Core Knowledge, Language, and Number

The natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a third proposal that builds on aspects of these views but rejects one tenet that they share: the thesis that counting is central to number. I suggest that children discover the natural numbers when they learn a natural language: especially nouns, number words, and the rules that compose quantified noun phrases. This learning, in turn, depends both on cognitive systems that are innate and shared by other animals, and on our species-specific language faculty. Thus, natural number concepts are unique to humans and culturally universal, yet they are learned. Natural number concepts may be our simplest abstract ideas. These concepts are exceedingly useful, serving as a basis for measurement, money, and mathematics, orienting us in space and time, and structuring activities from sports to elections. Likely because of their simplicity and ubiquity, the development of natural number concepts has been richly studied since the landmark research of Jean Piaget (1952) and the enduring challenges to his theory that followed (e.g., Gelman, 1972; Mehler & Bever, 1967; Siegal, 1999). Despite the simplicity of these concepts and the large body of research probing their development, however, the psychological foundations of natural number concepts continue to be debated. In the context of this debate, it is useful to characterize the natural number system in three interconnected ways. First, there is a minimal unit, ONE, that corresponds to the smallest distance separating distinct numbers (hereafter, the UNIT principle). Second, natural numbers can be generated by successive addition of one (hereafter, the principle of SUCCESSION). Third, two sets whose members can be placed in one-to-one correspondence have the same cardinal value: they are equal in number (hereafter, the principle of EXACT EQUALITY). Here I consider three general accounts of the development of this system of concepts, in relation to research that focuses on key aspects of these principles. According to the first account, natural number concepts are part of human nature. They are ancient: they evolved in distant ancestors, and so we share them with other animals. They are innate and begin functioning early in development: their emergence is not shaped by encounters with sets CONTACT Elizabeth S. Spelke spelke@wjh.harvard.edu Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138. In the literature on numerical development in children, the acquisition of “the successor principle” refers to a milestone in children’s mastery of counting: the point at which children understand that every word in their verbal count list refers to a cardinal value that is one larger than the value designated by the previous word. In some discussions, mastery of the successor principle of verbal counting is considered as a criterion for mastery of the natural number system (e.g., Sarnecka, 2016). Use of this criterion, however, would beg the present question. Because numerical language is learned, natural number concepts that were defined by children’s learning of number word meanings necessarily would be learned, ruling out Gelman and Gallistel’s nativist claims by definition and rendering Carey’s claims true by definition. On pain of circularity, experiments testing these theories and others require characterizations of numerical concepts that are independent of children’s learning of language. © 2017 Taylor & Francis Group, LLC LANGUAGE LEARNING AND DEVELOPMENT 2017, VOL. 13, NO. 2, 147–170 http://dx.doi.org/10.1080/15475441.2016.1263572 D ow nl oa de d by [ Sm ith so ni an A st ro ph ys ic s O bs er va to ry ] at 1 4: 27 2 5 A ug us t 2 01 7 and their transformations but instead serves to structure those encounters. And, these concepts are present in all human cultures: later development builds on them but does not overturn them, so they are available to children and adults everywhere. According to the second account, natural number concepts depend on a specific product of culture: a counting procedure. These concepts are recent and unique to humans, because the first counting procedure appears to have been invented relatively late in human prehistory. They are learned: indeed, contemporary children master counting procedures slowly and with difficulty. And they are culturally variable: different human groups count in different ways and to different extents, and some groups do not count at all. Drawing on old and new research, I suggest that neither of these theories captures the development of natural number concepts, and I sketch a third account of their emergence. I propose that natural number concepts arise through the productive combination of representations from a set of innate, ancient, and developmentally invariant cognitive systems: systems of core knowledge. In particular, natural number concepts depend on a system for representing sets and their approximate numerical magnitudes (hereafter, the Approximate Number System (ANS)) and a set of systems that collectively serve to represent objects as members of kinds. None of these core systems is unique to humans, but their productive combination depends on the acquisition and use of a natural language. Because both the core systems and the language faculty are universal across humans, and because children master their native language spontaneously, natural number concepts emerge universally, with no formal or informal instruction. Because language is unique to humans, so is our grasp of the natural numbers. Finally, because specific natural languages are learned, the system of natural number concepts is neither innate nor present in the youngest children. I begin by describing the first core system, the ANS. Then I turn to Gelman and Gallistel’s theory, considering both its virtues and the problems it faces in accounting for some prominent limits to the numerical reasoning of young children. These limits suggest that children’s earliest numerical representations fail to capture two of the three key principles that characterize the natural numbers: principles of exact equality and succession. That suggestion, in turn, motivates Carey’s account of the emergence of natural number concepts. After considering some strengths of this account, I turn to findings that raise problems for it. Then I suggest how language learning, rather than either the innate unfolding of a genetic program or the acquisition of culture-specific counting devices, might allow children to discover the natural numbers. A core system of number From newborn infants to professional mathematicians, humans spontaneously represent the cardinal values of sets of objects or sequences of events with ratio-limited precision (see Dehaene, 2011, for review). Experiments conducted on newborn infants serve to illustrate this ability (Izard, Sann, Spelke, & Streri, 2009). Infants in a maternity hospital were familiarized with sequences of syllables. The particular syllables changed from one sequence to the next, as did the duration of the syllable sequences, but the number of syllables was constant—4 long syllables per sequence for half the infants and 12 shorter syllables per sequence for the others. While infants listened to the sequences, alternating visual arrays appeared, containing 4 or 12 objects of variable but comparable shapes and sizes. Infants looked reliably longer at the object array that corresponded in number to the sequence of sounds. Because the visual and auditory arrays differed in modality and format, and could not be matched on the basis of continuous, extensive, or intensive quantities such as contour length, sequence duration, or item size, these looking patterns provide evidence that infants detected the numerical correspondence between these arrays, and therefore the numerical distinction between 4 and 12. Further experiments revealed that newborn infants also distinguish 6 from 18 but not 4 from 8. Beginning in the first days after birth, the ability to discriminate between two numbers depends on their ratio. Do newborn infants relate sequences of syllables to arrays of visual forms through noisy processes of one-to-one correspondence, applied to the individual members of these sets? Alternatively, do 148 E. S. SPELKE D ow nl oa de d by [ Sm ith so ni an A st ro ph ys ic s O bs er va to ry ] at 1 4: 27 2 5 A ug us t 2 01 7 infants represent the sequences and arrays as ensembles with approximate numerical magnitudes, matching sets with similar magnitudes? Several findings indicate that the latter representations underlie infants’ performance. Processing of individual objects shows a set size limit—most adults cannot hold more than four objects in mind at once, and infants are limited to fewer objects than this (Oakes, Baumgartner, Barrett, Messenger, & Luck, 2013)–but infants respond to number in much larger arrays. Indeed, infants often fail to represent the numerical sizes of very small sets of objects. For example, newborn infants fail to match sequences of 2 sounds to arrays of 2 rather than 6 objects when tested under the same conditions that reveal matching of sequences of 4 sounds to arrays of 4 rather than 12 objects (Coubart, Izard, Spelke, Marie, & Streri, 2014). When presented with very small sets, infants and adults alike typically focus attention on the individual objects and suppress representations of the set’s numerical magnitude (Hyde & Spelke, 2009, 2011), although both adults and infants can focus on the numerical values of small sets if presentation conditions make individual objects difficult to track (Hyde & Wood, 2011; Starr, Libertus, & Brannon, 20