources. (Sterelny 2010:467) Similarly, given the fact that cooking requires fire, and that keeping a fire going while hunting prey is not a task fit for a single individual, one could argue that the shape of our jaw and the ensuing increase in brain size that followed the jaw’s gradual evolution are the results of enculturation, since such changes to our cooking practices could only have occurred in societies where the right tools and social hierarchies were developed. And yet, despite the fact that our digestive system is entirely dependent on technology that has allowed us to pre-process our food outside our body for thousands of generations, this information is not required to explain why certain foods cause heartburn, or other facts about how our digestive system works: “there is no explanatory mileage in treating my soup pot as part of my digestive system, once its importance as a scaffold is recognised” (Sterelny 2010: 468). Nor does our essential reliance on external artefacts for our food intake mean that such technology is part of our digestive system. As Sterelny put it in this delicious passage: [o]ur digestion is, then, technologically supported in profound and pervasive ways…We have engineered our gustatory niche; we have transformed both our food sources and the process of eating itself. Our under-powered jaws, short gut, small teeth and mouth fit our niche because we eat soft, rich and easily digested food. Our digestive system is 23 Sterelny mentions that while estimates locate the advent of cooking food anywhere between one and two million years ago, there is no doubt that we have been cooking our food for at least 400 000 years. 18 environmentally scaffolded. But is my soup pot, my food processor and my fine collection of choppers part of my digestive system? As far as I know, no one has defended an extended stomach hypothesis, treating routine kitchen equipment as part of an agent’s digestive system; indeed “extended stomach” and “extended digestion” on Google Scholar return some very strange hits. (Sterelny 2010:467-8) 9 Conclusion In this paper I have tried to argue that research into one of the most important puzzles in the study of numerical cognition – how systems like the ANS and the OFS allow us to develop an understanding of what numbers are – should focus on what goes on inside our head rather than on our cultural background. To argue for this, I discussed research into SFON and its possible relation to individual differences in arithmetic abilities and argued that cultural factors are not explanatorily useful in this domain. While the focus of SFON research is not directly aimed at explaining how the Induction goes through, it does certainly support the claim that an intracranial process – in this case, attention to numerosity – is positively related to the development of arithmetical abilities. It also illustrates how, in some cases, in studying what happens inside the head, we don’t need to look outside. I argued that the Induction that allows us to bridge the gap is one such case, given that many aspects of the learning trajectory children go through in learning what numbers are are culture-independent, and that individual differences within cultures are also culture-independent. Of course, most externalists does not deny the central role played by the brain.24 But externalism seems like a counterproductive approach when it says that we can appeal to culturally-inherited cognitive aids to identify what bridges the gap, or that we bridge the gap because of cultural factors. As I tried to argue, while culture can indeed increase our chances of developing numerical content, the reasons for culture’s influence on how we bridge the gap can be explained using vocabulary about how the brain works, and how attention to numerosity affects systems like the ANS and the OFS. If I am right, while it is true that we rely on external objects (and people) for our daily cognitive regime, including learning and using numerical content, this does not mean that we can explain the emergence of novel content in a person’s head by appealing to these general facts about how our minds work. It is not always helpful to appeal to the external aspects of our minds to explain particular phenomena, even if our minds are indeed extended, embodied, or enculturated. 24 Menary (2015) and Malafouris (2013) can be read as denying the central importance of the brain in cognition. 19 REFERENCES Allport, A. (2011). Attention and Integration. In Mole, C., Smithies, D., & Wu, W. 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