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Summary The theory of computation is a mathematical theory about the properties of abstract computational objects, such as algorithms and Turing machines. They are abstract in the sense that they ignore or leave out considerations about by features of physical implementations, such as finite memory.  In contrast, computations are done by physical systems: concrete machines made of silicon and metal, or brains made of biological materials, can run algorithms or implement Turing machines. This area is concerned with questions about how the abstract objects that are in the purview of the theory of computation relate to physical systems.
Key works The relationship between abstract computation and physical systems such as brains is a central issue in philosophy of mind, particularly given the rise of computational functionalism as a foundation for the study of the mind.  Here the work of Chalmers 1996 provides a good starting point for bridging the theory of computation with theories of physical systems by means of an implementation relation. 
Introductions A good introduction is Piccinini 2010
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Analog and Digital Computation
  1. Kazuyuki Aihara & Jun Kyung Ryeu (2001). Chaotic Neurons and Analog Computation. Behavioral and Brain Sciences 24 (5):810-811.
    Chaotic dynamics can be related to analog computation. A possibility of electronically implementing the chaos-driven contracting system in the target article is explored with an analog electronic circuit with inevitable noise from the viewpoint of analog computation with chaotic neurons.
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  2. Kenneth Aizawa, It is Not All About Turing-Equivalent Computation.
    One account of the history of computation might begin in the 1930’s with some of the work of Alonzo Church, Alan Turing, and Emil Post. One might say that this is where something like the core concept of computation was first formally articulated. Here were the first attempts to formalize an informal notion of an algorithm or effective procedure by which a mathematician might decide one or another logico-mathematical question. As each of these formalisms was shown to compute the same (...)
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  3. Francesco Berto & Jacopo Tagliabue (2014). The World is Either Digital or Analogue. Synthese 191 (3):481-497.
    We address an argument by Floridi (Synthese 168(1):151–178, 2009; 2011a), to the effect that digital and analogue are not features of reality, only of modes of presentation of reality. One can therefore have an informational ontology, like Floridi’s Informational Structural Realism, without commitment to a supposedly digital or analogue world. After introducing the topic in Sect. 1, in Sect. 2 we explain what the proposition expressed by the title of our paper means. In Sect. 3, we describe Floridi’s argument. In (...)
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  4. Francesco Berto & Jacopo Tagliabue (2012). Cellular Automata. Stanford Encyclopedia of Philosophy.
    Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogeneous, simple units, the atoms or cells. At each time unit, the cells instantiate one of a finite set of states. They evolve in parallel at discrete time steps, following (...)
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  5. James A. Blachowicz (1997). Analog Representation Beyond Mental Imagery. Journal of Philosophy 94 (2):55-84.
  6. David J. Chalmers, Analog Vs. Digital Computation.
    It is fairly well-known that certain hard computational problems (that is, 'difficult' problems for a digital processor to solve) can in fact be solved much more easily with an analog machine. This raises questions about the true nature of the distinction between analog and digital computation (if such a distinction exists). I try to analyze the source of the observed difference in terms of (1) expanding parallelism and (2) more generally, infinite-state Turing machines. The issue of discreteness vs continuity will (...)
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  7. Carol E. Cleland (2002). On Effective Procedures. Minds and Machines 12 (2):159-179.
    Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses (...)
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  8. Jack Copeland, Even Turing Machines Can Compute Uncomputable Functions.
    Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability.
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  9. William Demopoulos (1987). On Some Fundamental Distinctions of Computationalism. Synthese 70 (January):79-96.
    The following paper presents a characterization of three distinctions fundamental to computationalism, viz., the distinction between analog and digital machines, representation and nonrepresentation-using systems, and direct and indirect perceptual processes. Each distinction is shown to rest on nothing more than the methodological principles which justify the explanatory framework of the special sciences.
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  10. Chris Eliasmith (2000). Is the Brain Analog or Digital? Cognitive Science Quarterly 1 (2):147-170.
    It will always remain a remarkable phenomenon in the history of philosophy, that there was a time, when even mathematicians, who at the same time were philosophers, began to doubt, not of the accuracy of their geometrical propositions so far as they concerned space, but of their objective validity and the applicability of this concept itself, and of all its corollaries, to nature. They showed much concern whether a line in nature might not consist of physical points, and consequently that (...)
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  11. Nir Fresco & Marty J. Wolf (2014). The Instructional Information Processing Account of Digital Computation. Synthese 191 (7):1469-1492.
    What is nontrivial digital computation? It is the processing of discrete data through discrete state transitions in accordance with finite instructional information. The motivation for our account is that many previous attempts to answer this question are inadequate, and also that this account accords with the common intuition that digital computation is a type of information processing. We use the notion of reachability in a graph to defend this characterization in memory-based systems and underscore the importance of instructional information for (...)
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  12. Nelson Goodman (1968). Languages of Art. Bobbs-Merrill.
    . . . Unlike Dewey, he has provided detailed incisive argumentation, and has shown just where the dogmas and dualisms break down." -- Richard Rorty, The Yale Review.
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  13. John Haugeland (1981). Analog and Analog. Philosophical Topics 12 (1):213-226.
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  14. Matthew Katz (2008). Analog and Digital Representation. Minds and Machines 18 (3):403-408.
    In this paper, I argue for three claims. The first is that the difference between analog and digital representation lies in the format and not the medium of representation. The second is that whether a given system is analog or digital will sometimes depend on facts about the user of that system. The third is that the first two claims are implicit in Haugeland's (1998) account of the distinction.
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  15. David Lewis (1971). Analog and Digital. Noûs 5 (3):321-327.
  16. Bruce J. MacLennan (1994). Words Lie in Our Way. Minds and Machines 4 (4):421-37.
    The central claim of computationalism is generally taken to be that the brain is a computer, and that any computer implementing the appropriate program would ipso facto have a mind. In this paper I argue for the following propositions: (1) The central claim of computationalism is not about computers, a concept too imprecise for a scientific claim of this sort, but is about physical calculi (instantiated discrete formal systems). (2) In matters of formality, interpretability, and so forth, analog computation and (...)
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  17. Bruce J. MacLennan (1993). Grounding Analog Computers. Philosophical Explorations 2:8-51.
    In this commentary on Harnad's "Grounding Symbols in the Analog World with Neural Nets: A Hybrid Model," the issues of symbol grounding and analog (continuous) computation are separated, it is argued that symbol graounding is as important an issue for analog cognitive models as for digital (discrete) models. The similarities and differences between continuous and discrete computation are discussed, as well as the grounding of continuous representations. A continuous analog of the Chinese Room is presented.
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  18. Corey J. Maley (2011). Analog and Digital, Continuous and Discrete. Philosophical Studies 155 (1):117-131.
    Representation is central to contemporary theorizing about the mind/brain. But the nature of representation--both in the mind/brain and more generally--is a source of ongoing controversy. One way of categorizing representational types is to distinguish between the analog and the digital: the received view is that analog representations vary smoothly, while digital representations vary in a step-wise manner. I argue that this characterization is inadequate to account for the ways in which representation is used in cognitive science; in its place, I (...)
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  19. Jacques Mallah (forthcoming). Structure and Dynamics in Implementation of Computations. In Yasemin J. Erden (ed.), Proceedings of the 7th AISB Symposium on Computing and Philosophy:. AISB.
    Without a proper restriction on mappings, virtually any system could be seen as implementing any computation. That would not allow characterization of systems in terms of implemented computations and is not compatible with a computationalist philosophy of mind. Information-based criteria for independence of substates within structured states are proposed as a solution. Objections to the use of requirements for transitions in counterfactual states are addressed, in part using the partial-brain argument as a general counterargument to neural replacement arguments.
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  20. Drew McDermott (2001). The Digital Computer as Red Herring. Psycoloquy 12 (54).
    Stevan Harnad correctly perceives a deep problem in computationalism, the hypothesis that cognition is computation, namely, that the symbols manipulated by a computational entity do not automatically mean anything. Perhaps, he proposes, transducers and neural nets will not have this problem. His analysis goes wrong from the start, because computationalism is not as rigid a set of theories as he thinks. Transducers and neural nets are just two kinds of computational system, among many, and any solution to the semantic problem (...)
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  21. Gualtiero Piccinini (2008). Computers. Pacific Philosophical Quarterly 89 (1):32–73.
    I offer an explication of the notion of computer, grounded in the practices of computability theorists and computer scientists. I begin by explaining what distinguishes computers from calculators. Then, I offer a systematic taxonomy of kinds of computer, including hard-wired versus programmable, general-purpose versus special-purpose, analog versus digital, and serial versus parallel, giving explicit criteria for each kind. My account is mechanistic: which class a system belongs in, and which functions are computable by which system, depends on the system's mechanistic (...)
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  22. Lee A. Rubel (1989). Digital Simulation of Analog Computation and Church's Thesis. Journal of Symbolic Logic 54 (3):1011-1017.
  23. Hava T. Siegelmann (2003). Neural and Super-Turing Computing. Minds and Machines 13 (1):103-114.
    ``Neural computing'' is a research field based on perceiving the human brain as an information system. This system reads its input continuously via the different senses, encodes data into various biophysical variables such as membrane potentials or neural firing rates, stores information using different kinds of memories (e.g., short-term memory, long-term memory, associative memory), performs some operations called ``computation'', and outputs onto various channels, including motor control commands, decisions, thoughts, and feelings. We show a natural model of neural computing that (...)
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  24. Eric Steinhart (2003). Supermachines and Superminds. Minds and Machines 13 (1):155-186.
    If the computational theory of mind is right, then minds are realized by machines. There is an ordered complexity hierarchy of machines. Some finite machines realize finitely complex minds; some Turing machines realize potentially infinitely complex minds. There are many logically possible machines whose powers exceed the Church–Turing limit (e.g. accelerating Turing machines). Some of these supermachines realize superminds. Superminds perform cognitive supertasks. Their thoughts are formed in infinitary languages. They perceive and manipulate the infinite detail of fractal objects. They (...)
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  25. Russell Trenholme (1994). Analog Simulation. Philosophy of Science 61 (1):115-131.
    The distinction between analog and digital representation is reexamined; it emerges that a more fundamental distinction is that between symbolic and analog simulation. Analog simulation is analyzed in terms of a (near) isomorphism of causal structures between a simulating and a simulated process. It is then argued that a core concept, naturalistic analog simulation, may play a role in a bottom-up theory of adaptive behavior which provides an alternative to representational analyses. The appendix discusses some formal conditions for naturalistic analog (...)
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  26. Jonathan A. Waskan (forthcoming). A Virtual Solution to the Frame Problem. Proceedings of the First IEEE-RAS International Conference on Humanoid Robots.
    We humans often respond effectively when faced with novel circumstances. This is because we are able to predict how particular alterations to the world will play out. Philosophers, psychologists, and computational modelers have long favored an account of this process that takes its inspiration from the truth-preserving powers of formal deduction techniques. There is, however, an alternative hypothesis that is better able to account for the human capacity to predict the consequences worldly alterations. This alternative takes its inspiration from the (...)
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Computers
  1. Gro Bjerknes & Tone Bratteteig (1988). Computers — Utensils or Epaulets? The Application Perspective Revisited. AI and Society 2 (3):258-266.
    This paper is a discussion about how the Application Perspective works in practice.1 We talk about values and attitudes to system development and computer systems, and we illustrate how they have been carried out in practice by examples from the Florence project.2 The metaphors ‘utensil’ and ‘epaulet’ refer to questions about how we conceive the computer system we are to design in the system development process. Our experience is that, in the scientific community, technical challenges mean making computer systems that (...)
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  2. Zach Blas & Micha Cárdenas (2013). Imaginary Computational Systems: Queer Technologies and Transreal Aesthetics. [REVIEW] AI and Society 28 (4):559-566.
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  3. Carol E. Cleland (2002). On Effective Procedures. Minds and Machines 12 (2):159-179.
    Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses (...)
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  4. Carol E. Cleland (1995). Effective Procedures and Computable Functions. Minds and Machines 5 (1):9-23.
    Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept (...)
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  5. Carol E. Cleland (1993). Is the Church-Turing Thesis True? Minds and Machines 3 (3):283-312.
    The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930''s. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn''t be computed by a Turing machine but could be computed by means of some other kind of effective (...)
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  6. B. Jack Copeland (2002). Accelerating Turing Machines. Minds and Machines 12 (2):281-300.
    Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...)
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  7. B. Jack Copeland (2002). Hypercomputation. Minds and Machines 12 (4):461-502.
    A survey of the field of hypercomputation, including discussion of a variety of objections.
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  8. B. Jack Copeland (1996). What is Computation? Synthese 108 (3):335-59.
    To compute is to execute an algorithm. More precisely, to say that a device or organ computes is to say that there exists a modelling relationship of a certain kind between it and a formal specification of an algorithm and supporting architecture. The key issue is to delimit the phrase of a certain kind. I call this the problem of distinguishing between standard and nonstandard models of computation. The successful drawing of this distinction guards Turing's 1936 analysis of computation against (...)
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  9. B. Jack Copeland & Oron Shagrir (2007). Physical Computation: How General Are Gandy's Principles for Mechanisms? [REVIEW] Minds and Machines 17 (2):217-231.
    What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We (...)
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  10. Jack Copeland, Even Turing Machines Can Compute Uncomputable Functions.
    Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability.
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  11. Amnon H. Eden (2007). Three Paradigms of Computer Science. Minds and Machines 17 (2):135-167.
    We examine the philosophical disputes among computer scientists concerning methodological, ontological, and epistemological questions: Is computer science a branch of mathematics, an engineering discipline, or a natural science? Should knowledge about the behaviour of programs proceed deductively or empirically? Are computer programs on a par with mathematical objects, with mere data, or with mental processes? We conclude that distinct positions taken in regard to these questions emanate from distinct sets of received beliefs or paradigms within the discipline: – The rationalist (...)
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  12. Andrew A. Fingelkurts, Alexander A. Fingelkurts & Carlos F. H. Neves (2009). Brain and Mind Operational Architectonics and Man-Made “Machine” Consciousness. Cognitive Processing 10 (2):105-111.
    To build a true conscious robot requires that a robot’s “brain” be capable of supporting the phenomenal consciousness as human’s brain enjoys. Operational Architectonics framework through exploration of the temporal structure of information flow and inter-area interactions within the network of functional neuronal populations [by examining topographic sharp transition processes in the scalp electroencephalogram (EEG) on the millisecond scale] reveals and describes the EEG architecture which is analogous to the architecture of the phenomenal world. This suggests that the task of (...)
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  13. Luciano Floridi (2011). The Construction of Personal Identities Online. Minds and Machines 21 (4):477-479.
    The Construction of Personal Identities Online Content Type Journal Article Category Introduction Pages 1-3 DOI 10.1007/s11023-011-9254-y Authors Luciano Floridi, Department of Philosophy, University of Hertfordshire, de Havilland Campus, Hatfield, Hertfordshire, AL10 9AB UK Journal Minds and Machines Online ISSN 1572-8641 Print ISSN 0924-6495.
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  14. Nir Fresco (2008). An Analysis of the Criteria for Evaluating Adequate Theories of Computation. Minds and Machines 18 (3):379-401.
  15. Vinod Goel (1991). Notationality and the Information Processing Mind. Minds and Machines 1 (2):129-166.
    Cognitive science uses the notion of computational information processing to explain cognitive information processing. Some philosophers have argued that anything can be described as doing computational information processing; if so, it is a vacuous notion for explanatory purposes.An attempt is made to explicate the notions of cognitive information processing and computational information processing and to specify the relationship between them. It is demonstrated that the resulting notion of computational information processing can only be realized in a restrictive class of dynamical (...)
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  16. Joanna Golinska-Pilarek, Angel Mora & Emilio Munoz Velasco (2008). An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-Closeness and Distance. In Tu-Bao Ho & Zhi-Hua Zhou (eds.), PRICAI 2008: Trends in Artificial Intelligence. Springer. 128--139.
    We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some (...)
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  17. Stefan Gruner (2013). Eric Winsberg: Science in the Age of Computer Simulation. [REVIEW] Minds and Machines 23 (2):251-254.
  18. Hartmut Haberland (1996). Cognitive Technology and Pragmatics: Analogies and (Non-)Alignments. [REVIEW] AI and Society 10 (3-4):303-308.
    This paper presents some considerations about the relationship between languages and computer systems from a pragmatic, user-centered point of view.
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  19. Boaz Miller & Isaac Record (2013). Justified Belief in a Digital Age: On the Epistemic Implications of Secret Internet Technologies. Episteme 10 (02):117 - 134.
    People increasingly form beliefs based on information gained from automatically filtered Internet ‎sources such as search engines. However, the workings of such sources are often opaque, preventing ‎subjects from knowing whether the information provided is biased or incomplete. Users’ reliance on ‎Internet technologies whose modes of operation are concealed from them raises serious concerns about ‎the justificatory status of the beliefs they end up forming. Yet it is unclear how to address these concerns ‎within standard theories of knowledge and justification. (...)
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  20. Gualtiero Piccinini, Computation in Physical Systems. Stanford Encyclopedia of Philosophy.
  21. Gualtiero Piccinini (2008). Computers. Pacific Philosophical Quarterly 89 (1):32–73.
    I offer an explication of the notion of computer, grounded in the practices of computability theorists and computer scientists. I begin by explaining what distinguishes computers from calculators. Then, I offer a systematic taxonomy of kinds of computer, including hard-wired versus programmable, general-purpose versus special-purpose, analog versus digital, and serial versus parallel, giving explicit criteria for each kind. My account is mechanistic: which class a system belongs in, and which functions are computable by which system, depends on the system's mechanistic (...)
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  22. Gualtiero Piccinini (2008). Some Neural Networks Compute, Others Don't. Neural Networks 21 (2-3):311-321.
    I address whether neural networks perform computations in the sense of computability theory and computer science. I explicate and defend
    the following theses. (1) Many neural networks compute—they perform computations. (2) Some neural networks compute in a classical way.
    Ordinary digital computers, which are very large networks of logic gates, belong in this class of neural networks. (3) Other neural networks
    compute in a non-classical way. (4) Yet other neural networks do not perform computations. Brains may well fall into this last class.
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  23. Gualtiero Piccinini (2007). Computing Mechanisms. Philosophy of Science 74 (4):501-526.
    This paper offers an account of what it is for a physical system to be a computing mechanism—a system that performs computations. A computing mechanism is a mechanism whose function is to generate output strings from input strings and (possibly) internal states, in accordance with a general rule that applies to all relevant strings and depends on the input strings and (possibly) internal states for its application. This account is motivated by reasons endogenous to the philosophy of computing, namely, doing (...)
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  24. Ricardo Restrepo (2013). Realismo científico, computacionalismo y la máxima pragmática. In Douglas Anderson, Ricardo Restrepo, Victor Hugo Chica & Diana Patricia Carmona (eds.), El pragmatismo norteamericano. IAEN.
    Se identifica el argumento de que la teoría de que hay propiedades computacionales suficientes para propiedades mentales es una teoría o falsa o vacía, ya que las propiedades computacionales no son empíricamente descubriles, intrínsecas ni causales, como sí lo son las propiedades mentales. Es un argumento que se puede destilar de los problemas que John Searle imputa a la ciencia cognitiva computacional, pero encuentra su correlato antecedente en el argumento que Max Newman utilizó para refutar el estructuralismo físico de Bertrand (...)
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