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Interweavings of Alan Turing＇s Mathematics and Sociology of Knowledge

Journal of Mathematics and System Science 2012年第7期2卷 420-429页

作者： Isabel Cafezeiro Ivan da Costa Marques Instituto de Computagao Universidade Federal Fluminense Rua Passo da Patria 24210-240 Brazil and Programa dePos-Graduagao em Historia das Ciencias e das Ttecnicas e Epistemologia Universidade Federal do Rio de Janeiro programa de pos-graduagao em historia das ciencias e das ttecnicas e epistemologia Universidade Federal do Rio de JaneiroAv Atlantica 22010-000 Brazil

This paper starts from the analysis of how Alan Turing proceeded to build the notion of computability in his famous 1936 text ＂On computable numbers, with an application to the entscheidungsproblem＂. Looking in detail at his stepwise construction, which starts from the materialities to achieve a satisfactory level of abstraction, it is considered how his way of doing mathematics was one that constructs mathematical knowledge by evading a definite separation between matter and form; in this way, making the world and language come together. Following the same line of reasoning, it is argued in this paper that the abstract and the concrete, the deduction and the induction, the technical and the social as well as the objective and the subjective are unthinkable as pure entities. By considering the controversies and discussions from the mid-nineteenth century until now, it is shown that local （social） elements necessarily participate in what is usually considered ＂technical content＂ or ＂objectivity＂. While Alan Turing was a precursor of what today might be said to be an ＂anthropological approach to mathematical culture＂, unveiling and reviving approaches that enable the axis of authority for mathematics, logic and computing to be shifted, he also opened different paths for the construction of a variety of mathematical knowledge as well.

关键词： Computability Alan Tttring knowledge construction.

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