How humans come to possess such striking cognitive abilities and rich conceptual repertoires is perhaps the question of cognitive science. Susan Carey has explored this question in part by identifying specific areas – such as number – in which humans demonstrate unique and impressive ability, and exploring their development in great depth (Carey Reference Carey2009). In her treatment of number, Carey argues that children gain an understanding of the natural numbers through a process of mapping the ordinal structure of the number list to quantity. We agree with this proposal, but we suggest (1) that analogy interacts with language in several additional and distinct ways to support the acquisition of number; and (2) that arriving at the analogy from ordered numerals to ordered quantities probably requires more than a single leap (Gentner Reference Gentner2010).
Carey and others have provided substantial evidence suggesting that two core capacities enter into numerical cognition: the analog magnitude system, which allows approximate judgments of quantity; and a system for keeping track of small numbers of items (up to three or four). Neither is sufficient for representing large exact numbers—nor indeed for representing the natural number sequence at all. As Carey reviews, two lines of evidence suggest that language is key in this achievement: (1) cross-linguistic studies of Amazonian peoples whose languages – Pirahã and Mundurukú – lack a full counting system, and who show marked deficiencies in dealing with exact numerosities greater than 3 (Everett Reference Everett2005; Frank et al. Reference Frank, Everett, Fedorenko and Gibson2008; Gordon Reference Gordon2004; Pica et al. Reference Pica, Lemer, Izard and Dehaene2004); and (2) developmental evidence that children at first learn the linguistic count sequence as a kind of social routine (Fuson Reference Fuson1988), and that learning this sequence is instrumental in their acquisition of the conceptual structure of the natural numbers.
We believe language is instrumental in achieving conceptual mastery of all sorts; indeed, we have proposed that language, combined with powerful analogical processes, is crucial to humans' remarkable abilities (Gentner & Christie Reference Gentner and Christie2008). Analogical processes conspire with language in several ways in conceptual development, including the acquisition of number. First, common labels invite comparison and subsequent abstraction (Gentner & Medina Reference Gentner and Medina1998). Hearing the count label “3” applied to three pears and three apples prompts comparison across the sets and abstraction of their common set size (Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005). Second, the repeated use of the same numerals for the same quantities helps stabilize numerical representation. English speakers consistently assign “3” to the same quantity, whether counting up from 1 or down from 10. This uniform usage might be taken for granted, except that Pirahã speakers, astoundingly, assign their numeral terms to different quantities when naming increasing vs. decreasing set sizes. A third way in which analogy interacts with language to support cognitive development is that linguistic structure invites corresponding conceptual structure (Gentner Reference Gentner, Gentner and Goldin-Meadow2003). For example, learning and using the spatial ordinal series “top, middle, bottom” invites preschoolers to represent space in an ordered vertical pattern (Loewenstein & Gentner Reference Loewenstein and Gentner2005).
This brings us to Carey's (Reference Carey2004; Reference Carey2009) bold proposal that learning the natural numbers relies on an analogical mapping from ordinal linguistic structure to ordinal conceptual structure. One symptom of this analogical insight is a sudden change in the pace of learning. As Carey reviews, children first learn the count sequence as a social routine. Despite their fluency with this linguistic sequence, children may show only minimal insight into the binding to numerical quantity. A typical 2-year-old can count from “1” to “10,” but cannot produce a set of five items on request. The binding of small numerals to quantities is slow, piecemeal, and context-specific (Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005; Wynn Reference Wynn1990). But once a child binds “3” or “4” to the appropriate quantity, the pattern changes; the child rapidly binds the succeeding numbers to their cardinalities. Further, the child shows understanding of the successor principle, that every natural number has a successor whose cardinality is greater by one.
How does the analogy between numeral order and quantity order emerge? The correspondence between counting one further in the linguistic sequence and increasing by one in set size is highly abstract. We suggest that children arrive at this insight in a stepwise fashion, roughly as follows (for simplicity, we consider the case where the insight occurs after “3” is bound to 3):
When “1”, “2” and “3” are bound to their respective quantities, the child has two instances in which further-by-one in count goes with greater-by-one in set size: 1→2 and 2→3. Should the alert child wonder whether this parallel continues to hold, s/he will find immediate confirmation: counting from “3” to “4” indeed goes with a set size increase 3→4. At this point the child has a very productive rule of thumb:
Over repeated use of this highly productive rule, the child re-represents the two parallel relations as the same (more abstract) relation – a successor relation – applying to different dimensions, such as:
At this point the analogy has revealed a powerful abstraction: the common relational structure required for the successor function.
Bertrand Russell (Reference Russell1920) memorably stated: “It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstraction involved is far from easy.” Though English speakers may see the natural numbers as obvious, the evidence from the Pirahã bears out Russell's speculation: a conception of “two-ness” is not inevitable in human cognition. Carey's proposal provides a route by which this insight can be acquired.
How humans come to possess such striking cognitive abilities and rich conceptual repertoires is perhaps the question of cognitive science. Susan Carey has explored this question in part by identifying specific areas – such as number – in which humans demonstrate unique and impressive ability, and exploring their development in great depth (Carey Reference Carey2009). In her treatment of number, Carey argues that children gain an understanding of the natural numbers through a process of mapping the ordinal structure of the number list to quantity. We agree with this proposal, but we suggest (1) that analogy interacts with language in several additional and distinct ways to support the acquisition of number; and (2) that arriving at the analogy from ordered numerals to ordered quantities probably requires more than a single leap (Gentner Reference Gentner2010).
Carey and others have provided substantial evidence suggesting that two core capacities enter into numerical cognition: the analog magnitude system, which allows approximate judgments of quantity; and a system for keeping track of small numbers of items (up to three or four). Neither is sufficient for representing large exact numbers—nor indeed for representing the natural number sequence at all. As Carey reviews, two lines of evidence suggest that language is key in this achievement: (1) cross-linguistic studies of Amazonian peoples whose languages – Pirahã and Mundurukú – lack a full counting system, and who show marked deficiencies in dealing with exact numerosities greater than 3 (Everett Reference Everett2005; Frank et al. Reference Frank, Everett, Fedorenko and Gibson2008; Gordon Reference Gordon2004; Pica et al. Reference Pica, Lemer, Izard and Dehaene2004); and (2) developmental evidence that children at first learn the linguistic count sequence as a kind of social routine (Fuson Reference Fuson1988), and that learning this sequence is instrumental in their acquisition of the conceptual structure of the natural numbers.
We believe language is instrumental in achieving conceptual mastery of all sorts; indeed, we have proposed that language, combined with powerful analogical processes, is crucial to humans' remarkable abilities (Gentner & Christie Reference Gentner and Christie2008). Analogical processes conspire with language in several ways in conceptual development, including the acquisition of number. First, common labels invite comparison and subsequent abstraction (Gentner & Medina Reference Gentner and Medina1998). Hearing the count label “3” applied to three pears and three apples prompts comparison across the sets and abstraction of their common set size (Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005). Second, the repeated use of the same numerals for the same quantities helps stabilize numerical representation. English speakers consistently assign “3” to the same quantity, whether counting up from 1 or down from 10. This uniform usage might be taken for granted, except that Pirahã speakers, astoundingly, assign their numeral terms to different quantities when naming increasing vs. decreasing set sizes. A third way in which analogy interacts with language to support cognitive development is that linguistic structure invites corresponding conceptual structure (Gentner Reference Gentner, Gentner and Goldin-Meadow2003). For example, learning and using the spatial ordinal series “top, middle, bottom” invites preschoolers to represent space in an ordered vertical pattern (Loewenstein & Gentner Reference Loewenstein and Gentner2005).
This brings us to Carey's (Reference Carey2004; Reference Carey2009) bold proposal that learning the natural numbers relies on an analogical mapping from ordinal linguistic structure to ordinal conceptual structure. One symptom of this analogical insight is a sudden change in the pace of learning. As Carey reviews, children first learn the count sequence as a social routine. Despite their fluency with this linguistic sequence, children may show only minimal insight into the binding to numerical quantity. A typical 2-year-old can count from “1” to “10,” but cannot produce a set of five items on request. The binding of small numerals to quantities is slow, piecemeal, and context-specific (Mix et al. Reference Mix, Sandhofer, Baroody and Kail2005; Wynn Reference Wynn1990). But once a child binds “3” or “4” to the appropriate quantity, the pattern changes; the child rapidly binds the succeeding numbers to their cardinalities. Further, the child shows understanding of the successor principle, that every natural number has a successor whose cardinality is greater by one.
How does the analogy between numeral order and quantity order emerge? The correspondence between counting one further in the linguistic sequence and increasing by one in set size is highly abstract. We suggest that children arrive at this insight in a stepwise fashion, roughly as follows (for simplicity, we consider the case where the insight occurs after “3” is bound to 3):
When “1”, “2” and “3” are bound to their respective quantities, the child has two instances in which further-by-one in count goes with greater-by-one in set size: 1→2 and 2→3. Should the alert child wonder whether this parallel continues to hold, s/he will find immediate confirmation: counting from “3” to “4” indeed goes with a set size increase 3→4. At this point the child has a very productive rule of thumb:
Over repeated use of this highly productive rule, the child re-represents the two parallel relations as the same (more abstract) relation – a successor relation – applying to different dimensions, such as:
At this point the analogy has revealed a powerful abstraction: the common relational structure required for the successor function.
Bertrand Russell (Reference Russell1920) memorably stated: “It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstraction involved is far from easy.” Though English speakers may see the natural numbers as obvious, the evidence from the Pirahã bears out Russell's speculation: a conception of “two-ness” is not inevitable in human cognition. Carey's proposal provides a route by which this insight can be acquired.