Gödel’s first incompleteness theorem is sometimes said to refute mechanism about the mind. §1 contains a discussion of mechanism. We look into its origins, motivations and commitments, both in general and with regard to the human mind, and ask about the place of modern computers and modern cognitive science within the general mechanistic paradigm. In §2 we give a sharp formulation of a mechanistic thesis about the mind in terms of the mathematical notion of computability. We present the argument (...) from Gödel’s theorem against mechanism in terms of this formulation and raise two objections, one of which is known but is here given a more precise formulation, and the other is new and based on the discussion in §1. (shrink)

We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.

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 the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics. (shrink)

Penrose proved that a computational or formalizable theory of the brainís cognitive functioning is impossible, but suggested that a physical non-computational and non-formalizable one may be viable. Arguments as to why Penroseís program is unrealizable are presented. The main argument is that a non-formalizable theory should be verbal. However, verbal paradoxes based on Cantorís diagonal processes show the impossibility of a consistent verbal theory of the brain comprising its arithmetical cognition. It is suggested that (...) comprehensive theories of the human brain and of physical experience are Kantian rather than Platonic ideas. This suggestion is likewise based on arguments related to diagonal processes. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined (...) with Gödel’s results and accepted theorems of recursion theory, does provide the basis for an apparent paradox. The difficulty arises when such an algorithm is embedded within a computer program of sufficient arithmetic power. The required computer program (an AI system) is described herein, and the paradox is derived. A solution to the paradox is proposed, which, it is argued, illuminates the truth status of axioms in formal models of programs and Turing machines. (shrink)

According to the computational theory of mind , to think is to compute. But what is meant by the word 'compute'? The generally given answer is this: Every case of computing is a case of manipulating symbols, but not vice versa - a manipulation of symbols must be driven exclusively by the formal properties of those symbols if it is qualify as a computation. In this paper, I will present the following argument. Words like 'form' and 'formal' are (...) ambiguous, as they can refer to form in either the syntactic or the morphological sense. CTM fails on each disambiguation, and the arguments for CTM immediately cease to be compelling once we register that ambiguity. The terms 'mechanical' and 'automatic' are comparably ambiguous. Once these ambiguities are exposed, it turns out that there is no possibility of mechanizing thought, even if we confine ourselves to domains where all problems can be settled through decision-procedures. The impossibility of mechanizing thought thus has nothing to do with recherché mathematical theorems, such as those proven by Gödel and Rosser. A related point is that CTM involves, and is guilty of reinforcing, a misunderstanding of the concept of an algorithm. (shrink)

"The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...) of an algorithm as central to AI, whereas it is argued here that for the purpose of explaining mental capabilities the architecture of an intelligent system is more important than the concept of an algorithm, using the premise that what makes something intelligent is not what it does but how it does it. What needs to be explained is also unclear: Penrose thinks we all know what consciousness is and claims that the ability to judge Go "del's formula to be true depends on it. He also suggests that quantum phenomena underly consciousness. This is rebutted by arguing that our existing concept of "consciousness" is too vague and muddled to be of use in science. This and related concepts will gradually be replaced by a more powerful theory-based taxonomy of types of mental states and processes. The central argument offered by Penrose against the strong AI thesis depends on a tempting but unjustified interpretation of Goedel's incompleteness theorem. Some critics are shown to have missed the point of his argument. A stronger criticism is mounted, and the relevance of mathematical Platonism analysed. Architectural requirements for intelligence are discussed and differences between serial and parallel implementations analysed. (shrink)

In this book, Ben-Yami reassesses the way Descartes developed and justified some of his revolutionary philosophical ideas. The first part of the book shows that one of Descartes' most innovative and influential ideas was that of representation without resemblance. Ben-Yami shows how Descartes transfers insights originating in his work on analytic geometry to his theory of perception. The second part shows how Descartes was influenced by the technology of the period, notably clockwork automata, in holding life to be a (...) mechanical phenomenon, reducing the soul to the mind and considering it immaterial. Ben-Yami explores the later role of the digital computer in Turing's criticism of Descartes' ideas. The last part discusses the Meditations: far from starting everything afresh without presupposing anything that can be doubted, Descartes' innovations in the dream argument, the cogito and elsewhere are modifications of old ideas based upon considerations issuing from his separately developed theories, formed under the influence of the technology, mathematics and science of his age. (shrink)

In this paper I discuss whether Gödel's incompleteness theorems have any implications for studies in Artificial Life (AL). Since Gödel's incompleteness theorems have been used to argue against certain mechanistic theories of the mind, it seems natural to attempt to apply the theorems to certain strong mechanistic arguments postulated by some AL theorists. -/- We find that an argument using the incompleteness theorems can not be constructed that will block the hard AL claim, specifically in the field (...) of robotics. However, we will see that the beginnings of an argument casting doubt on our ability to create living systems entirely resident in a computer environment might be suggested by looking at the incompleteness theorems from the point of view of Gödel's belief in mathematical realism. (shrink)

Hubert Dreyfus once noted that it would be difficult to ascertain whether Edmund Husserl had a computational theory of mind. I provide evidence that he had one. Both Steven Pinker and Steven Horst think that the computational theory of mind must have two components: a representational-symbolic component and a causal component. Bearing this in mind, we proceed to a close-reading of the sections of “On the Logic of Signs” wherein Husserl presents, if I’m correct, his (...) class='Hi'>computational theory of mind embedded in a language of thought. My argument goes like this: the computational theory of mind is the idea, following Haugeland, that the mind comes prepackaged as, or is endogenously constrained to be, an automatic formal system; this explains, according to Husserl, why automatic trains of thought without logical intent resemble arguments exhibiting deductive structure with logical intent. In general, an automatic formal system yields true results provided that the syntactic symbols with which they compute are univocal and are semantically evaluable, and the mechanized inferences they perform are valid and preserve truth. These two conditions describe a computational cognitive process: the first condition connects representations to syntax, and the second condition uses the syntax, in inauthentic judging, to arrive at true conclusions through blind causality. Now, in point of textual fact, these are the conditions which Husserl attributes to our “natural psychological mechanism of symbolic inference” which typically yields true results. Since a formal system attributed to the “internal structure” of the mind, and guided by blind causality, just is the computational theory of mind, it follows, I think, that Husserl had a computational theory of mind. This computational theory is, moreover, embedded in a language of thought, since Husserl attributes a language-like form to our thoughts so that they may be mechanically processed. I conclude with a discussion of my results. (shrink)

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of (...) forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)

Marco Buzzoni Gödel, Searle, and the Computational Theory of the (Other) Mind According to Sergio Galvan, some of the arguments offered by Lucas and Penrose are somewhat obscure or even logically invalid, but he accepts their fundamental idea that a human mind does not work as a computational machine. His main point is that there is a qualitative difference between the principles of the logic of provability and those of the logic of evidence and belief. To evaluate (...) this suggestion, I shall first compare it with Searle’s concept of “intentionality”, and then introduce a distinction between two different senses of intentionality: a reflexive-transcendental sense and a positive (that is, historical empirical or formal logical) one. In the first of these senses, the nature of human reason is such that we have no idea how a real material system – or the corresponding formal one – could instantiate it. However, although this will turn out to be an important element of truth in Searle’s and Galvan’s conception, it does not exclude the opposite truth of Turing’s functionalism: because intentionality, intuition, vision or insight – taken in their reflexive-transcendental sense – are simply invisible to the scientific eye, a man and a machine (or a robot) that is and one that is not endowed with intentionality are de facto indistinguishable from a strictly scientific point of view. For this reason, we might eventually be entitled, or even — by the practical precautionary principle – morally obliged, to attribute minds to machines. (shrink)

I argue that Stich's Syntactic Theory of Mind (STM) and a naturalistic narrow content functionalism run on a Language of Though story have the same exact structure. I elaborate on the argument that narrow content functionalism is either irremediably holistic in a rather destructive sense, or else doesn't have the resources for individuating contents interpersonally. So I show that, contrary to his own advertisement, Stich's STM has exactly the same problems (like holism, vagueness, observer-relativity, etc.) that he claims plague (...) content-based psychologies. So STM can't be any better than the Representational Theory of Mind (RTM) in its prospects for forming the foundations of a scientifically respectable psychology, whether or not RTM has the problems that Stich claims it does. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

According to the computational theory of mind (CTM), mental capacities are explained by inner computations, which in biological organisms are realized in the brain. Computational explanation is so popular and entrenched that it’s common for scientists and philosophers to assume CTM without argument.

Technological advances in computer science have secured the computer metaphor status of a heuristic methodological tool used to answer the question about the nature of mind. Nevertheless, some philosophers strongly support opposite opinions. Anti-computationalism in the philosophy of mind is a methodological program that uses extremely heterogeneous grounds for argumentation, deserving analysis and discussion. This article provides an overview and interpretation of the traditional criticism of the computational theory of mind ; its basic theses have been formed in (...) Western philosophy in the last quarter of the 20th century. The main goal is to reveal the content of the arguments of typical anti-computationalist programs and expand their application to the framework of the semantic problems of the Classic Computational Theory of Mind. The main fault of the symbolic ap-proach in the classical computationalism is the absence of a full-fledged theory of semantic properties. The relevance of considering these seemingly outdated problems is justified by the fact that the problem of meaning remains in the core of the latest developments in various areas of AI and the principles of human-computer interaction. (shrink)

Technological advances in computer science have secured the computer metaphor status of a heuristic methodological tool used to answer the question about the nature of mind. Nevertheless, some philosophers strongly support opposite opinions. Anti-computationalism in the philosophy of mind is a methodological program that uses extremely heterogeneous grounds for argumentation, deserving analysis and discussion. This article provides an overview and interpretation of the traditional criticism of the computational theory of mind ; its basic theses have been formed in (...) Western philosophy in the last quarter of the 20th century. The main goal is to reveal the content of the arguments of typical anti-computationalist programs and expand their application to the framework of the semantic problems of the Classic Computational Theory of Mind. The main fault of the symbolic ap-proach in the classical computationalism is the absence of a full-fledged theory of semantic properties. The relevance of considering these seemingly outdated problems is justified by the fact that the problem of meaning remains in the core of the latest developments in various areas of AI and the principles of human-computer interaction. (shrink)

Abstract Mental representations, Swiatczak (Minds Mach 21:19–32, 2011) argues, are fundamentally biochemical and their operations depend on consciousness; hence the computational theory of mind, based as it is on multiple realisability and purely syntactic operations, must be wrong. Swiatczak, however, is mistaken. Computation, properly understood, can afford descriptions/explanations of any physical process, and since Swiatczak accepts that consciousness has a physical basis, his argument against computationalism must fail. Of course, we may not have much idea how consciousness (itself (...) a rather unclear plurality of notions) might be implemented, but we do have a hypothesis—that all of our mental life, including consciousness, is the result of computational processes and so not tied to a biochemical substrate. Like it or not, the computational theory of mind remains the only game in town. Content Type Journal Article Pages 1-8 DOI 10.1007/s11023-012-9271-5 Authors David Davenport, Computer Engineering Department, Bilkent University, 06800 Ankara, Turkey Journal Minds and Machines Online ISSN 1572-8641 Print ISSN 0924-6495. (shrink)

In The Mind Doesn’t Work that Way, Jerry Fodor argues that mental representations have context sensitive features relevant to cognition, and that, therefore, the Classical Computational Theory of Mind (CTM) is mistaken. We call this the Globality Argument. This is an in principle argument against CTM. We argue that it is self-defeating. We consider an alternative argument constructed from materials in the discussion, which avoids the pitfalls of the official argument. We argue that it is also unsound and (...) that, while it is an empirical issue whether context sensitive features of mental representations are relevant to cognition, it is empirically implausible. (shrink)

Over the past several decades, the philosophical community has witnessed the emergence of an important new paradigm for understanding the mind.1 The paradigm is that of machine computation, and its influence has been felt not only in philosophy, but also in all of the empirical disciplines devoted to the study of cognition. Of the several strategies for applying the resources provided by computer and cognitive science to the philosophy of mind, the one that has gained the most attention from philosophers (...) has been the Computational Theory of Mind (CTM). CTM was first articulated by Hilary Putnam (1960, 1961), but finds perhaps its most consistent and enduring advocate in Jerry Fodor (1975, 1980, 1981, 1987, 1990, 1994). It is this theory, and not any broader interpretations of what it would be for the mind to be a computer, that I wish to address in this paper. What I shall argue here is that the notion of symbolic representation employed by CTM is fundamentally unsuited to providing an explanation of the intentionality of mental states (a major goal of CTM), and that this result undercuts a second major goal of CTM, sometimes refered to as the vindication of intentional psychology. This line of argument is related to the discussions of derived intentionality by Searle (1980, 1983, 1984) and Sayre (1986, 1987). But whereas those discussions seem to be concerned with the causal dependence of familiar sorts of symbolic representation upon meaning-bestowing acts, my claim is rather that there is not one but several notions of meaning to be had, and that the notions that are applicable to symbols are conceptually dependent upon the notion that is applicable to mental states in the fashion that Aristotle refered to as paronymy. That is, an analysis of the notions of meaning applicable to symbols reveals that they contain presuppositions about meaningful mental states, much as Aristotle's analysis of the sense of healthy that is applied to foods reveals that it means conducive to having a healthy body, and hence any attempt to explain mental semantics in terms of the semantics of symbols is doomed to circularity and regress. I shall argue, however, that this does not have the consequence that computationalism is bankrupt as a paradigm for cognitive science, as it is possible to reconstruct CTM in a fashion that avoids these difficulties and makes it a viable research framework for psychology, albeit at the cost of losing its claims to explain intentionality and to vindicate intentional psychology. I have argued elsewhere (Horst, 1996) that local special sciences such as psychology do not require vindication in the form of demonstrating their reducibility to more fundamental theories, and hence failure to make good on these philosophical promises need not compromise the broad range of work in empirical cognitive science motivated by the computer paradigm in ways that do not depend on these problematic treatments of symbols. (shrink)

The Computational Theory of Mind (CTM) holds that the mind is a computer and that cognition is the manipulation of representations. CTM is commonly viewed as the main hypothesis in cognitive science, with classical CTM (related to the Language of Thought Hypothesis) being the most popular variant. However, other computational accounts of the mind either reject LOTH or do not subscribe to RTM. CTM proponents argue that it clarifies how thought and content are causally relevant in the (...) physical world, and they frame this argument in terms of the physical symbol hypothesis, formalism, or syntactic engines. This article focuses on specific problems with CTM, with four main sections covering the introduction of the three main variants of CTM, the most important conceptions of computational explanation in cognitive science, skeptical arguments against CTM raised by Hilary Putnam, and common objections to CTM. (shrink)

The frame problem is a fundamental challenge in AI, and the Lucas-Penrose argument is supposed to show a limitation of AI if it is successful at all. Here we discuss both of them from a unified Gödelian point of view. We give an informational reformulation of the frame problem, which turns out to be tightly intertwined with the nature of Gödelian incompleteness in the sense that they both hinge upon the finitarity condition of agents or systems, without which their (...) alleged limitations can readily be overcome, and that they can both be seen as instances of the fundamental discrepancy between finitary beings and infinitary reality. We then revisit the Lucas-Penrose argument, elaborating a version of it which indicates the impossibility of information physics or the computational theory of the universe. It turns out through a finer analysis that if the Lucas-Penrose argument is accepted then information physics is impossible too; the possibility of AI or the computational theory of the mind is thus linked with the possibility of information physics or the computational theory of the universe. We finally reconsider the Penrose’s Quantum Mind Thesis in light of recent advances in quantum modelling of cognition, giving a structural reformulation of it and thereby shedding new light on what is problematic in the Quantum Mind Thesis. Overall, we consider it promising to link the computational theory of the mind with the computational theory of the universe; their integration would allow us to go beyond the Cartesian dualism, giving, in particular, an incarnation of Chalmers’ double-aspect theory of information. (shrink)

Gödel argued that his incompleteness theorems imply that either “the mind cannot be mechanized” or “there are absolutely undecidable sentences.” In the precursor to this paper I examined the early arguments for the first disjunct. In the present paper I examine the most sophisticated argument for the first disjunct, namely, Penrose’s new argument. It turns out that Penrose’s argument requires a type-free notion of truth and a type-free notion of absolute provability. I show that there is a natural such (...) system, DTK. I prove a series of results which show that Gödel’s disjunction is provable in the system, Penrose’s argument is invalid in the system, there can be no proof or refutation of either disjunct in the system, the independence results are robust in that they persist when one strengthens the principles governing absolute provability, and there are reasons to believe that the situation will not improve under any plausible alteration of the underlying theory of truth. (shrink)

Given any simply consistent formal theory F of the state complexity L(S) of finite binary sequences S as computed by 3-tape-symbol Turing machines, there exists a natural number L(F ) such that L(S) > n is provable in F only if n L(F ). The proof resembles Berry’s..

In the field of computability and algorithmicity, there have recently been two essays that are of great interest: Peter Slezak's "Descartes's Diagonal Deduction," and David Deutsch's "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer." In brief, the former shows that Descartes' Cogito argument is structurally similar to Godel's proof that there are statements true but cannot be proven within a formal system such as Principia Mathematica, while Deutsch provides strong arguments for believing that the universe can (...) be represented as a Turing machine. King contends that the conjoining of Slezak's analysis with Deutsch's provides a perspective from which it is possible to argue that a scientific theology can be taken a little more seriously at present than in the past. , , , , In the field of computability and algorithmicity, there have recently been two essays that are of great interest: Peter Slezak's "Descartes's Diagonal Deduction," and David Deutsch's "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer." In brief, the former shows that Descartes' Cogito argument is structurally similar to Godel's proof that there are statements true but cannot be proven within a formal system such as Principia Mathematica, while Deutsch provides strong arguments for believing that the universe can be represented as a Turing machine. King contends that the conjoining of Slezak's analysis with Deutsch's provides a perspective from which it is possible to argue that a scientific theology can be taken a little more seriously at present than in the past. (shrink)

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known (...) mathematics must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound. (shrink)

Suppose the principles explaining how the human mind (brain) reaches logical conclusions and judgments were different from – and independent of – thoseinvolved innormatively valid reasoning. Then such principles should affect both conclusion generation and recognition that particular conclusions are or are not justified. People, however, demonstrate a discrepancy between impaired performance in generating logical conclusions as opposed to rather impressive competence in recognizing rational (versus irrational) ones. This discrepancy is hypothesized to arise from often generating an incomplete specification of (...) a logical or judgmental problem when attempting to solve it – versus a recognition of such incompleteness when it is pointed out. The basic argument is developed, with common examples, in the context of specifying or failing to specify all possible combinations in simple logical arguments and is then extended to probabilistic reasoning, where complete versus incomplete specification corresponds to attending to all or to only some components of Bayes theorem-based reasoning. (shrink)

Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T−1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T−1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based (...) theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution. As a preparation of Banach’s Inverse Mapping Theorem we also study the Open Mapping Theorem and we show that the uniform versions of both theorems are limit computable, which means that they are effectively -measurable with respect to the effective Borel hierarchy. (shrink)

Suppose the principles explaining how the human mind (brain) reaches logical conclusions and judgments were different from – and independent of – thoseinvolved innormatively valid reasoning. Then such principles should affect both conclusion generation and recognition that particular conclusions are or are not justified. People, however, demonstrate a discrepancy between impaired performance in generating logical conclusions as opposed to rather impressive competence in recognizing rational (versus irrational) ones. This discrepancy is hypothesized to arise from often generating an incomplete specification of (...) a logical or judgmental problem when attempting to solve it – versus a recognition of such incompleteness when it is pointed out. The basic argument is developed, with common examples, in the context of specifying or failing to specify all possible combinations in simple logical arguments and is then extended to probabilistic reasoning, where complete versus incomplete specification corresponds to attending to all or to only some components of Bayes theorem-based reasoning. (shrink)

Kirsten Besheer has recently considered Descartes’ doubting appropriately in the context of his physiological theories in the spirit of recent important re-appraisals of his natural philosophy. However, Besheer does not address the notorious indubitability and its source that Descartes claims to have discovered. David Cunning has remarked that Descartes’ insistence on the indubitability of his existence presents “an intractable problem of interpretation” in the light of passages that suggest his existence is “just as dubitable as anything else”. However, although the (...) cogito argument is widely thought to be central to the force of Descartes’ indubitability, for his part, Cunning does not consider its relevance and force. Accordingly, this article is concerned with the cogito argument and the question central to Hintikka’s seminal contribution, described by Cottingham as “Perhaps the most debated question,” namely, whether or not the cogito can be construed as a logical inference. Clearly, an inferential account has the potential to explain the certainty of Descartes’ conclusion that he exists. Recently, Sarkar offers what he characterizes as “novel and fairly conclusive reasons why the cogito cannot be construed as an argument,” asserting “the discovery of the cogito can only be an intuition not a deduction.” Obviously, it would greatly support the opposing inferential construal if a remotely plausible logical argument could be proposed. Toward this end, I defend the virtues of my ‘Diagonal’ account of Descartes’ cogito Above all, I show how my analysis meets the requirement that any satisfactory solution to the problem of the cogito would reconcile Descartes’ claim that the cogito is a certain inference with his claim that it is an intuitive kind of knowledge. Through a critical discussion of analyses such as that of Gallois , I show that it is possible to provide a textually faithful analysis that permits seeing the cogito as both inference and intuition because it may be seen as an exercise in the mathematical method of Analysis. Above all, as Feldman requires, I show that the Diagonal account is not only textually elegant, but permits crediting Descartes with a worthy insight, thereby resolving the tension between what Howell has termed the Humean and Cartesian problems, namely, the elusiveness and the certainty of the self. (shrink)

Descartes argued that productivity, namely our ability to generate an unlimited number of new thoughts or ideas from previous ones, derives from a single undividable source in the human soul. Cognitive scientists, in contrast, have viewed productivity as a modular phenomenon. According to this latter view, syntactic, semantic, musical or visual productivity emerges each from their own generative engines in the human brain. Recent evidence has, however, led some authors to revitalize the Cartesian theory. According to this view, a (...) single source or a single mechanism in the human brain produces productivity in every cognitive domain, whether in the domain of music, semantics or syntax. In this article, we will address recent evidence concerning the single source hypothesis from brain-imaging studies, linguistics, cognitive theories of music perception, biology of cognition and cognitive development, along with several objections that have been presented against this hypothesis. We formulate two versions of the Cartesian theory which combine the more recent computational theory of cognition with Descartes' view on productivity. (shrink)

It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...) pictures a monadology already via its form. And secondly, because Gödel wanted to establish in his Philosophical Remarks a modern monadology in adapting the philosophical principles of Leibniz’s monadology to modern science. This article will only deal with the first aspect that has been mentioned namely that a single remark by Gödel (very often) mirrors his whole philosophy (at the time) for example: set theory, perception and intuition of mathematical objects and axioms, the nature of the human mind, the concept and existence of God, and so on. This renders it extremely difficult to interpret Gödel’s philosophical remarks but interpreting it provides also a deep insight in Gödel’s philosophical thoughts. (shrink)

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The (...) usual ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

ABSTRACT Usually, the problems in AI may be many times related to Philosophy of Mind, and perhaps because this reason may be in essence very disputable. So, for instance, the famous question: Can a machine think? It was proposed by Alan Turing [16]. And it may be the more decisive question, but for many people it would be a nonsense. So, two of the very fundamental and more confronted positions usually considered according this line include the Connectionism and the (...) class='Hi'>Computational Theory of Mind. We analyze here its content, with their past disputes, and current situation. (shrink)

Crispin Wright joins the ranks of those who have sought to refute mechanist theories of mind by invoking Gödel's Incompleteness Theorems. His predecessors include Gödel himself, J. R. Lucas and, most recently, Roger Penrose. The aim of this essay is to show that, like his predecessors, Wright, too, fails to make his case, and that, indeed, he fails to do so even when judged by standards of success which he himself lays down.

Following Post program, we will propose a linguistic and empirical interpretation of Gödel’s incompleteness theorem and related ones on unsolvability by Church and Turing. All these theorems use the diagonal argument by Cantor in order to find limitations in finitary systems, as human language, which can make “infinite use of finite means”. The linguistic version of the incompleteness theorem says that every Turing complete language is Gödel incomplete. We conclude that the incompleteness and unsolvability theorems find (...) limitations in our finitary tool, which is our complete language. (shrink)

John Searle’s Chinese room argument is a celebrated thought experiment designed to refute the hypothesis, popular among artificial intelligence scientists and philosophers of mind, that “the appropriately programmed computer really is a mind”. Since its publication in 1980, the CRA has evoked an enormous amount of debate about its implications for machine intelligence, the functionalist philosophy of mind, theories of consciousness, etc. Although the general consensus among commentators is that the CRA is flawed, and not withstanding the popularity of the (...) systems reply in some quarters, there is remarkably little agreement on exactly how and why it is flawed. A newcomer to the controversy could be forgiven for thinking that the bewildering collection of diverse replies to Searle betrays a tendency to unprincipled, ad hoc argumentation and, thereby, a weakness in the opposition’s case. In this paper, treating the CRA as a prototypical example of a ‘destructive’ thought experiment, I attempt to set it in a logical framework, which allows us to systematise and classify the various objections. Since thought experiments are always posed in narrative form, formal logic by itself cannot fully capture the controversy. On the contrary, much also hinges on how one translates between the informal everyday language in which the CRA was initially framed and formal logic and, in particular, on the specific conception of possibility that one reads into the logical formalism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Descartes’s Theory of MindEnrique Chávez-ArvizoDesmond M. Clarke. Descartes’s Theory of Mind. Oxford: Clarendon Press, 2003. Pp. viii + 267. Cloth, $49.95.Desmond Clarke, commentator on Cartesian natural philosophy, has now published an interpretation of Descartes's dualism, a theme which can hardly be said to be underrepresented in the literature. The monograph is divided into nine chapters concerned with explanation, sensation, imagination and memory, the passions, the will, (...) language, thought, substance, and property dualism—yes, property dualism.The book pushes forward a reading of Descartes's theory of mind, made from a perspective which Clarke has adopted over the last three decades or so, that maintains that Descartes was primarily (and willingly) a scientist and incidentally (and reluctantly) a philosopher. From this viewpoint, according to Clarke, natural philosophy provides the paradigm for genuine explanation in Descartes's metaphysics. Furthermore, given that Descartes rejects Scholastic substantial forms and real qualities as explanatorily infertile and since substances are equally unfruitful, substances play no explanatory role in his system of philosophy. Substance dualism is not at the center of Descartes's philosophy of mind; his account of explanation is. This makes Descartes not a substance dualist; but a property dualist. So argues, in very broad strokes, Clarke. [End Page 116]Unlike us, Descartes did not draw a sharp distinction between philosophy and science. For him, these two realms of enquiry were part of a unified realm of knowledge, what he deemed scientia or philosophy in the wider sense. Hence his use of the instructive metaphor of philosophy as a tree, whose roots are metaphysics, the trunk physics, and the branches the other sciences which he reduces to three principal ones, namely medicine, mechanics, and morals (Letter Preface to Picot, French translation of the Principles of Philosophy, Oeuvres de Descartes, ed. C. Adam and P. Tannery [Paris: Vrin, 1989], 14). However, Descartes did distinguish the roots from the trunk, that is, metaphysics (or "first philosophy") from physics (or "natural philosophy"). Commentators on Descartes's system of philosophy ignore his natural philosophy at their peril, for it does shed considerable light on his metaphysics. But, likewise, Descartes's philosophy greatly illuminates his science. It is for this reason that I want to cast doubt on Clarke's apparent scientific imperialism in which Descartes's methodological, epistemological, and metaphysical concerns are forced to play second fiddle to his scientific views. We cannot neatly fit Descartes's metaphysical views into Clarke's scientific mold, nor can we force them in, without incurring serious error. I do not think that one has to choose between one and the other as being primary, between the tree of knowledge and the inverted tree of knowledge. We need to keep in mind that in Descartes's system of philosophy, both science and philosophy are interconnected and that both parts elucidate each other.Clarke's attempt to eliminate substance from Descartes's account of mind is problematic. True, the term 'substance' is an extremely ambiguous one and Descartes's comments about it are often blurred and puzzling. On the one hand, one must concede that the concept of substance plays a small role in Descartes's natural philosophy, in which although the material universe is depicted in terms of extended substance or res extensa, physics is ultimately unwrapped by way of laws describing the behavior of particles in virtue of their size, shape, and motion. But, in contrast, substance plays a far more prominent role in his metaphysics, not to mention in his theology. This leads to a related point. Clarke overlooks the importance of theology for Descartes's philosophical project and the role of God in Cartesian natural philosophy.The level of argument drops considerably in the final chapter, where Clarke offers an uncritical and not very clear survey of naming and necessity, the identity theory, anomalous monism, and property dualism. This chapter seems in tension with his disapproval of anachronistic readings of Descartes (8-9) and contains some serious slips. Clarke's answer to his own question "Was Descartes, then, a substance dualist" is a mind boggling "Yes and no" (258), and his claim that Descartes was a property dualist, collapses under... (shrink)

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a (...) Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)

In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there (...) exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)

This book is dedicated to a classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents (...) sufficient formal logic to give a full development of Gödel's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics". (shrink)

Berkeley has been notoriously charged with inconsistency because he held that spiritual substance exists, Although he argued against the existence of material substance. Berkeley is only inconsistent on the assumption that his argument in favor of spiritual substance parallels the rejected argument for material substance. I show that berkeley is relying on quite a different argument, One perfectly consistent with his theory of ideas, Based on presuppositions the germs of which can be found in the thought of his predecessors (...) in the theory of ideas, Descartes and locke. (shrink)

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception (...) through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms. (shrink)

Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that canno be a computation.

This chapter contains sections titled: Descartes' Cogito Augustine's “Si fallor, sum” Argument (If I Am Mistaken, I Exist).

Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that cannot be a computation.

Easy to understand philosophy papers in all areas. Table of contents: Three Short Philosophy Papers on Human Freedom The Paradox of Religions Institutions Different Perspectives on Religious Belief: O’Reilly v. Dawkins. v. James v. Clifford Schopenhauer on Suicide Schopenhauer’s Fractal Conception of Reality Theodore Roszak’s Views on Bicameral Consciousness Philosophy Exam Questions and Answers Locke, Aristotle and Kant on Virtue Logic Lecture for Erika Kant’s Ethics Van Cleve on Epistemic Circularity Plato’s Theory of Forms Can we trust our senses? (...) Yes we can Descartes on What He Believes Himself to Be The Role of Values in Science Modern Science Kant’s Moral Philosophy Plato’s Republic as Pol Potist Bureaucracy Schopenhauer on Human Suffering Bertrand Russell on the Value of Philosophy The Philosophical Value of Uncertainty Logic Homework: Theorems and Models Searle vs. Turing on the Imitation Game Hume, Frankfurt, and Holbach on Personal Freedom Manifesto of the University of Wisconsin, Madison Secular Society Michael’s Analysis of the Limits of Civil Protections Bentham and Mill on Different Types of Pleasure Set Theory Homework Aristotle on Virtue Nagel On the Hard Problem Wittgenstein on Language and Thought Camus and Schopenhauer on the Meaning of Life Camus’ Hero as Rebel without a Cause My Little Finger: Camus’ Absurdism Illustrated Are Late-term Abortions Ethical? Does Mathematics Assume the Truth of Platonism? The Self-defeating Nature of Utilitarianism and Consequentialism Generally What is The Good Life? Bentham and Mill regarding types of pleasures Kant’s Moral Philosophy Five Short Papers on Mind-body Dualism Tracy Latimer’s Father had the Right to Kill Her: Towards a doctrine of generalized self-defense Arguments Concerning God and Morality Goldman, Rousseau and von Hayek on the Ideal State J.S Mill on Liberty and Personal Freedom A Kantian Analysis of a Borderline Date-rape Situation Living Well as Flourishing: Aristotle’s Conception of the Good Life Three Essays on Medical Ethics: Answers to Exam Questions on Elective Amputation, Vaccination, and Informed Consent Hobbes, Marx, Rousseau, Nietzsche: Their Central Themes De Tocqueville on Egoism Mill vs. Hobbes on Liberty Exam-Essays on the Moral Systems of Mill, Bentham, and Kant Kant’s Moral System Aristotle on Virtue Plato’s Cave Allegory An Ethical Quandary Superorganisms The Tuskegee Experiment A Rawlsian Analysis Why Moore’s Proof of an External World Fails A Defense of Nagel’s Argument Against Materialism A Utilitarian Analysis of a Case of Theft The Paradox of the Self-aware Wretch: An Analysis of Pascal’s Moral Philosophy Jean-Paul Sartre: Decline and Fall of a Marxist Sell-out A One Page Proof of Plato’s Theory of Forms Plato’s Republic as Pol Potist Bureaucracy The problem of the one and the many Four Short Essays on Truth and Knowledge What is ‘the Good Life?’ The Ontological Argument Different Political Philosophies: Plato, Locke, Madison, Rousseau, Hayek, and Mill on the State What do I know with certainty? Skepticism about skepticism Neuroscience and Freewill Operant Conditioning What makes us special? Are Late-term Abortions Ethical? No Two Papers on Epistemology: Gettier and Bostrom Examination Nietzsche on Punishment God’s Foreknowledge and Moral Responsibility . (shrink)

The thesis examines the philosophical implications of the computational theory of early vision developed by Marr. According to Marr, early visual processes consist of sequences of "modular" computational mechanisms. These processes rely on functional relations between rates of change in stimulus magnitudes which result from certain contingent, global properties--natural constraints--of the physical world. ;Marr argues that explanations of early vision must have three distinct levels of description: computational, algorithmic and physical. In Chapter 1 I defend the (...) explanatory significance of this distinction in levels. In fulfilling its role in describing the dependence of visual processes on natural constraints, the computational level forms an autonomous level of description in the sense that it is unaffected by the computational steps at the other levels. ;In Chapter 2 I discuss the implications of natural constraints for the issues of individualism and methodological solipsism. I conclude that Mart's theory is nonindividualistic in the sense that visual content does not supervene on neural properties. However, this merely reflects the fact that different computational theories may be selected for the same system. Importantly, Marr's theory does not violate methodological solipsism since interpretations within theories must supervene on neural properties. ;In Chapter 3 I argue from the results of Chapters 1 and 2 that psychological explanation does not reduce to neurophysiology. This conclusion does not follow from the functionalist argument against physicalism, which is based on an incorrect account of computational theories. Rather the conclusion reflects the explanatory incompleteness of neurophysiological theories given the autonomous role of the computational level. ;In Chapter 4 I look in detail at the arguments for a "language of thought" as they apply to early vision. I distinguish two versions of the language of thought hypothesis, one weaker than the other. I conclude that the stronger version, which claims that a cognitive system is "program-using", is false of early vision because of the role of natural constraints. The weaker claim that cognitive processes employ symbolic transformations is true of the computational-level theory of early vision, but there is insufficient evidence to establish the claim at the algorithmic level. (shrink)

The number of studies related to natural and artificial mechanisms of learning rapidly increases. However, there is no general theory of learning that could provide a unifying basis for exploring different directions in this growing field. For a long time the development of such a theory has been hindered by nativists' belief that the development of a biological organism during ontogeny should be viewed as parameterization of an innate, encoded in the genome structure by an innate algorithm, and (...) nothing essentially new is created during this process. Noam Chomsky has claimed, therefore, that the creation of a non-trivial general mathematical theory of learning is not feasible, since any algorithm cannot produce a more complex algorithm. This study refutes the above argumentation by developing a counter-example based on the mathematical theory of algorithms and computable functions. It introduces a novel concept of a Universal Learning System (ULS) capable of learning to control in an optimal way any given constructive system from a certain class. The necessary conditions for the existence of a ULS and its main functional properties are investigated. The impossibility of building an algorithmic ULS for a sufficiently complex class of controlled objects is shown, and a proof of the existence of a non-algorithmic ULS based on the axioms of classical mathematics is presented. It is argued that a non-algorithmic ULS is a legitimate object of not only mathematics, but also the world of nature. These results indicate that an algorithmic description of the organization and adaptive development of biological systems in general is not sufficient. At the same time, it is possible to create a rigorous non-algorithmic general theory of learning as a theory of ULS. The utilization of this framework for integrating learning-related studies is discussed. (shrink)