This system allows one to compare and discriminate small numbers of individuals. The second is a system of precise representation of distinct small numbers: 1, 2, 3, possibly 4. The first is a system of approximate representation of numerical magnitude, which allows one to compare and discriminate large, approximate numerical magnitudes. There has been much behavioral and neuropsychological evidence that there are two core systems of number cognition that are innately available ( Xu and Spelke, 2000 Carey, 2001, 2009 Xu, 2003 Feigenson et al., 2004 McCrink and Wynn, 2004).
This leads us to a central question of this article: how natural numbers are linguistically represented and how such representations are related to other cognitive systems, if any. As I will show, however, representations of number in natural languages do not reveal any straightforward trace of the successor function. In natural languages, number most clearly emerges in two domains: grammatical number ( Corbett, 2000 and references therein) and numerals ( Stampe, 1976 Corbett, 1977 Greenberg, 1978 Comrie, 2005a, b Kayne, 2010). 15) also suggest that “in parallel with the faculty of language, our capacities for number rely on a recursive computation.” “Operating without bounds, Merge yields a discrete infinity of structured expressions” ( Chomsky, 2007a, p. 139) hypothesizes that Merge can give rise to the successor function (i.e., every numerosity N has a unique successor, N + 1) in a set-theoretic fashion (1 = one, 2 = ). Some developmental psychologists have suggested that the concepts of natural numbers are innate to humans ( Gelman and Gallistel, 1978 Wynn, 1992b 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 Gelman and Butterworth, 2005). Common to both faculties is the use of finite means to achieve discrete infinity, that is, an open-ended array of discrete expressions ( von Humboldt, 1836 Chomsky, 1965, 2007a, b, 2008, 2010). This unique capacity is obviously what has made the development of sophisticated mathematics possible ( Hauser and Watumull, in press). Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity ( Hauser et al., 2002 Chomsky, 1982, 1986 for studies on the capacity for number, see Gelman and Gallistel, 1978 Wynn, 1992a, b Dehaene, 1993, 1997 Butterworth, 1999 Pica et al., 2004). Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions ( Chomsky, 1995 Miyagawa et al., 2013, 2014 Berwick and Chomsky, 2015). To the extent that my arguments are correct, linguistic representations of number, grammatical number, and numerals do not incorporate anything like the successor function. In contrast, numeral systems arise from integrating the pre-existing two core systems of number and the human language faculty. Following behavioral and neuropsychological evidence that there are two core systems of number cognition innately available, a core system of representation of large, approximate numerical magnitudes and a core system of precise representation of distinct small numbers ( Feigenson et al., 2004), I argue that grammatical number reflects the core system of precise representation of distinct small numbers alone.
However, a careful look at two domains in language, grammatical number and numerals, reveals no trace of the successor function. (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. 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).
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). Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions ( Hauser et al., 2002 Berwick and Chomsky, 2015).