Quantum Information Theory and the Foundations of Quantum Mechanics is a conceptual analysis of one of the most prominent and exciting new areas of physics, providing the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. -/- Beginning from a careful, revisionary, analysis of the concepts of information in the everyday and classical information-theory settings, Christopher G. Timpson argues for an ontologically deflationary account of the nature of quantum information. (...) Against what many have supposed, quantum information can be clearly defined (it is not a primitive or vague notion) but it is not part of the material contents of the world. Timpson's account sheds light on the nature of nonlocality and information flow in the presence of entanglement and, in particular, dissolves puzzles surrounding the remarkable process of quantum teleportation. In addition it permits a clear view of what the ontological and methodological lessons provided by quantum information theory are; lessons which bear on the gripping question of what role a concept like information has to play in fundamental physics. Topics discussed include the slogan 'Information is Physical', the prospects for an informational immaterialism (the view that information rather than matter might fundamentally constitute the world), and the status of the Church-Turing hypothesis in light of quantum computation. -/- With a clear grasp of the concept of information in hand, Timpson turns his attention to the pressing question of whether advances in quantum information theory pave the way for the resolution of the traditional conceptual problems of quantum mechanics: the deep problems which loom over measurement, nonlocality and the general nature of quantum ontology. He marks out a number of common pitfalls to be avoided before analysing in detail some concrete proposals, including the radical quantum Bayesian programme of Caves, Fuchs, and Schack. One central moral which is drawn is that, for all the interest that the quantum information-inspired approaches hold, no cheap resolutions to the traditional problems of quantum mechanics are to be had. (shrink)

Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version (...) of the Church–Turing Thesis is unaffected by SAD computation. (shrink)

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T 0 -spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a (...) totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets. (shrink)

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We (...) focus on the countable distributive lattices P w and P s of weak and strong degrees of mass problems given by nonempty Π 1 0 subsets of 2 ω . Using an abstract Gödel/Rosser incompleteness property, we characterize the Π 1 0 subsets of 2 ω whose associated mass problems are of top degree in P w and P s , respectively. Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within P w . Namely, we characterize r as the unique largest weak degree of a Π 1 0 subset of 2 ω of positive measure. Within P w we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect Π 1 0 subsets of 2 ω . In addition, we present other natural examples of intermediate degrees in P w . We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory. (shrink)

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.

The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism. The purpose of (...) this article is to begin the articulation of this relationship, pointing out its sources and its limitations. (shrink)

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 (...) of physically realizable machines. (shrink)

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were (...) replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem, which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:f is recursive,f is computable on an abstract machine,f is definable in the untyped λ-calculus.These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance. (shrink)

We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.

In 1960, E. P. Wigner, a joint winner of the 1963 Nobel Prize for Physics, published a paper titled On the Unreasonable Effectiveness of Mathematics in the Natural Sciences [61]. This paper can be construed as an examination and affirmation of Galileo's tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of mathematics in accurately describing physical phenomena. Wigner viewed these examples as (...) illustrations of what he called the empirical law of epistemology, which asserts that the mathematical formulation of the laws of nature is both appropriate and accurate, and that mathematics is actually the correct language for formulating the laws of nature. At the same time, Wigner pointed out that the reasons for the success of mathematics in the natural sciences are not completely understood; in fact, he went as far as asserting that “… the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”. (shrink)

We identify the computational complexity of the satisfiability problem for FO 2 , the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO 2 has the finite-model property, which means that if an FO 2 -sentence is satisfiable, then it has a finite (...) model. Moreover, Mortimer showed that every satisfiable FO 2 -sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO 2 -sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO 2 is NEXPTIME-complete. (shrink)

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 (...) paradigm, which was common among theoretical computer scientists, defines computer science as a branch of mathematics, treats programs on a par with mathematical objects, and seeks certain, a priori knowledge about their ‘correctness’ by means of deductive reasoning. – The technocratic paradigm, promulgated mainly by software engineers and has come to dominate much of the discipline, defines computer science as an engineering discipline, treats programs as mere data, and seeks probable, a posteriori knowledge about their reliability empirically using testing suites. – The scientific paradigm, prevalent in the branches of artificial intelligence, defines computer science as a natural (empirical) science, takes programs to be entities on a par with mental processes, and seeks a priori and a posteriori knowledge about them by combining formal deduction and scientific experimentation. We demonstrate evidence corroborating the tenets of the scientific paradigm, in particular the claim that program-processes are on a par with mental processes. We conclude with a discussion in the influence that the technocratic paradigm has been having over computer science. (shrink)

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 (...) will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable. (shrink)

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 (...) to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

A survey of the field of hypercomputation, including discussion of a variety of objections.

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 (...) of action(- types) from the particular courses of action(-tokens) that actually instantiate them and the causal processes and/or interpretations that ultimately make them effective. On my analysis, effectiveness is not just a matter of logical form; `content' matters. The analysis I provide has the advantage of applying to ordinary, everyday procedures such as recipes and methods, as well as the more refined procedures of mathematics and computer science. It also has the virtue of making better sense of the physical possibilities for hypercomputation than the received view and its extensions, e.g. Turing's o-machines, accelerating machines. (shrink)

Cognitive psychologists, like biologists, frequently describe mechanisms when explaining phenomena. Unlike biologists, who can often trace material transformations to identify operations, psychologists face a more daunting task in identifying operations that transform information. Behavior provides little guidance as to the nature of the operations involved. While not itself revealing the operations, identification of brain areas involved in psychological mechanisms can help constrain attempts to characterize the operations. In current memory research, evidence that the same brain areas are involved in what (...) are often taken to be different memory phenomena or in other cognitive phenomena is playing such a heuristic function. †To contact the author, please write to: Department of Philosophy, 0119, University of California, San Diego, La Jolla, CA 92093‐0119; e‐mail: [email protected]. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Chaos-related obstructions to predictability have been used to challenge accounts of theory validation based on the agreement between theoretical predictions and experimental data. These challenges are incomplete in two respects: they do not show that chaotic regimes are unpredictable in principle and, as a result, that there is something conceptually wrong with idealized expectations of correct predictions from acceptable theories, and they do not explore whether chaos-induced predictive failures of deterministic models can be remedied by stochastic modeling. In this paper (...) we appeal to an asymptotic analysis of state space trajectories and their numerical approximations to show that chaotic regimes are deterministically unpredictable even with unbounded resources. Additionally, we explain why stochastic models of chaotic systems, while predictively successful in some cases, are in general predictively as limited as deterministic ones. We conclude by suggesting that the way in which scientists deal with such principled obstructions to predictability calls for a more comprehensive approach to theory validation, on which experimental testing is augmented by a multifaceted mathematical analysis of theoretical models, capable of identifying chaos-related predictive failures as due to principled limitations which the world itself imposes on any less-than-omniscient epistemic access to some natural systems. (shrink)

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

As much as assumptions about mechanisms and mechanistic explanation have deeply affected psychology, they have received disproportionately little analysis in philosophy. After a historical survey of the influences of mechanistic approaches to explanation of psychological phenomena, we specify the nature of mechanisms and mechanistic explanation. Contrary to some treatments of mechanistic explanation, we maintain that explanation is an epistemic activity that involves representing and reasoning about mechanisms. We discuss the manner in which mechanistic approaches serve to bridge levels rather than (...) reduce them, as well as the different ways in which mechanisms are discovered. Finally, we offer a more detailed example of an important psychological phenomenon for which mechanistic explanation has provided the main source of scientific understanding. (shrink)

trying to explain these reactions in terms of changes in ele- began trying to characterize physiological processes in.

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 (...) justice to the practices of computer scientists and computability theorists. It is also an application of recent literature on mechanisms, because it assimilates computational explanation to mechanistic explanation. The account can be used to individuate computing mechanisms and the functions they compute and to taxonomize computing mechanisms based on their computing power. (shrink)

After briefly discussing the relevance of the notions computation and implementation for cognitive science, I summarize some of the problems that have been found in their most common interpretations. In particular, I argue that standard notions of computation together with a state-to-state correspondence view of implementation cannot overcome difficulties posed by Putnam's Realization Theorem and that, therefore, a different approach to implementation is required. The notion realization of a function, developed out of physical theories, is then introduced as a replacement (...) for the notional pair computation-implementation. After gradual refinement, taking practical constraints into account, this notion gives rise to the notion digital system which singles out physical systems that could be actually used, and possibly even built. (shrink)

This paper challenges two orthodox theses: (a) that computational processes must be algorithmic; and (b) that all computed functions must be Turing-computable. Section 2 advances the claim that the works in computability theory, including Turing's analysis of the effective computable functions, do not substantiate the two theses. It is then shown (Section 3) that we can describe a system that computes a number-theoretic function which is not Turing-computable. The argument against the first thesis proceeds in two stages. It is first (...) shown (Section 4) that whether a process is algorithmic depends on the way we describe the process. It is then argued (Section 5) that systems compute even if their processes are not described as algorithmic. The paper concludes with a suggestion for a semantic approach to computation. (shrink)

Because it is time-dependent, parallel computation is fundamentally different from sequential computation. Parallel programs are non-deterministic and are not effective procedures. Given the brain operates in parallel, this casts doubt on AI's attempt to make sequential computers intelligent.

To clarify the notion of computation and its role in cognitive science, we need an account of implementation, the nexus between abstract computations and physical systems. I provide such an account, based on the idea that a physical system implements a computation if the causal structure of the system mirrors the formal structure of the computation. The account is developed for the class of combinatorial-state automata, but is sufficiently general to cover all other discrete computational formalisms. The implementation relation is (...) non-vacuous, so that criticisms by Searle and others fail. This account of computation can be extended to justify the foundational role of computation in artificial intelligence and cognitive science. (shrink)

We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine - a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'Church-Turing thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by (...) Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of _nonclassical_ computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a number of foundational arguments that are commonly rehearsed in cognitive science, and gesture towards a new class of cognitive models. (shrink)

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 (...) a difficulty that has persistently been raised against it, and undercuts various objections that have been made to the computational theory of mind. (shrink)

We critically discuss Cleland''s analysis of effective procedures as mundane effective procedures. She argues that Turing machines cannot carry out mundane procedures, since Turing machines are abstract entities and therefore cannot generate the causal processes that are generated by mundane procedures. We argue that if Turing machines cannot enter the physical world, then it is hard to see how Cleland''s mundane procedures can enter the world of numbers. Hence her arguments against versions of the Church-Turing thesis for number theoretic functions (...) miss the mark. (shrink)

Over the past thirty years, it is been common to hear the mind likened to a digital computer. This essay is concerned with a particular philosophical view that holds that the mind literally is a digital computer (in a specific sense of “computer” to be developed), and that thought literally is a kind of computation. This view—which will be called the “Computational Theory of Mind” (CTM)—is thus to be distinguished from other and broader attempts to connect the mind with computation, (...) including (a) various enterprises at modeling features of the mind using computational modeling techniques, and (b) employing some feature or features of production-model computers (such as the stored program concept, or the distinction between hardware and software) merely as a guiding metaphor for understanding some feature of the mind. This entry is therefore concerned solely with the Computational Theory of Mind (CTM) proposed by Hilary Putnam [1961] and developed most notably for philosophers by Jerry Fodor [1975, 1980, 1987, 1993]. The senses of ‘computer’ and ‘computation’ employed here are technical; the main tasks of this entry will therefore be to elucidate: (a) the technical sense of ‘computation’ that is at issue, (b) the ways in which it is claimed to be applicable to the mind, (c) the philosophical problems this understanding of the mind is claimed to solve, and (d) the major criticisms that have accrued to this view. (shrink)

The computational view of mind rests on certain intuitions regarding the fundamental similarity between computation and cognition. We examine some of these intuitions and suggest that they derive from the fact that computers and human organisms are both physical systems whose behavior is correctly described as being governed by rules acting on symbolic representations. Some of the implications of this view are discussed. It is suggested that a fundamental hypothesis of this approach is that there is a natural domain of (...) human functioning that can be addressed exclusively in terms of a formal symbolic or algorithmic vocabulary or level of analysis. Much of the paper elaborates various conditions that need to be met if a literal view of mental activity as computation is to serve as the basis for explanatory theories. The coherence of such a view depends on there being a principled distinction between functions whose explanation requires that we posit internal representations and those that we can appropriately describe as merely instantiating causal physical or biological laws. In this paper the distinction is empirically grounded in a methodological criterion called the " cognitive impenetrability condition." Functions are said to be cognitively impenetrable if they cannot be influenced by such purely cognitive factors as goals, beliefs, inferences, tacit knowledge, and so on. Such a criterion makes it possible to empirically separate the fixed capacities of mind from the particular representations and algorithms used on specific occasions. In order for computational theories to avoid being ad hoc, they must deal effectively with the "degrees of freedom" problem by constraining the extent to which they can be arbitrarily adjusted post hoc to fit some particular set of observations. This in turn requires that the fixed architectural function and the algorithms be independently validated. It is argued that the architectural assumptions implicit in many contemporary models run afoul of the cognitive impenetrability condition, since the required fixed functions are demonstrably sensitive to tacit knowledge and goals. The paper concludes with some tactical suggestions for the development of computational cognitive theories. (shrink)

According to the dominant computational approach in cognitive science, cognitive agents are digital computers; according to the alternative approach, they are dynamical systems. This target article attempts to articulate and support the dynamical hypothesis. The dynamical hypothesis has two major components: the nature hypothesis (cognitive agents are dynamical systems) and the knowledge hypothesis (cognitive agents can be understood dynamically). A wide range of objections to this hypothesis can be rebutted. The conclusion is that cognitive systems may well be dynamical systems, (...) and only sustained empirical research in cognitive science will determine the extent to which that is true. (shrink)

Traditional approaches to modeling cognitive systems are computational, based on utilizing the standard tools and concepts of the theory of computation. More recently, a number of philosophers have argued that cognition is too subtle or complex for these tools to handle. These philosophers propose an alternative based on dynamical systems theory. Proponents of this view characterize dynamical systems as (i) utilizing continuous rather than discrete mathematics, and, as a result, (ii) being computationally more powerful than traditional computational automata. Indeed, the (...) logical possibility of such super-powerful systems has been demonstrated in the form of analog artificial neural networks. In this paper I consider three arguments against the nomological possibility of these automata. While the first two arguments fail, the third succeeds. In particular, the presence of noise reduces the computational power of analog networks to that of traditional computational automata, and noise is a pervasive feature of information processing in biological systems. Consequently, as an empirical thesis, the proposed dynamical alternative is under-motivated: What is required is an account of how continuously valued systems could be realized in physical systems despite the ubiquity of noise. (shrink)

This is an amended version of material that first appeared in A. Clark, Microcognition: Philosophy, Cognitive Science, and Parallel Distributed Processing (MIT Press, Cambridge, MA, 1989), Ch. 1, 2, and 6. It appears in German translation in Metzinger,T (Ed) DAS LEIB-SEELE-PROBLEM IN DER ZWEITEN HELFTE DES 20 JAHRHUNDERTS (Frankfurt am Main: Suhrkamp. 1999).

It is not widely realised that Turing was probably the first person to consider building computing machines out of simple, neuron-like elements connected together into networks in a largely random manner. Turing called his networks unorganised machines. By the application of what he described as appropriate interference, mimicking education an unorganised machine can be trained to perform any task that a Turing machine can carry out, provided the number of neurons is sufficient. Turing proposed simulating both the behaviour of the (...) network and the training process by means of a computer program. We outline Turing's connectionist project of 1948. (shrink)

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models. (shrink)

According to the conventional wisdom, Turing said that computing machines can be intelligent. I don't believe it. I think that what Turing really said was that computing machines –- computers limited to computing –- can only fake intelligence. If we want computers to become genuinelyintelligent, we will have to give them enough “initiative” to do more than compute. In this paper, I want to try to develop this idea. I want to explain how giving computers more ``initiative'' can allow them (...) to do more than compute. And I want to say why I believe that they will have to go beyond computation before they can become genuinely intelligent. (shrink)

In his PhD thesis (1938) Turing introduced what he described as 'a new kind of machine'. He called these 'O-machines'. The present paper employs Turing's concept against a number of currently fashionable positions in the philosophy of mind.

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 (...) the question, "Can machines think?" is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A." The interrogator is allowed to put questions to A and B. We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?". (shrink)

I advocate a theory of syntactic semantics as a way of understanding how computers can think (and how the Chinese-Room-Argument objection to the Turing Test can be overcome): (1) Semantics, considered as the study of relations between symbols and meanings, can be turned into syntax – a study of relations among symbols (including meanings) – and hence syntax (i.e., symbol manipulation) can suffice for the semantical enterprise (contra Searle). (2) Semantics, considered as the process of understanding one domain (by modeling (...) it) in terms of another, can be viewed recursively: The base case of semantic understanding –understanding a domain in terms of itself – is syntactic understanding. (3) An internal (or narrow ), first-person point of view makes an external (or wide ), third-person point of view otiose for purposes of understanding cognition. (shrink)

Turing''s test has been much misunderstood. Recently unpublished material by Turing casts fresh light on his thinking and dispels a number of philosophical myths concerning the Turing test. Properly understood, the Turing test withstands objections that are popularly believed to be fatal.