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Citation Hiraiwa, Ken. “The Faculty of Language Integrates the Two Core Systems of Number.” Frontiers in Psychology 8 (March 2017): 351 © 2017 Hiraiwa As Published http://dx.doi.org/10.3389/fpsyg.2017.00351 Publisher Frontiers Research Foundation Version Final published version Citable link http://hdl.handle.net/1721.1/110094 Terms of Use Creative Commons Attribution 4.0 International License Detailed Terms http://creativecommons.org/licenses/by/4.0/ PERSPECTIVE published: 16 March 2017 doi: 10.3389/fpsyg.2017.00351 The Faculty of Language Integrates the Two Core Systems of Number Ken Hiraiwa 1, 2* 1 Department of English, Meiji Gakuin University, Tokyo, Japan, 2 Department of Linguistics and Philosophy, Massachusetts Institute of Technology, Cambridge, MA, USA Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions (Hauser et al., 2002; Berwick and Chomsky, 2015). Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity (Gelman and Gallistel, 1978; Dehaene, 1997). Thus, the origin of this numerical capacity and its relation to language have been of much interdisciplinary interest in developmental and behavioral psychology, cognitive neuroscience, and linguistics (Dehaene, 1997; Hauser et al., 2002; Pica et al., 2004). Hauser et al. (2002) and Chomsky (2008) hypothesize that a recursive generative operation that is central to the computational system of language (called Merge) can give rise to the successor function in a set-theoretic fashion, from which capacities for discretely infinite natural numbers may be derived. However, a careful look at two domains in language, grammatical number and numerals, reveals no trace of the successor function. Following behavioral and neuropsychological evidence that there are two core systems of number cognition Edited by: Carlo Semenza, innately available, a core system of representation of large, approximate numerical University of Padua, Italy magnitudes and a core system of precise representation of distinct small numbers Reviewed by: (Feigenson et al., 2004), I argue that grammatical number reflects the core system of Frank Domahs, Philipps University of Marburg, precise representation of distinct small numbers alone. In contrast, numeral systems Germany arise from integrating the pre-existing two core systems of number and the human Xinlin Zhou, language faculty. To the extent that my arguments are correct, linguistic representations Beijing Normal University, China of number, grammatical number, and numerals do not incorporate anything like the *Correspondence: Ken Hiraiwa successor function. [email protected] Keywords: natural language, number, natural numbers, numerals, core systems of number, grammatical number, syntax, linguistics Specialty section: This article was submitted to Language Sciences, a section of the journal INTRODUCTION Frontiers in Psychology Only humans possess the faculty of language that allows an infinite array of hierarchically Received: 16 December 2016 structured expressions (Chomsky, 1995; Miyagawa et al., 2013, 2014; Berwick and Chomsky, Accepted: 23 February 2017 2015). Similarly, humans have a capacity for infinite natural numbers, while all other species seem Published: 16 March 2017 to lack such a capacity (Hauser et al., 2002; Chomsky, 1982, 1986; for studies on the capacity Citation: for number, see Gelman and Gallistel, 1978; Wynn, 1992a,b; Dehaene, 1993, 1997; Butterworth, Hiraiwa K (2017) The Faculty of Language Integrates the Two Core 1999; Pica et al., 2004). This unique capacity is obviously what has made the development of Systems of Number. sophisticated mathematics possible (Hauser and Watumull, in press). Common to both faculties Front. Psychol. 8:351. is the use of finite means to achieve discrete infinity, that is, an open-ended array of discrete doi: 10.3389/fpsyg.2017.00351 expressions (von Humboldt, 1836; Chomsky, 1965, 2007a,b, 2008, 2010). Thus, the origin of this Frontiers in Psychology | www.frontiersin.org 1 March 2017 | Volume 8 | Article 351 Hiraiwa The Faculty of Language and Number numerical capacity and its relation to language have been of arisen by integrating the two pre-existing core systems with the much interdisciplinary interest in developmental and behavioral recursive combinatorial computation Merge, which is unique to psychology, cognitive neuroscience, and linguistics (Dehaene, the human language faculty. 1997; Hauser et al., 2002; Pica et al., 2004; Gelman and (1) a. Grammatical number reflects the core system of precise Butterworth, 2005). representation of distinct small numbers alone. Some developmental psychologists have suggested that the b. Numeral systems reflect both of the core systems of concepts of natural numbers are innate to humans (Gelman and number and Merge. Gallistel, 1978; Wynn, 1992b; Dehaene, 1997). Chomsky (2008, p. 139) hypothesizes that Merge can give rise to the successor Consequently, the concepts of natural numbers and its realization function (i.e., every numerosity N has a unique successor, N + 1) in language are distinct and language interfaces with the two core in a set-theoretic fashion (1 = one, 2 = {one}, 3={one, {one}}, ...) systems of number. and that the capacity for discretely infinite natural numbers may be derived from this. Merge is central to the generative system NUMERICAL NOTATIONS: A SIMPLE of language (Chomsky, 2008). It is a set-theoretic recursive combinatorial operation that takes two objects X and Y and forms EXAMPLE OF THE CORE SYSTEM OF {X, Y} (e.g., two items the and dog are combined to form a set PRECISE REPRESENTATION OF SMALL {the, dog}, which can then be further combined with another NUMBERS item saw to form a set {saw, {the, dog}}). “Operating without bounds, Merge yields a discrete infinity of structured expressions” Around the world, natural numbers have often been represented (Chomsky, 2007a, p. 5). Hauser et al. (2002, p. 15) also suggest by numerical-notation systems (Menninger, 1969; Ifrah, 1985, that “in parallel with the faculty of language, our capacities for 2000). Dehaene (1997, p. 54) observes that many numerical number rely on a recursive computation.” notation systems denote the first three or four numbers by a In natural languages, number most clearly emerges in two specific analog number of identical marks, and the following domains: grammatical number (Corbett, 2000 and references numbers by essentially arbitrary symbols (e.g., I, II, III, IV, V and therein) and numerals (Stampe, 1976; Corbett, 1977; Greenberg, 一, 二, 三, 四, 五 in Roman and Chinese/Japanese notations for 1978; Comrie, 2005a,b; Kayne, 2010). As I will show, however, “1”–“5”). He argues that the limit to “3” or “4” follows from the representations of number in natural languages do not reveal core system of precise representation of distinct small numbers. any straightforward trace of the successor function. This leads It would not be expected if numerical notations reflected the us to a central question of this article: how natural numbers successor function. are linguistically represented and how such representations are related to other cognitive systems, if any. GRAMMATICAL NUMBER ALSO There has been much behavioral and neuropsychological evidence that there are two core systems of number cognition that REFLECTS THE CORE SYSTEM OF are innately available (Xu and Spelke, 2000; Carey, 2001, 2009; PRECISE REPRESENTATION OF SMALL Xu, 2003; Feigenson et al., 2004; McCrink and Wynn, 2004). NUMBERS The first is a system of approximate representation of numerical magnitude, which allows one to compare and discriminate large, If the two core systems of number are innate to humans, one approximate numerical magnitudes. The second is a system of also expects to find some similar trace of these systems within precise representation of distinct small numbers: 1, 2, 3, possibly natural-language syntax. 4. This system allows one to compare and discriminate small Grammatical number is the grammatical coding of numerical numbers of individuals. The analog approximate-magnitude quantity. Some languages overtly mark number on nouns (e.g., system has a ratio limit of 1:2. Experiments show that 6-month- I saw the dog (singular) vs. I saw the dog-s (plural) in English). old infants can discriminate numerosities of 8 and 16 and According to Corbett (2000), grammatical-number systems only numerosities of 16 and 32, where the ratio is 1:2, but they fail come in three varieties. English represents the most common, to discriminate numerosities of 8 and 12 and of 16 and 24, where a singular-plural system that distinguishes “1” and “more than the ratio is 2:3 (Xu and Spelke, 2000; Barth et al., 2003; Xu, 2003; 1.” The second most common system is a singular-dual-plural Feigenson, 2007). In contrast, the small-numbers system has a set system, which distinguishes “1,” “2,” and “more than 2” (as in size limit of 3 or 4. Experiments show that 10- and 12-month- Hopi; see Hale, 1997). Finally, a trial system, although quite old infants can identify the larger of 1 and 2 and the larger of rare cross-linguistically, involves a four-way distinction, singular- 2 and 3, while they fail to discriminate between large numbers dual-trial-plural, distinguishing “1,” “2,” “3,” and “more than 3” (Starkey and Cooper, 1980; Feigenson et al., 2002a,b; Xu, 2003; (as in Larike; see Laidig and Laidig, 1990; Corbett, 2000). Carey, 2004; also Barner et al., 2008). A “quadral” system, in which the precise cardinalities I propose that the grammatical-number system and the 1 through 4 are all distinguished, is reported but quite numeral system constitute linguistic evidence for these two core controversial: according to Corbett (2000, p. 30) and Dixon systems of number representation. The grammatical-number (2010), the “quadral” number in such systems should actually system reflects the system of precise representation of distinct be analyzed as paucal (i.e., denoting an approximate, relatively small numbers alone.