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Rips et al. argue that the concept of natural number, which includes formal properties such as the successor function and commutativity, is not grounded in non-symbolic (nonverbal) numerical representations involving object files and internal magnitudes. Rather, the natural numbers are constructed “top-down” on the basis of innate constraints on processing (e.g., recursion) that lead to “math schemas,” which encompass various formal properties. Although we agree with Rips et al. (and others) that nonverbal numerical representations alone will not allow for the construction of the concept of natural number, we disagree with two claims central to their proposal: (1) the discontinuity claim that there is no continuity between early numerical representations and natural number, and (2) the input claim that particular experiences (e.g., cardinality-related talk and object-based activities) do not support natural number construction.
The discontinuity claim
Although Rips et al. acknowledge that adults use magnitude representations on tasks such as numerical estimation and comparison, they argue that this does not provide evidence for continuity between internal magnitudes and natural number, as these tasks could engage the magnitude system alone and not abstract mathematical knowledge. Furthermore, they emphasize that magnitude representations do not serve as precursors to natural number because they do not instantiate key principles such as the successor function. However, evidence that abstract mathematical reasoning might be influenced by magnitude-related information would lend support to greater continuity. In fact, Landy and Goldstone (2007) have shown that adults' success in solving...