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  1. Alan Turing and the Mathematical Objection.Gualtiero Piccinini - 2003 - Minds and Machines 13 (1):23-48.
    This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for (...)
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  • Physical Hypercomputation and the Church–Turing Thesis.Oron Shagrir & Itamar Pitowsky - 2003 - Minds and Machines 13 (1):87-101.
    We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy's thesis.
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  • Effective Computation by Humans and Machines.Shagrir Oron - 2002 - Minds and Machines 12 (2):221-240.
    There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument (...)
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  • What is Categorical Structuralism?Geoffrey Hellman - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 151--161.
  • Computational Complexity Theory and the Philosophy of Mathematics†.Walter Dean - 2019 - Philosophia Mathematica 27 (3):381-439.
    Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...)
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  • Kurt Gödel and Computability Theory.Richard Zach - 2006 - In Arnold Beckmann, Ulrich Berger, Benedikt Löwe & John V. Tucker (eds.), Logical Approaches to Computational Barriers. Second Conference on Computability in Europe, CiE 2006, Swansea. Proceedings. Berlin: Springer. pp. 575--583.
    Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar (...)
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  • Three Dogmas of First-Order Logic and Some Evidence-Based Consequences for Constructive Mathematics of Differentiating Between Hilbertian Theism, Brouwerian Atheism and Finitary Agnosticism.Bhupinder Singh Anand - manuscript
    We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...)
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  • Enciclopédia de Termos Lógico-Filosóficos.João Branquinho, Desidério Murcho & Nelson Gonçalves Gomes (eds.) - 2006 - São Paulo, SP, Brasil: Martins Fontes.
    Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...)
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  • Philosophy of Mathematics and Computer Science.Kazimierz Trzęsicki - 2010 - Studies in Logic, Grammar and Rhetoric 22 (35).
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  • On Communication and Computation.Paul Bohan Broderick - 2004 - Minds and Machines 14 (1):1-19.
    Comparing technical notions of communication and computation leads to a surprising result, these notions are often not conceptually distinguishable. This paper will show how the two notions may fail to be clearly distinguished from each other. The most famous models of computation and communication, Turing Machines and (Shannon-style) information sources, are considered. The most significant difference lies in the types of state-transitions allowed in each sort of model. This difference does not correspond to the difference that would be expected after (...)
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  • Paper Machines.Daniele Mundici & Wilfried Seig - 1995 - Philosophia Mathematica 3 (1):5-30.
    Machines were introduced as calculating devices to simulate operations carried out by human computers following fixed algorithms. The mathematical study of (paper) machines is the topic of our essay. The first three sections provide necessary logical background, examine the analyses of effective calculability given in the thirties, and describe results that are central to recursion theory, reinforcing the conceptual analyses. In the final section we pursue our investigation in a quite different way and focus on principles that govern the operations (...)
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  • Informal Versus Formal Mathematics.Francisco Antonio Doria - 2007 - Synthese 154 (3):401-415.
    We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.
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  • Scientific Phenomena and Patterns in Data.Pascal Ströing - 2018 - Dissertation, LMU München
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  • Kalmár's Argument Against the Plausibility of Church's Thesis.Máté Szabó - 2018 - History and Philosophy of Logic 39 (2):140-157.
    In his famous paper, An Unsolvable Problem of Elementary Number Theory, Alonzo Church identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his (...)
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  • Computing Machinery and Intelligence.Alan M. Turing - 1950 - Mind 59 (October):433-60.
    I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to (...)
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  • On Constructivity and the Rosser Property: A Closer Look at Some Gödelean Proofs.Saeed Salehi & Payam Seraji - 2018 - Annals of Pure and Applied Logic 169 (10):971-980.
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  • Generality’s Price: Inescapable Deficiencies in Machine-Learned Programs.John Case, Keh-Jiann Chen, Sanjay Jain, Wolfgang Merkle & James S. Royer - 2006 - Annals of Pure and Applied Logic 139 (1):303-326.
    This paper investigates some delicate tradeoffs between the generality of an algorithmic learning device and the quality of the programs it learns successfully. There are results to the effect that, thanks to small increases in generality of a learning device, the computational complexity of some successfully learned programs is provably unalterably suboptimal. There are also results in which the complexity of successfully learned programs is asymptotically optimal and the learning device is general, but, still thanks to the generality, some of (...)
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  • Computation in Cognitive Science: It is Not All About Turing-Equivalent Computation.Kenneth Aizawa - 2010 - Studies in History and Philosophy of Science Part A 41 (3):227-236.
    It is sometimes suggested that the history of computation in cognitive science is one in which the formal apparatus of Turing-equivalent computation, or effective computability, was exported from mathematical logic to ever wider areas of cognitive science and its environs. This paper, however, indicates some respects in which this suggestion is inaccurate. Computability theory has not been focused exclusively on Turing-equivalent computation. Many essential features of Turing-equivalent computation are not captured in definitions of computation as symbol manipulation. Turing-equivalent computation did (...)
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  • Diagonalisation and Church's Thesis: Kleene's Homework.Enrique Alonso & Maria Manzano - 2005 - History and Philosophy of Logic 26 (2):93-113.
    In this paper we will discuss the active part played by certain diagonal arguments in the genesis of computability theory. 1?In some cases it is enough to assume the enumerability of Y while in others the effective enumerability is a substantial demand. These enigmatical words by Kleene were our point of departure: When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the ??definable functions. But, quickly realizing that the diagonalization cannot be (...)
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  • Alonzo Church:His Life, His Work and Some of His Miracles.Maía Manzano - 1997 - History and Philosophy of Logic 18 (4):211-232.
    This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...)
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  • General Iteration and Unary Functions.G. M. Germano & S. Mazzanti - 1991 - Annals of Pure and Applied Logic 54 (2):137-178.
    Programming practice suggests a notion of general iteration corresponding to the while-do construct. This leads to new characterizations of general computable unary functions usable in computer science.
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  • Louis Joly as a Platonist Painter?Roger Pouivet - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 337--341.
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