Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.

Memory for numbers improves with age. One source of this improvement may be learning linear spatial-numeric associations, but previous evidence for this hypothesis likely confounded memory span with quality of numerical magnitude representations and failed to distinguish spatial-numeric mappings from other numeric abilities, such as counting or number word-cardinality mapping. To obviate the influence of memory span on numerical memory, we examined 39 3- to 5-year-olds’ ability to recall one spontaneously produced number (1-20) after a delay, and the relation between (...) numeric recall (controlling for non-numeric recall) and quality of mapping between symbolic and non-symbolic quantities using number-line estimation, give-a-number estimation, and counting tasks. Consistent with previous reports, mapping of numerals to space, to discrete quantities, and to numbers in memory displayed a logarithmic-to-linear shift. Also, linearity of spatial-numeric mapping correlated strongly with multiple measures of numeric recall (percent correct and percent absolute error), even when controlling for age and non-numeric memory. Results suggest that linear spatial-numeric mappings may aid memory for number over and above children’s other numeric skills. (shrink)

Analogical inferences are an important consequence of the way semantic knowledge is represented, that is, with relations as explicit structures that can take arguments. We review evidence that this feature of semantic cognition successfully predicts how quickly and broadly children's concepts change with experience and show that Rogers & McClelland's (R&M's) parallel distributed processing (PDP) model fails to simulate these cognitive changes due to its handling of relational information.

It is unclear how an argumentative environment would select for better reasoning given three general findings. First, argument rationality typically fails to persuade poor reasoners. Second, reasoned argumentation competes with more persuasive and less rational arguments for limited cognitive resources. Third, those poor at reasoning fail to distinguish between valid and invalid arguments. Reasoning, therefore, is poorly designed for argument.