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A taxonomy of deviant encodings

Paula Quinon

In F. Manea, R. Miller & D. Nowotka, Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science, vol 10936. Springer. pp. 338-348 (2018)

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On the Invariance of Gödel’s Second Theorem with Regard to Numberings.Balthasar Grabmayr - 2021 - Review of Symbolic Logic 14 (1):51-84.details The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these (...) Download Export citation Bookmark 10 citations
On Two Different Kinds of Computational Indeterminacy.Philippos Papayannopoulos, Nir Fresco & Oron Shagrir - 2022 - The Monist 105 (2):229-246.details It is often indeterminate what function a given computational system computes. This phenomenon has been referred to as “computational indeterminacy” or “multiplicity of computations.” In this paper, we argue that what has typically been considered and referred to as the challenge of computational indeterminacy in fact subsumes two distinct phenomena, which are typically bundled together and should be teased apart. One kind of indeterminacy concerns a functional characterization of the system’s relevant behavior. Another kind concerns the manner in which the (...) Download Export citation Bookmark 4 citations
On Algorithms, Effective Procedures, and Their Definitions.Philippos Papayannopoulos - 2023 - Philosophia Mathematica 31 (3):291-329.details I examine the classical idea of ‘algorithm’ as a sequential, step-by-step, deterministic procedure (i.e., the idea of ‘algorithm’ that was already in use by the 1930s), with respect to three themes, its relation to the notion of an ‘effective procedure’, its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that ‘algorithm’ has been conceptualized and used in contrasting ways in the above (...) Download Export citation Bookmark 1 citation
Generalization of Shapiro’s theorem to higher arities and noninjective notations.Dariusz Kalociński & Michał Wrocławski - 2022 - Archive for Mathematical Logic 62 (1):257-288.details 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 (...) Download Export citation Bookmark 1 citation
Implicit and Explicit Examples of the Phenomenon of Deviant Encodings.Paula Quinon - 2020 - Studies in Logic, Grammar and Rhetoric 63 (1):53-67.details 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 (...) Download Export citation Bookmark 1 citation
The Anti-Mechanist Argument Based on Gödel’s Incompleteness Theorems, Indescribability of the Concept of Natural Number and Deviant Encodings.Paula Quinon - 2020 - Studia Semiotyczne 34 (1):243-266.details This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that (...) Download Export citation Bookmark 1 citation

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