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This chapter begins by considering how the child who uses counting to represent number (i.e., is a cardinal-principle-knower) integrates the count list with analog magnitude representations, and what this integration buys the child. The goal is to illustrate further the meaning-making capacity of Quinian bootstrapping. It then presents evidence for a second discontinuity in the development of mathematical cognition: the construction of representations of fractions.

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain.

A new realistic account of an ontology of extensive magnitudes is developed, formulated in Seven Principles. The principles are defended by the role of magnitudes in scientific explanation and in counterfactuals. Scientific laws can be formulated using this ontology of magnitudes. A metaphysics-first view of the perception of magnitudes is then defended by using this metaphysics of magnitudes. The metaphysics-first treatment permits explanation of features of the perception of extensive magnitudes. Notions of analogue computation, analogue representation, and analogue content are explained using this apparatus. Deployment of the resulting theory allows the development, against Kuhn, of a case for the objectivity of analogue perceptual content.

This chapter argues that input-output modeling is an essential element of computing, at least in current computational approaches in cognitive neuroscience. A model, in the context of this work, is a representational system that preserves patterns of relations of the target domain. A process input-output models a given target when its input-output function and some relation in the target have a shared formal structure. Finally, the chapter shows that modeling is often associated with computing, that it plays a major methodological role in discovering what function is being computed, and that it enhances a distinctive account of computational explanation.