Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

The nosological diagnosis is a particular type of nontheoretical diagnosis consisting of identifying the disease that afflicts the patient without explaining the underlying etiopathological mechanisms. Its origins are within the essentialist point of view on the nature of diseases, which dates back at least to 18th-century taxonomy studies. In this article, we propose a model of nosological diagnosis as a two-phase process composed of the categorization of inductive inferences and argumentations by analogy. In the inductive phase, disease entities are identified (...) by means of typicality-based categorization processes, and meaningful clinical samples are learned ; in the subsequent phase, those samples are used as the bases of argumentations by analogy to obtain a diagnosis for a given patient. This model extends the prototype resemblance theory of disease including also the exemplar theory proposed in cognitive science and, moreover, it frames the clinical activity of nosological diagnosis and how it can be explained within the theory of argumentation. According to it, diagnosis based on the recognition of a typical syndrome is explained in terms of the prototype theory of categorization and the antisymmetrical argumentation by analogy, while diagnosis based on a comparison with a previous clinical case is explained by the exemplar theory of categorization and by the symmetrical argumentation by analogy. (shrink)

I present the traditional debate about the so called explanation of Russell’s paradox and propose a new way to solve the contradiction that arises in Frege’s system. I briefly examine two alternative explanatory proposals—the Predicativist explanation and the Cantorian one—presupposed by almost all the proposed solutions of Russell’s Paradox. From the discussion about these proposals a controversial conclusion emerges. Then, I examine some particular zig zag solutions and I propose a third explanation, presupposed by them, in which I emphasise the (...) role of an implicit premise in the derivation of the paradox. In conclusion, I propose a different zig zag solution obtained by the adoption of a negative free logic. (shrink)