Perhaps the time has come to re-examine the basic notions of cognitive science. Together with previous challenges against associationism, the target article should be viewed as a call to arms to re-evaluate the empirical basis for contemporary conceptualizations of human learning and the notion of “mental representation,” a concept that has become too imprecise for describing the elements of cognition.

We challenge the arguments of Cohen Kadosh & Walsh (CK&W) on two grounds. First, interactions between number form (e.g., notation, format, modality) and an experimental factor do not show that the notations/formats/modalities are processed separately. Second, we discuss evidence that numbers are coded abstractly, also when not required by task demands and processed unintentionally, thus challenging the authors' dual-code account

The issue of abstractness raises two distinct questions. First, is there a format-independent magnitude representation? Second, does analogue magnitude really play a crucial role in the development of human mathematics? We suggest that neither developmental nor cultural studies support this notion. The field needs to redefine the properties of the core number representation as used in human arithmetic.

Mitchell et al. present a lucid and provocative challenge to the claim that links between mental representations are formed automatically. However, the propositional approach they offer requires clearer specification, especially with regard to how propositions and memories interact. A definition of a system would also clarify the debate, as might an alternative technique for assessing task “dissociations.”

The proposal to recruit available formal structures to build an algorithmic model of all learning falters on close examination of its essential assumption: that the input and output of the model are propositional in structure. After giving three framework considerations, I describe three possibly fatal problems with this assumption, concluding each with a question that needs answering to avoid fatality.

We agree that the default numerical representation is best accessed by probing automatic processing. The locus of this representation is apparently at the horizontal intraparietal sulcus (HIPS), the convergence zone of magnitude information. The parietal lobes are the right place to look for non-abstract representation of magnitude, yet the proof for that is still to be found.

The target article undermines the existence of a shared unitary numerical format, illustrating a variety of representations. The “abstract”/“not-abstract” dichotomy does not capture their specific features. These representations are “supramodal” with respect to the sensory modality of the stimulus, and independent of its specific notation, with a main role of spatial codes, both related and unrelated to the mental number line.

Mitchell et al. propose that associative learning in humans and other animals requires the formation of propositions by means of conscious and controlled reasoning. This approach neglects important aspects of current thinking in evolutionary biology and neuroscience that support the claim that learning, here exemplified by fear learning, neither needs to be conscious nor controlled.

We are big fans of propositions. But we are not big fans of the “propositional approach” proposed by Mitchell et al. The authors ignore the critical role played by implicit, non-inferential processes in biological cognition, overestimate the work that propositions alone can do, and gloss over substantial differences in how different kinds of animals and different kinds of cognitive processes approximate propositional representations.

We discuss methodological problems and present our own empirical data on calculation tasks in toddlers. We propose to develop enriching theoretical models concerning quantity representations, based on empirical findings from developmental psychology. A revitalization of the debate is worthy, because it is reminiscent of the philosophical dispute on universal entities in scholasticism and Plato's theory of ideal numbers.

The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.

An attentional-associative model (Schmajuk et al. 1996), previously evaluated against multiple sets of classical conditioning data, is applied to causal learning. In agreement with Mitchell et al.'s suggestion, according to the model associative learning can be a conscious, controlled process. However, whereas our model correctly predicts blocking following or preceding subadditive training, the propositional approach cannot account for those results.

To support their main claim, Mitchell et al. broach the issue of the relationship between the learning performance of human and nonhuman animals. We show that their argumentation is problematic both theoretically and empirically. In fact, results from learning studies with humans and honey-bees strongly suggest that human learning is not entirely propositional.

The extent to which human learning should be thought of in terms of elementary, automatic versus controlled, cognitive processes is unresolved after nearly a century of often fierce debate. Mitchell et al. provide a persuasive review of evidence against automatic, unconscious links. Indeed, unconscious processes seem to play a negligible role in any form of learning, not just in Pavlovian conditioning. But a modern connectionist framework, in which “cognitive” phenomena are emergent properties, is likely to offer a fuller account of human learning than the propositional framework Mitchell et al. propose.

Mitchell et al. describe many fascinating studies, and in the process, propose what they consider to be a unified framework for human learning in which effortful, controlled learning results in propositional knowledge. However, it is unclear how any of their findings privilege a propositional account, and we remain concerned that embedding all knowledge in propositional representations obscures the tight interdependence between learning from experiences and the use of the results of learning as a basis for action.

Contrary to the target article's claims, social cognition research shows considerable learning (about other people) that is relatively automatic. Some of this learning is propositional (spontaneous trait inferences) and some is associative (spontaneous trait transference). Other dichotomies – for example, between learning explicit and implicit attitudes – are also important. However conceived, human conditioning is not synonymous with human learning.

We argue that the propositional and link-based approaches to human contingency learning represent different levels of analysis because propositional reasoning requires a basis, which is plausibly provided by a link-based architecture. Moreover, in their attempt to compare two general classes of models (link-based and propositional), Mitchell et al. refer to only two generic models and ignore the large variety of different models within each class.