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Abstract: Mathematics shares with language an essential reliance on the human capacity for recursion, permitting the generation of an infinite range of embedded expressions from a finite set of symbols. We studied the role of syntax in arithmetic thinking, a neglected component of numerical cognition, by examining eye movement sequences during the calculation of arithmetic expressions. Specifically, we investigated whether, similar to language, an expression has to be scanned sequentially while the nested syntactic structure is being computed or, alternatively, whether this structure can be extracted quickly and in parallel. Our data provide evidence for the latter: fixations sequences were stereotypically organized in clusters that reflected a fast identification of syntactic embeddings. A syntactically relevant pattern of eye movement was observed even when syntax was defined by implicit procedural rules (precedence of multiplication over addition) rather than explicit parentheses. While the total number of fixations was determined by syntax, the duration of each fixation varied with the complexity of the arithmetic operation at each step. These findings provide strong evidence for a syntactic organization for arithmetic thinking, paving the way for further comparative analysis of differences and coincidences in the instantiation of recursion in language and mathematics. [Copyright &y& Elsevier]

Cantor's theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same 'size' in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the 'size' of A should be less than the 'size' of B (part-whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part-whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part-whole principle, envisaged by Bolzano (Kitcher). [ABSTRACT FROM AUTHOR]

We investigate connections between the syntactic and semantic distance of programs on an abstract, recursion theoretic level. For a certain rather restrictive notion of interdependency of the two kinds of distances, there remain only few and “unnatural” numberings allowing such close relationship. Weakening the requirements leads to the discovery of universal metrics such that for an arbitrary recursively enumerable family of functions a numbering compatible with such a metric can uniformly be constructed. We conclude our considerations with some implications on learning theory. [ABSTRACT FROM AUTHOR]

The article presents a study investigating minimal degree for recursively inseparable sets. It is found that "effectively inseparable" could not be replaced by "recursively inseparable" as the set which separates a disjoint pair of recursively inseparable sets was unable to bound a minimal degree. It is also mentioned that, every degree of Peano Arithmetic (PA) model bounds a minimal degree.

The article presents a study which focuses on the development of an arithmetical measure theory based on the approach of J.H. Lutz. The arithmetic measure yielded same notion of measure 0 set as considered before the Martin-Löf randomness. The location of the class of r.e. sets through common recursion-theoritic reducibilities is highlighted,

The article discusses the properties needed in proving various results from recursion theory. It states that the usual setting for recursion theory is the standard model for second order arithmetic. It identifies relatively recursively enumerable versus a certain relatively formula in models of Peano Arithmetic (PA). It defines the set of forcing conditions.