Clarke and Beck's defense of the theoretical construct “approximate number system” (ANS) is flawed in serious ways – from biological misconceptions to mathematical naïveté. The authors misunderstand behavioral/psychological technical concepts, such as numerosity and quantical cognition, which they disdain as “exotic.” Additionally, their characterization of rational numbers is blind to the essential role of symbolic reference in the emergence of number.

The neural realization of number in abstract form is implausible, but from this it doesn't follow that numbers are not abstract. Clear definitions of abstraction are needed so they can be applied homogenously to numerical and non-numerical cognition. To achieve a better understanding of the neural substrate of abstraction, productive cognition – not just comprehension and perception – must be investigated.

Early quantitative skills cannot be directly extended to provide the richness, precision, and sophistication of the concept of natural number. These skills must interact with top-down mathematical schemas, which can be explained by bodily grounded everyday mechanisms for abstraction and imagination (e.g., conceptual metaphor, blending) that are both biologically plausible and culturally shaped (established beyond the child's mind).