In Carey's proposal, core cognitive processes have a strictly circumscribed conceptual domain: “A dedicated input analyzer computes representations of one kind of entity in the world and only that kind” (Carey Reference Carey2009, p. 451). This leads Carey to assume that conceptual change transforms representations within a specified domain. So, for example, Carey proposes that the natural numbers are constructed from number-based core processes: either parallel representation of individuals belonging to small sets, or analog magnitude representations. On this view, a CS2 “transcends” a CS1 – two conceptual systems cover the same domain, but one covers that domain more completely and more richly than the other.
Substantial empirical evidence suggests that instead, concepts in learned theories are often built out of processes and representational vehicles taken from widely different domains. This is especially so when the topic is abstract, as in the case of mathematics. For example, Longo and Lourenco (Reference Longo and Lourenco2007) present evidence that overlapping mechanisms modulate attention in numerical and spatial tasks (see also Hubbard et al. Reference Hubbard, Piazza, Pinel and Dehaene2005). In their experiments, participants who show a high degree of left-side pseudoneglect in a physical bisection task also showed a high degree of small-number pseudoneglect in a numerical bisection task. Furthermore, this is unlikely to result just from an analogical mapping between number and space used during learning, because numerical bisection biases, like physical biases, depend on whether numbers are physically presented in near or far space (Longo & Lourenco Reference Longo and Lourenco2009). This fact suggests an online connection between spatial and numerical attention. Longo and Lourenco (Reference Longo and Lourenco2009) interpret these results in terms of shared mechanisms: Foundational processes that guide attention in physical space are recycled to guide attention in numerical space.
Moreover, there is ample evidence for the reuse of motor-control systems in mathematical processing. For example, Andress et al. (Reference Andres, Seron and Oliver2007) report that hand motor circuits are activated by a dot-counting task; Badets and Pesenti (Reference Badets and Pesenti2010) demonstrated that observing grip-closure movements (but not nonbiological closure motions) interferes with numerical magnitude processing; and Goldin-Meadow (Reference Goldin-Meadow2003) recounts the many ways in which gesturing aids in the acquisition of mathematical concepts. These examples suggest that the motor system offers representational resources to disparate domains.
Similarly, several authors have reported that algebraic reasoning co-opts mechanisms involved in perception and manipulation of physical objects (Dörfler Reference Dörfler and Weigand2004; Kirshner Reference Kirshner1989; Landy & Goldstone Reference Landy and Goldstone2007; Landy & Goldstone Reference Landy, Jones, Goldstone, Love, McRae and Sloutsky2008). For example, Landy and Goldstone (Reference Landy and Goldstone2007) report that reasoners systematically utilize processes of perceptual grouping to proxy for the ordering of algebraic operations. Again, the relationship does not appear to be merely analogical. As Carey emphasizes, one expects analogies to occur over extended durations. In contrast, Kirshner and Awtry (Reference Kirshner and Awtry2004) report that, at least, the use of spatial perception in interpreting equation structure happens immediately upon exposure to the spatially regular algebra notation, and must be unlearned through the process of acquiring sophisticated algebraic knowledge. The most natural interpretation is that relevant computations are performed directly on spatial representations of symbol systems, and tend to work not because of developed internal analogies but because the symbolic notation itself generally aligns physical and abstract properties.
In basic arithmetic knowledge, and in the algebraic understanding of abstract relations, distinctly perceptual-motor processing is applied to do conceptual work in a widely different domain. Despite Carey's assumption that cognitive resources are strongly typed – some are domain-specific input analyzers, some are components of core knowledge, others are parts of richer domain theories – it appears that at least some basic cognitive resources are used promiscuously in a variety of domains, and applied to a variety of contents. We suggest that this is possible for two reasons. First, on an evolutionary timescale it is more efficient to repurpose or replicate preexisting neural structures than to build entirely new ones. Second, many new symbolic environments, such as a math class utilizing a number line, or algebraic notation, form rich and multimodal experiences, which can themselves be analyzed using preexisting cognitive processes (“core” or not). Whenever such analyses yield largely successful results, a learner is likely to incorporate the relevant constraints and computational systems into the conceptual apparatus (Clark Reference Clark2008). Therefore, initially dedicated mechanisms such as those governing perceptual grouping and attention, can be co-opted, given an appropriate cultural context, into performing highly abstract and conceptual functions.
When we make the claim that perceptual processes are co-opted for mathematical reasoning, for example, that automatically computed spatial arrangements of physical symbols are used as proxies for understanding generic relations, this is not a return to old-fashioned empiricism. It is not our view that all mathematical content can be reduced to perceptual content. Nor is it our view that because humans are able to co-opt perceptual processes to do mathematics, that this implies that the content of mathematical claims can be exhaustively reduced to perceptual primitives. Indeed, it is important to our story that they are not so reduced. We wish to point out that cognitive resources that are used for perceptual and motor reasoning in one domain are often usefully exploited for conceptual understanding in another domain. In fact, given the mounting evidence for the reuse of neural systems across the boundaries of traditional cognitive domains (Anderson Reference Anderson2010), it would be very surprising if many of our most important cognitive resources were domain-bound in the manner of Carey's core processes.
In short, whether on a cultural or evolutionary timescale, learning systems apply any available resources to the understanding of new symbol systems, without regard for whether that old system is domain-specific, “perceptual,” or “conceptual.” One important role of culturally constructed symbol systems is to serve, themselves, as rich environmental structures that can be the target of pre-existing cognitive mechanisms (“core” or otherwise). Carey's “Quinian bootstrapping,” which treats novel symbol systems as mere placeholders, with no properties beyond conceptual role – that is, their inferential relationship to other symbols in their set – simplifies the process of learning new symbol systems at the cost of missing much of their value.
In Carey's proposal, core cognitive processes have a strictly circumscribed conceptual domain: “A dedicated input analyzer computes representations of one kind of entity in the world and only that kind” (Carey Reference Carey2009, p. 451). This leads Carey to assume that conceptual change transforms representations within a specified domain. So, for example, Carey proposes that the natural numbers are constructed from number-based core processes: either parallel representation of individuals belonging to small sets, or analog magnitude representations. On this view, a CS2 “transcends” a CS1 – two conceptual systems cover the same domain, but one covers that domain more completely and more richly than the other.
Substantial empirical evidence suggests that instead, concepts in learned theories are often built out of processes and representational vehicles taken from widely different domains. This is especially so when the topic is abstract, as in the case of mathematics. For example, Longo and Lourenco (Reference Longo and Lourenco2007) present evidence that overlapping mechanisms modulate attention in numerical and spatial tasks (see also Hubbard et al. Reference Hubbard, Piazza, Pinel and Dehaene2005). In their experiments, participants who show a high degree of left-side pseudoneglect in a physical bisection task also showed a high degree of small-number pseudoneglect in a numerical bisection task. Furthermore, this is unlikely to result just from an analogical mapping between number and space used during learning, because numerical bisection biases, like physical biases, depend on whether numbers are physically presented in near or far space (Longo & Lourenco Reference Longo and Lourenco2009). This fact suggests an online connection between spatial and numerical attention. Longo and Lourenco (Reference Longo and Lourenco2009) interpret these results in terms of shared mechanisms: Foundational processes that guide attention in physical space are recycled to guide attention in numerical space.
Moreover, there is ample evidence for the reuse of motor-control systems in mathematical processing. For example, Andress et al. (Reference Andres, Seron and Oliver2007) report that hand motor circuits are activated by a dot-counting task; Badets and Pesenti (Reference Badets and Pesenti2010) demonstrated that observing grip-closure movements (but not nonbiological closure motions) interferes with numerical magnitude processing; and Goldin-Meadow (Reference Goldin-Meadow2003) recounts the many ways in which gesturing aids in the acquisition of mathematical concepts. These examples suggest that the motor system offers representational resources to disparate domains.
Similarly, several authors have reported that algebraic reasoning co-opts mechanisms involved in perception and manipulation of physical objects (Dörfler Reference Dörfler and Weigand2004; Kirshner Reference Kirshner1989; Landy & Goldstone Reference Landy and Goldstone2007; Landy & Goldstone Reference Landy, Jones, Goldstone, Love, McRae and Sloutsky2008). For example, Landy and Goldstone (Reference Landy and Goldstone2007) report that reasoners systematically utilize processes of perceptual grouping to proxy for the ordering of algebraic operations. Again, the relationship does not appear to be merely analogical. As Carey emphasizes, one expects analogies to occur over extended durations. In contrast, Kirshner and Awtry (Reference Kirshner and Awtry2004) report that, at least, the use of spatial perception in interpreting equation structure happens immediately upon exposure to the spatially regular algebra notation, and must be unlearned through the process of acquiring sophisticated algebraic knowledge. The most natural interpretation is that relevant computations are performed directly on spatial representations of symbol systems, and tend to work not because of developed internal analogies but because the symbolic notation itself generally aligns physical and abstract properties.
In basic arithmetic knowledge, and in the algebraic understanding of abstract relations, distinctly perceptual-motor processing is applied to do conceptual work in a widely different domain. Despite Carey's assumption that cognitive resources are strongly typed – some are domain-specific input analyzers, some are components of core knowledge, others are parts of richer domain theories – it appears that at least some basic cognitive resources are used promiscuously in a variety of domains, and applied to a variety of contents. We suggest that this is possible for two reasons. First, on an evolutionary timescale it is more efficient to repurpose or replicate preexisting neural structures than to build entirely new ones. Second, many new symbolic environments, such as a math class utilizing a number line, or algebraic notation, form rich and multimodal experiences, which can themselves be analyzed using preexisting cognitive processes (“core” or not). Whenever such analyses yield largely successful results, a learner is likely to incorporate the relevant constraints and computational systems into the conceptual apparatus (Clark Reference Clark2008). Therefore, initially dedicated mechanisms such as those governing perceptual grouping and attention, can be co-opted, given an appropriate cultural context, into performing highly abstract and conceptual functions.
When we make the claim that perceptual processes are co-opted for mathematical reasoning, for example, that automatically computed spatial arrangements of physical symbols are used as proxies for understanding generic relations, this is not a return to old-fashioned empiricism. It is not our view that all mathematical content can be reduced to perceptual content. Nor is it our view that because humans are able to co-opt perceptual processes to do mathematics, that this implies that the content of mathematical claims can be exhaustively reduced to perceptual primitives. Indeed, it is important to our story that they are not so reduced. We wish to point out that cognitive resources that are used for perceptual and motor reasoning in one domain are often usefully exploited for conceptual understanding in another domain. In fact, given the mounting evidence for the reuse of neural systems across the boundaries of traditional cognitive domains (Anderson Reference Anderson2010), it would be very surprising if many of our most important cognitive resources were domain-bound in the manner of Carey's core processes.
In short, whether on a cultural or evolutionary timescale, learning systems apply any available resources to the understanding of new symbol systems, without regard for whether that old system is domain-specific, “perceptual,” or “conceptual.” One important role of culturally constructed symbol systems is to serve, themselves, as rich environmental structures that can be the target of pre-existing cognitive mechanisms (“core” or otherwise). Carey's “Quinian bootstrapping,” which treats novel symbol systems as mere placeholders, with no properties beyond conceptual role – that is, their inferential relationship to other symbols in their set – simplifies the process of learning new symbol systems at the cost of missing much of their value.