We contrapose computational models using representations of numbers in parietal cortical activity patterns (abstract or not) with dynamic models, whereby prefrontal cortex (PFC) orchestrates neural operators. The neural operators under PFC control are activity patterns that mobilize synaptic matrices formed by learning into textured oscillations we observe through the electroencephalogram from the scalp (EEG) and the electrocorticogram from the cortical surface (ECoG). We postulate that specialized operators produce symbolic representations existing only outside of brains.

Evans & Levinson (E&L) advocate taking linguistic diversity at “face value.” Their argument consists of a list of diverse phenomena and the assertion that no non-vacuous theory could possibly uncover a meaningful unity underlying them. I argue, with evidence from Tlingit and Warlpiri, that E&L's list itself should not be taken at face value – and that the actual research record already demonstrates unity amidst diversity.

Oaksford & Chater (O&C) would need to define rationality if they want to argue that stomachs are not rational. The question of rationality, anyhow, is orthogonal to the debate concerning whether humans use classical deductive logic or probabilistic reasoning.

Mitchell et al. contemplate the possibility of rats being capable of propositional reasoning. We suggest that this is an unlikely and unsubstantiated possibility. Nonhuman animals and human infants do learn about the contingencies in the world; however, such learning seems not to be based on propositional reasoning, but on more elementary associative processes.

Because the addition of the conditional premise tends to increase modus ponens (MP) inferences, Oaksford & Chater argue that the additional knowledge is assimilated to world knowledge before the Ramsey test is carried out to evaluate P(q|p), so that the process of applying the Ramsey test could become indistinguishable from the process of applying the second-step conditionalization.

The goal of the present commentary is to show that past results on automatic numerical processing in different notations are consistent with the idea of an abstract numerical representation. This is done by reviewing the relevant studies and giving alternative explanations to the ones proposed in the target article.

While endorsing Evans & Levinson's (E&L's) call for rigorous documentation of variation, we defend the idea of Universal Grammar as a toolkit of language acquisition mechanisms. The authors exaggerate diversity by ignoring the space of conceivable but nonexistent languages, trivializing major design universals, conflating quantitative with qualitative variation, and assuming that the utility of a linguistic feature suffices to explain how children acquire it.

Mitchell et al. argue that conditioning phenomena may be better explained by high-level, rational processes, rather than by non-cognitive associative mechanisms. This commentary argues that this viewpoint is compatible with neuroscientific data, may extend to nonhuman animals, and casts computational models of reinforcement learning in a new light.

Oaksford & Chater's (O&C's) account of deductive reasoning is parsimonious at a local level (because a rational model is used to explain a wide range of behavior) and at a global level (because their Bayesian approach connects to other areas of research). Their emphasis on environmental structure is especially important, and the power of their approach is seen at both the computational and algorithmic levels.

Evans & Levinson's (E&L's) major point is that human languages are intriguingly diverse rather than (like animal communication systems) uniform within the species. This does not establish a “myth” about language universals, or advance the ill-framed pseudo-debate over universal grammar. The target article does, however, repeat a troublesome myth about Fitch and Hauser's (2004) work on pattern learning in cotton-top tamarins.

Additional support is presented for the necessity of awareness of the CS-US relation in human autonomic conditioning. However, possible limitations and exceptions regarding this general rule are discussed. Limitations include the lack of relationship between conditioned response (CR) strength and degree of awareness, and an important exception may be the finding of conditioning with backwardly masked CSs of a biologically prepared nature.

Much evidence cited by Cohen Kadosh & Walsh (CK&W) in support of their notation-specific representation hypothesis is based on tasks requiring automatic number processing. Several of these findings can be alternatively explained by differential expertise in mapping numerical symbols onto semantic magnitude representations. The importance of considering symbol-referent mapping expertise in theories on numerical representations is highlighted.

Modern linguistics has highlighted the fundamental invariance of human language: A rich invariant structure has emerged from comparative studies nourished by sophisticated formal models; languages also differ along important dimensions, but variation is constrained in severe and systematic ways. I illustrate this research direction in the domains of island constraints, word order restrictions, and the expression of referential dependencies. Both language invariance and language variability within systematic limits are highly relevant for the cognitive sciences.

Mitchell et al. contend that there is no need to posit a contribution based on the formation of associative links to human learning. In order to sustain this argument, they have ignored evidence which is difficult to explain with propositional accounts; and they have mischaracterised the evidence they do cite by neglecting features of these experiments that contradict a propositional account.

I challenge two points in Cohen Kadosh & Walsh's (CK & W) argument: First, the definition of abstraction is too restricted; second, the distinction between representations and operations is too clear-cut. For example, taking Jean Piaget's “conservation of number task,” I propose that another way to avoid orthodoxy in the field of numerical cognition is to consider inhibition as an alternative idea of abstraction.

Is human cognition best described by optimal models, or by adaptive but suboptimal heuristic strategies? It is frequently hard to identify which theoretical model is normatively best justified. In the context of information search, naïve subjects' heuristic strategies are better motivated than some “optimal” models.

Drawing on our Associative-Propositional Evaluation (APE) Model, we argue for the usefulness of distinguishing between basic operating principles of learning processes (associative linking vs. propositional reasoning) and secondary features pertaining to the conditions of their operation (automatic vs. controlled). We review empirical evidence that supports the joint operation of associative and propositional processes in the formation of new associations.

Generative linguistics' search for linguistic universals (1) is not comparable to the vague explanatory suggestions of the article; (2) clearly merits a more central place than linguistic typology in cognitive science; (3) is fundamentally untouched by the article's empirical arguments; (4) best explains the important facts of linguistic diversity; and (5) illuminates the dominant component of language's “biocultural” nature: biology.

We delineate a developmental model of number representations. Notably, developmental dyscalculia (DD) is rarely associated with an all-or-none deficit in numerosity processing as would be expected if assuming abstract number representations. Finally, we suggest that the “generalist genes” view might be a plausible – though thus far speculative – explanatory framework for our model of how number representations develop.