A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation of that system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception.

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This study examined executive functioning and reading achievement in 106 6- to 8-year-old Brazilian children from a range of social backgrounds of whom approximately half lived below the poverty line. A particular focus was to explore the executive function profile of children whose classroom reading performance was judged below standard by their teachers and who were matched to controls on chronological age, sex, school type (private or public), domicile (Salvador/BA or São Paulo/SP) and socioeconomic status. Children completed a battery of 12 executive function tasks that were conceptual tapping cognitive flexibility, working memory, inhibition and selective attention. Each executive function domain was assessed by several tasks. Principal component analysis extracted four factors that were labeled “Working Memory/Cognitive Flexibility,” “Interference Suppression,” “Selective Attention,” and “Response Inhibition.” Individual differences in executive functioning components made differential contributions to early reading achievement. The Working Memory/Cognitive Flexibility factor emerged as the best predictor of reading. Group comparisons on computed factor scores showed that struggling readers displayed limitations in Working Memory/Cognitive Flexibility, but not in other executive function components, compared to more skilled readers. These results validate the account that working memory capacity provides a crucial building block for the development of early literacy skills and extends it to a population of early readers of Portuguese from Brazil. The study suggests that deficits in working memory/cognitive flexibility might represent one contributing factor to reading difficulties in early readers. This might have important implications for how educators might intervene with children at risk of academic under achievement.

We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose-Einstein statistics. This involved both the introduction of group theory at the top level, and some modeling at the “phenomenological” level, and thus provides a nice example of the relationships we are interested in. We conclude with a discussion of the “autonomy” of London's model.