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  1. The Representational Foundations of Computation.Michael Rescorla - 2015 - Philosophia Mathematica 23 (3):338-366.
    Turing computation over a non-linguistic domain presupposes a notation for the domain. Accordingly, computability theory studies notations for various non-linguistic domains. It illuminates how different ways of representing a domain support different finite mechanical procedures over that domain. Formal definitions and theorems yield a principled classification of notations based upon their computational properties. To understand computability theory, we must recognize that representation is a key target of mathematical inquiry. We must also recognize that computability theory is an intensional enterprise: it (...)
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  • Can Church’s thesis be viewed as a Carnapian explication?Paula Quinon - 2019 - Synthese 198 (Suppl 5):1047-1074.
    Turing and Church formulated two different formal accounts of computability that turned out to be extensionally equivalent. Since the accounts refer to different properties they cannot both be adequate conceptual analyses of the concept of computability. This insight has led to a discussion concerning which account is adequate. Some authors have suggested that this philosophical debate—which shows few signs of converging on one view—can be circumvented by regarding Church’s and Turing’s theses as explications. This move opens up the possibility that (...)
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  • Computing with Numbers and Other Non-syntactic Things: De re Knowledge of Abstract Objects.Stewart Shapiro - 2017 - Philosophia Mathematica 25 (2):268-281.
    ABSTRACT Michael Rescorla has argued that it makes sense to compute directly with numbers, and he faulted Turing for not giving an analysis of number-theoretic computability. However, in line with a later paper of his, it only makes sense to compute directly with syntactic entities, such as strings on a given alphabet. Computing with numbers goes via notation. This raises broader issues involving de re propositional attitudes towards numbers and other non-syntactic abstract entities.
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  • Computability, Notation, and de re Knowledge of Numbers.Stewart Shapiro, Eric Snyder & Richard Samuels - 2022 - Philosophies 1 (7):20.
    Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...)
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  • Generalization of Shapiro’s theorem to higher arities and noninjective notations.Dariusz Kalociński & Michał Wrocławski - 2022 - Archive for Mathematical Logic 62 (1):257-288.
    In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types (...)
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  • Implicit and Explicit Examples of the Phenomenon of Deviant Encodings.Paula Quinon - 2020 - Studies in Logic, Grammar and Rhetoric 63 (1):53-67.
    The core of the problem discussed in this paper is the following: the Church-Turing Thesis states that Turing Machines formally explicate the intuitive concept of computability. The description of Turing Machines requires description of the notation used for the input and for the output. Providing a general definition of notations acceptable in the process of computations causes problems. This is because a notation, or an encoding suitable for a computation, has to be computable. Yet, using the concept of computation, in (...)
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