Number Nativism is the view that humans innately represent precise natural numbers. Despite a long and venerable history, it is often considered hopelessly out of touch with the empirical record. I argue that this is a mistake. After clarifying Number Nativism and distancing it from related conjectures, I distinguish three arguments which have been seen to refute the view. I argue that, while popular, two of these arguments miss the mark, and fail to place pressure on Number Nativism. Meanwhile, a (...) third argument is best construed as a challenge: rather than refuting Number Nativism, it challenges its proponents to provide positive evidence for their thesis and show that this can be squared with apparent counterevidence. In response, I introduce psycholinguistic work on The Tolerance Principle (not yet considered in this context), propose that it is hard to make sense of without positing precise and innate representations of natural numbers, and argue that there is no obvious reason why these innate representations couldn’t serve as a basis for mature numeric conception. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...) along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...) initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind-Peano axioms in other terms. (shrink)

How can we acquire a grasp of cardinal numbers, even the first very small positive cardinal numbers, given that they are abstract mathematical entities? That problem of cognitive access is the main focus of this paper. All the major rival views about the nature and existence of cardinal numbers face difficulties; and the view most consonant with our normal thought and talk about numbers, the view that cardinal numbers are sizes of sets, runs into the cognitive access problem. The source (...) of the problem is the plausible assumption that cognitive access to something requires causal contact with it. It is argued that this assumption is in fact wrong, and that in this and similar cases we should accept that a certain recognise-and-distinguish capacity is sufficient for cognitive access. We can then go on to solve the cognitive access problem, and thereby support the set-size view of cardinal numbers, by paying attention to empirical findings about basic number abilities. To this end some selected studies of infants, pre-school children and a trained chimpanzee are briefly discussed. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. (...) By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

The lack of consensus on how to characterize humans’ capacity for belief reasoning has been brought into sharp focus by recent research. Children fail critical tests of belief reasoning before 3 to 4 years (Wellman, Cross, & Watson, 2001; Wimmer & Perner, 1983), yet infants apparently pass false belief tasks at 13 or 15 months (Onishi & Baillargeon, 2005; Surian, Caldi, & Sperber, 2007). Non-human animals also fail critical tests of belief reasoning but can show very complex social behaviour (e.g., (...) Call & Tomasello, 2005). Fluent social interaction in adult humans implies efficient processing of beliefs, yet direct tests suggest that belief reasoning is cognitively demanding, even for adults (e.g., Apperly, Samson & Humphreys, 2005). We interpret these findings by drawing an analogy with the domain of number cognition, where similarly contrasting results have been observed. We propose that the success of infants and non-human animals on some belief reasoning tasks may be best explained by a cognitively efficient but inflexible capacity for tracking belief-like states. In humans this capacity persists in parallel with later-developing, more flexible but more cognitively demanding theory of mind abilities. (shrink)

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)

Beck presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey. According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system, which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to (...) any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system. Since the ANS-based account offers a potential alternative for integer concept acquisition, I show that it provides a good reason to revisit the deviant-interpretation challenge. Finally, I will present a hybrid OTS-ANS model as the foundation of integer concept acquisition and the framework of enculturation as a solution to the challenge. (shrink)

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific (...) structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symbol‐symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots‐number word matching task—thought to be a measure of symbol‐magnitude associations (numerical magnitude processing)—a numeral‐ordering task that focuses on (...) symbol‐symbol associations (numerical order processing), and a digit‐number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain‐general factors (intellectual ability, working memory, inhibitory control, and non‐numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults’ arithmetic skills build upon symbol‐magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition. (shrink)

In this article we review the discussion over the thesis that language serves as an integrator of contents coming from different cognitive modules. After presenting the theoretical considerations, we examine two strands of empirical research that tested the hypothesis — spatial cognition and mathematical cognition. The idea shared by both of them is that each is composed of two separate modules processing information of a specific kind. For spatial thinking these are geometric information about the location of the object and (...) the information about the object’s properties such as color or size. For mathematical thinking, they are the absolute representation of small numbers and the approximate representation of numerosities. Language is said to integrate the two kinds of information within each of these domains, which the reviewed data demonstrates. In the final part of the paper, we offer some comments on the theoretical side of the discussion. (shrink)

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a (...) few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line. (shrink)

A theory of conceptual development must specify the innate representational primitives, must characterize the ways in which the initial state differs from the adult state, and must characterize the processes through which one is transformed into the other. The Origin of Concepts (henceforth TOOC) defends three theses. With respect to the initial state, the innate stock of primitives is not limited to sensory, perceptual, or sensorimotor representations; rather, there are also innate conceptual representations. With respect to developmental change, conceptual development (...) consists of episodes of qualitative change, resulting in systems of representation that are more powerful than, and sometimes incommensurable with, those from which they are built. With respect to a learning mechanism that achieves conceptual discontinuity, I offer Quinian bootstrapping. TOOC concludes with a discussion of how an understanding of conceptual development constrains a theory of concepts. (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist (...) hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...) 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end (...) states. A survey of these proposals, however, shows that they do not succeed in bridging the gap to knowledge of the integers. We suggest that a better theory depends on starting with primitives that are inherently structural and mathematical. (shrink)

In the paper, being the second part of the work entitled Noema and Noesis, the formal model of the noematic synthesis functions is presented. Together with functions of noetic synthesis, they are understood as components of functions of intentional reference, which are to be, in turn, formalizations of intentional acts of reference performed in the stream of consciousness. Noemata are understood as mental representations associated with mental worlds. The processes of their synthesis in the mind engage the work of many (...) noematic functions: generating predicative senses, producing noematic cores, and locating noemata in mental worlds (mental spaces). The constructed model is not intended as the faithful reconstruction of the Husserlian conception of noemata. Intuitions of the creator of phenomenology are treated as the only source of inspiration. (shrink)

The commentators raised issues relevant to all three important theses of The Origin of Concepts (henceforth TOOC). Some questioned the very existence of innate representational primitives, and others questioned my claims about their richness and whether they should be thought of as concepts. Some questioned the existence of conceptual discontinuity in the course of knowledge acquisition and others argued that discontinuity is much more common than was portrayed in TOOC. Some raised issues with my characterization of Quinian bootstrapping, and others (...) questioned the dual factor theory of concepts motivated by my picture of conceptual development. (shrink)

Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, (...) L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n=32) and adult speakers of English (n=32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language Acquisition, 11(4), 197-218]. In doing so, this work ties together research on the acquisition of the number vocabulary with a growing body of work on the development of quantification and sentence processing abilities in young children [Geurts, 2003; Lidz, J., Musolino, J. (2002). Children's command of quantification. Cognition, 84, 113-154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into children. Language Acquisition, 11(4), 277-291; Trueswell, J., Sekerina, I., Hilland, N., Logrip, M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young children. Cognition, 73, 89-134; Noveck, I. (2001). When children are more logical than adults: Experimental investigations of scalar implicature. Cognition, 78, 165-188; Noveck, I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every. . . not sentences. Journal of Semantics,24(1), 73-90. On a more general level, this work confirms the importance of integrating formal and developmental perspectives [Musolino, 2004], this time by highlighting the explanatory power of linguistically-based models of language acquisition and by showing that the complex structure postulated by linguists has important implications for developmental accounts of the number vocabulary. (shrink)

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I (...) argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality. (shrink)

The sociology of knowledge is a heterogeneous set of theories which generally focuses on the social origins of meaning. Strong arguments, epitomized by Durkheim's late work, have hypothesized that the very concepts our minds use to structure experience are constructed through social processes. This view has come under attack from theorists influenced by recent work in developmental psychology that has demonstrated some awareness of these categories in pre-socialized infants. However, further studies have shown that the innate abilities infants display differ (...) in systematic and theoretically significant ways from adults' explicit knowledge. This paper moves beyond the constructionist/nativist dichotomy by outlining the complex relationships between innate intelligence and explicit knowledge. I end by suggesting that there are four, distinct ways the social world influences thought- facilitation, division, specification, and construction. (shrink)

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of (...) contrasting alternatives, children assign them a meaning distinct from some. In the “Don’t Give-a-Number task”, children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast – plus knowledge of a number’s membership in a count list – enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children’s interpretation of unknown numbers is further constrained by understanding numerical entailment relations – that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number inbetween – who either has or doesn’t have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words. (shrink)

Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other mathematical concepts, and then fix the application (...) conditions based on those hypotheses—under the assumption that the neo‐logicist view on abstraction principles is correct. (shrink)

Rips et al.'s critique is misplaced when it faults the induction model for not explaining the acquisition of meta-numerical knowledge: This is something the model was never meant to explain. More importantly, the critique underestimates what children know, and what they have achieved, when they learn the cardinal meanings of the number words through.