In this brief commentary I compare and contrast two different views of evolution: one of limited (convergent) evolution and mathematical predictability, and one of unbounded diversity and no entailing laws. Clearly these opposing views cannot both be true at the same time. Their disagreement seems to rest on different underlying assumptions, and the challenge is to see if they can be reconciled.

Counterfactuals have become an important area of interdisciplinary interest, especially in logic, philosophy of language, epistemology, metaphysics, psychology, decision theory, and even artificial intelligence. In this study, we propose a new form of analysis for counterfactuals: analysis by algorithmic complexity. Inspired by Lewis-Stalnaker's Possible Worlds Semantics, the proposed method allows for a new interpretation of the debate between David Lewis and Robert Stalnaker regarding the Limit and Singularity assumptions. Besides other results, we offer a new way to answer the problems (...) raised by Goodman and Quine regarding vagueness, context-dependence, and the non- monotonicity of counterfactuals. Engaging in a dialogue with literature, this study will seek to bring new insights and tools to this debate. We hope our method of analysis can make counterfactuals more understandable in an intuitively plausible way, and a philosophically justifiable manner, aligned with the way we usually think about counterfactual propositions and our imaginative reasoning. (shrink)

A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds . By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine (...) codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful. (shrink)

Can the machines that play board games or recognize images only in the comfort of the virtual world be intelligent? To become reliable and convenient assistants to humans, machines need to learn how to act and communicate in the physical reality, just like people do. The authors propose two novel ways of designing and building Artificial General Intelligence (AGI). The first one seeks to unify all participants at any instance of the Turing test – the judge, the machine, the human (...) subject as well as the means of observation instead of building a separating wall. The second one aims to design AGI programs in such a way so that they can move in various environments. The authors of the article thoroughly discuss four areas of interaction for robots with AGI and introduce a new idea of techno-umwelt bridging artificial intelligence with biology in a new way. (shrink)

This paper discusses the epistemic status of biology from the standpoint of the systemic approach to living systems based on the notion of biological autonomy. This approach aims to provide an understanding of the distinctive character of biological systems and this paper analyses its theoretical and epistemological dimensions. The paper argues that, considered from this perspective, biological systems are examples of emergent phenomena, that the biological domain exhibits special features with respect to other domains, and that biology as a discipline (...) employs some core concepts, such as teleology, function, regulation among others, that are irreducible to those employed in physics and chemistry. It addresses the claim made by Jacques Monod that biology as a science is marginal. It argues that biology is general insofar as it constitutes a paradigmatic example of complexity science, both in terms of how it defines the theoretical object of study and of the epistemology and heuristics employed. As such, biology may provide lessons that can be applied more widely to develop an epistemology of complex systems. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and (...) to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. Chapter \textbf{12} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. The multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.}. (shrink)

The present PhD dissertation aims to examine the relation between modality and change in Aristotle’s metaphysics. -/- On the one hand, Aristotle supports his modal realism (i.e., worldly objects have modal properties - potentialities and essences - that ground the ascriptions of possibility and necessity) by arguing that the rejection of modal realism makes change inexplicable, or, worse, banishes it from the realm of reality. On the other hand, the Stagirite analyses processes by means of modal notions (‘change is the (...) actuality of what is virtual insofar as it is virtual’). In other words: to grasp what change is, one has to resort to the modal idiom of potentialities, while the fact that there is change is indicative of the fact that nature is full of modal properties. -/- Aristotle’s modal and kinetic realism finds a negative in the figure of the Megaric Diodorus Kronus. The polemical situation of Greek philosophy has indeed the dialectical advantage of not opposing the Aristotelian position to a sui generis straw man. Both in its reduction of modal judgements to temporal quantifications and in its dissolution of the reality of the state-of-being-in-motion in favour of a cinematographic view of processes, the philosophical figure of reductionist antirealism embodied by Diodorus constitutes an alternative that takes the opposite view of Aristotle’s realism. -/- The present study is structured as a discussion between these two positions. In the face of Diodorus’ antirealist challenges, Aristotle articulates modal and kinetic considerations: the answers he gives to Diodorus’ puzzles thus provide valuable insight into his metaphysics. Moreover, the examination of Aristotle’s and Diodorus’ metaphysics, because of their insight and originality, will not fail to interest the philosopher concerned with the metaphysical foundations of modern physics, insofar as the entanglement between modalities and processes is nowadays at the core of mechanics (phase spaces, path integrals, etc.). (shrink)

This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing iterated epistemic states on the model of fixed points in the modal $\mu$-calculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox -- i.e. the KK principle: $\square$$\phi$ $\rightarrow$ $\square$$\square$$\phi$ -- a hyperintensional epistemic $\mu$-automaton permits fixed points to entrain a principled means by which to iterate epistemic states and account (...) thereby for necessary conditions on self-knowledge. The hyperintensional epistemic $\mu$-calculus is applied to the iteration of the epistemic states of a single agent instead of the common knowledge of a group of agents, and is thus a novel contribution to the literature. (shrink)

The core of the problem discussed in this paper is the following: the Church-Turing Thesis states that Turing Machines formally explicate the intuitive concept of computability. The description of Turing Machines requires description of the notation used for the input and for the output. Providing a general definition of notations acceptable in the process of computations causes problems. This is because a notation, or an encoding suitable for a computation, has to be computable. Yet, using the concept of computation, in (...) a definition of a notation, which will be further used in a definition of the concept of computation yields an obvious vicious circle. The circularity of this definition causes trouble in distinguishing on the theoretical level, what is an acceptable notation from what is not an acceptable notation, or as it is usually referred to in the literature, “deviant encodings”. Deviant encodings appear explicitly in discussions about what is an adequate or correct conceptual analysis of the concept of computation. In this paper, I focus on philosophical examples where the phenomenon appears implicitly, in a “disguised” version. In particular, I present its use in the analysis of the concept of natural number. I also point at additional phenomena related to deviant encodings: conceptual fixed points and apparent “computability” of uncomputable functions. In parallel, I develop the idea that Carnapian explications provide a much more adequate framework for understanding the concept of computation, than the classical philosophical analysis. (shrink)

O eixo desta investigação é a noção de linguagem inata e interna, proposta por Jerry Fodor, que teve como principal desafio a tradição analítica do século XX, a qual se fundamentou no antimentalismo de Wittgenstein e na sua negação de uma linguagem privada. A hipótese da “linguagem do pensamento” pode ser equivocadamente associada ao conceito de linguagem natural. Por isso, Fodor sugere uma distinção entre tais linguagens, visto que a do pensamento deve ser compreendida como manipulação “interna” de símbolos. Esta (...) noção de “interno” tem gerado controvérsias quando confrontada com a tradição wittgensteiniana. Neste trabalho, apresentamos os alcances e limites da hipótese fodoriana de uma linguagem do pensamento, almejando especificamente discutir o impasse quanto à noção de “interno” suscitado por tal hipótese. (shrink)

This is a non-technical version of "The Decision Problem for Effective Procedures." The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of an (...) effective procedure--without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis--that the class of effective procedures is undecidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. The proof does not appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though this undecidability result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict. (shrink)

The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong to mathematics proper (in a narrow sense)—and even if (as many have asserted) for that reason the Church–Turing thesis is unprovable. It is proved logically that the class of effective procedures is not decidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. This (...) undecidability result is proved directly from the notion itself of an effective procedure, without reliance on any (partly) mathematical lemma, conjecture, or thesis invoking recursiveness or Turing-computability. In fact, there is no reliance on anything very mathematical. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of an effective procedure. Though the result that effectiveness is undecidable is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure invariably terminates with the correct verdict. (shrink)

Nous proposons une étude de la constitution du sens dans l'interaction humain-machine à partir des définitions que donnent Turing et Wittgenstein à propos de la pensée, la compréhension, et de la décision. Nous voulons montrer par l'analyse comparative des proximités et différences conceptuelles entre les deux auteurs que le sens commun entre humains et machines se co-constitue dans et à partir de l'action, et que c'est précisément dans cette co-constitution que réside la valeur sociale de leur interaction. Il s'agira pour (...) cela de problématiser l'interaction humain-machine autour de la question de ce que signifie « suivre une règle » pour définir et distinguer les modes interprétatifs et les comportements décisionnels de chacun. Nous en viendrons alors au constat que la mutualisation des signes qui s'opère à travers le dialogue humain-machine est au fondement de la constitution d'une société informatisée. (shrink)

Proceedings of the papers presented at the Symposium on "Revisiting Turing and his Test: Comprehensiveness, Qualia, and the Real World" at the 2012 AISB and IACAP Symposium that was held in the Turing year 2012, 2–6 July at the University of Birmingham, UK. Ten papers. - http://www.pt-ai.org/turing-test --- Daniel Devatman Hromada: From Taxonomy of Turing Test-Consistent Scenarios Towards Attribution of Legal Status to Meta-modular Artificial Autonomous Agents - Michael Zillich: My Robot is Smarter than Your Robot: On the Need for (...) a Total Turing Test for Robots - Adam Linson, Chris Dobbyn and Robin Laney: Interactive Intelligence: Behaviour-based AI, Musical HCI and the Turing Test - Javier Insa, Jose Hernandez-Orallo, Sergio España - David Dowe and M.Victoria Hernandez-Lloreda: The anYnt Project Intelligence Test (Demo) - Jose Hernandez-Orallo, Javier Insa, David Dowe and Bill Hibbard: Turing Machines and Recursive Turing Tests — Francesco Bianchini and Domenica Bruni: What Language for Turing Test in the Age of Qualia? - Paul Schweizer: Could there be a Turing Test for Qualia? - Antonio Chella and Riccardo Manzotti: Jazz and Machine Consciousness: Towards a New Turing Test - William York and Jerry Swan: Taking Turing Seriously (But Not Literally) - Hajo Greif: Laws of Form and the Force of Function: Variations on the Turing Test. (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).

Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's (...) Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and--in particular--the effectively-computable functions on string representations of numbers. (shrink)

The birth, growth, stabilization and subsequent understanding of a new field of practical and theoretical enquiry is always a conceptual process including several typologies of events, phenomena an...

In 1931 Kurt Gödel published his incompleteness results, and some years later Church and Turing showed that the decision problem for certain systems of symbolic logic has a negative solution. However, already in 1921 the young logician Emil Post worked on similar problems which resulted in what he called an “anticipation” of these results. For several reasons though he did not submit these results to a journal until 1941. This failure ‘to be the first’, did not discourage him: his contributions (...) to mathematical logic and its foundations should not be underestimated. It is the purpose of this article to show that an interest in the early work of Emil Post should be motivated not only by this historical fact, but also by the fact that Post’s approach and method differs substantially from those offered by Gödel, Turing and Church. In this paper it will be shown how this method evolved in his early work and how it finally led him to his results. (shrink)

ABSTRACTAlexander Wendt’s Quantum Mind and Social Science is an effort to establish foundations of social science based on the ontology of modern physics. The quantum revolution has deservedly had repercussions in many sciences, but it is unwise to ground social science on physical theories, which are subject to constant revision. Additionally, despite its empirical success, there is no agreed-upon interpretation of quantum theory. Finally, even if there were, the random indeterminacy intrinsic to the quantum world cannot account for the intentionality (...) of human action. Intentional agents act for reasons, not at random; nor are the reasons chosen at random. (shrink)

In recent years philosophers have been interested in the methodology of metaphysics. Most of these developments are related to formal work in logic or physics, often against the backdrop of the Carnap-Quine debate on ontology. Drawing on Quine’s later work, I argue that a psychological or cognitive perspective on metaphysical topics may be a valuable addition to contemporary metametaphysics. The method is illustrated by means of cognitive studies of the notions “identity,” “vagueness,” and “object” and is compared to other extant (...) metametaphysical positions. (shrink)

In recent years two different axiomatic characterizations of the intuitive concept of effective calculability have been proposed, one by Sieg and the other by Dershowitz and Gurevich. Analyzing them from the perspective of Carnapian explication, I argue that these two characterizations explicate the intuitive notion of effective calculability in two different ways. I will trace back these two ways to Turing’s and Kolmogorov’s informal analyses of the intuitive notion of calculability and to their respective outputs: the notion of computorability and (...) the notion of algorithmability. I will then argue that, in order to adequately capture the conceptual differences between these two notions, the classical two-step picture of explication is not enough. I will present a more fine-grained three-step version of Carnapian explication, showing how with its help the difference between these two notions can be better understood and explained. (shrink)

This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which (...) Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

Alan Turing is often portrayed as a materialist in secondary literature. In the present article, I suggest that Turing was instead an idealist, inspired by Cambridge scholars, Arthur Eddington, Ernest Hobson, James Jeans and John McTaggart. I outline Turing’s developing thoughts and his legacy in the USA to date. Specifically, I contrast Turing’s two notions of computability (both from 1936) and distinguish between Turing’s “machine intelligence” in the UK and the more well-known “artificial intelligence” in the USA. According to my (...) proposed historical interpretation, Turing did not view computations in the real world to be exhaustively and deterministically characterized by his automatic machines from 1936. (shrink)

In the popular imagination, the relevance of Turing's theoretical ideas to people producing actual machines was significant and appreciated by everybody involved in computing from the moment he published his 1936 paper ‘On Computable Numbers’. Careful historians are aware that this popular conception is deeply misleading. We know from previous work by Campbell-Kelly, Aspray, Akera, Olley, Priestley, Daylight, Mounier-Kuhn, Haigh, and others that several computing pioneers, including Aiken, Eckert, Mauchly, and Zuse, did not depend on Turing's 1936 universal-machine concept. Furthermore, (...) it is not clear whether any substance in von Neumann's celebrated 1945 ‘First Draft Report on the EDVAC’ is influenced in any identifiable way by Turing's work. This raises the questions: When does Turing enter the field? Why did the Association for Computing Machinery honor Turing by associating his name to ACM's most prestigious award, the Turing Award? Previous authors have.. (shrink)

Alan Turing’s 1950 imitation game has been widely understood as a means for testing if an entity is intelligent. Following a series of papers by Diane Proudfoot, I offer a socio-technological interpretation of Turing’s paper and present an alternative way of understanding both the imitation game and Turing’s concept of intelligence. Turing, I claim, saw intelligence as a social concept, meaning that possession of intelligence is a property determined by society’s attitude toward the entity. He realized that as long as (...) human society held a prejudiced attitude toward machinery—seeing machines a priori as mindless objects—machines could not be said to be intelligent, by definition. He also realized, though, that if humans’ a priori, chauvinistic attitude toward machinery changed, the existence of intelligent machines would become logically possible. Turing thought that such a change would eventually occur: He believed that when scientists overcome the technological challenge of constructing sophisticated machines that could imitate human verbal behavior—i.e., do well in the imitation game—humans’ prejudiced attitude toward machinery will have altered in such a way that machines could be said to be intelligent. The imitation game, for Turing, was not an intelligence test, but a technological aspiration whose realization would likely involve a change in society’s attitude toward machines. (shrink)

A skeptical worry known as ‘the indeterminacy of computation’ animates much recent philosophical reflection on the computational identity of physical systems. On the one hand, computational explanation seems to require that physical computing systems fall under a single, unique computational description at a time. On the other, if a physical system falls under any computational description, it seems to fall under many simultaneously. Absent some principled reason to take just one of these descriptions in particular as relevant for computational explanation, (...) widespread failure of computational explanation would appear to follow. This paper advances a new solution to the indeterminacy of computation. Very roughly, I argue that the computational identity of a physical system is determinate relative to a contextually specified way of regarding that system computationally—known as a labelling scheme. When a system simultaneously implements multiple computations, it does so relative to different labelling schemes. But relative to a fixed labelling scheme, a physical system has a unique computational identity. I argue that this relativistic conception of computational identity vindicates computational explanation in the face of simultaneous implementation. (shrink)

This article advertises a new account of computational implementation. According to the resemblance account, implementation is a matter of resembling a computational architecture. The resemblance account departs from previous theories by denying that computational architectures are exhausted by their formal, mathematical features. Instead, they are taken to be permeated with causality, spatiotemporality, and other nonmathematical features. I argue that this approach comports well with computer scientific practice and offers a novel response to so-called triviality arguments.

In this paper we defend structural representations, more specifically neural structural representation. We are not alone in this, many are currently engaged in this endeavor. The direction we take, however, diverges from the main road, a road paved by the mathematical theory of measure that, in the 1970s, established homomorphism as the way to map empirical domains of things in the world to the codomain of numbers. By adopting the mind as codomain, this mapping became a boon for all those (...) convinced that a representation system should bear similarities with what was being represented, but struggled to find a precise account of what such similarities mean. The euforia was brief, however, and soon homomorphism revealed itself to be affected by serious weaknesses, the primary one being that it included systems embarrassingly alien to representations. We find that the defense attempts that have followed, adopt strategies that share a common format: valid structural representations come as “homomorphism plus X”, with various “X”, provided in descriptive format only. Our alternative direction stems from the observation of the overlooked departure from homomorphism as used in the theory of measure and its later use in mental representations. In the former case, the codomain or the realm of numbers, is the most suited for developing theorems detailing the existence and uniqueness of homomorphism for a wide range of empirical domains. In the latter case, the codomain is the realm of the mind, possibly more vague and more ill-defined than the empirical domain itself. The time is ripe for articulating the mapping between represented domains and the mind in formal terms, by exploiting what is currently known about coding mechanisms in the brain. We provide a sketch of a possible development in this direction, one that adopts the theory of neural population coding as codomain. We will show that our framework is not only not in disagreement with the “plus X” proposals, but can lead to natural derivation of several of the “X”. (shrink)

Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.

A version of the Church-Turing Thesis states that every effectively realizable physical system can be simulated by Turing Machines (‘Thesis P’). In this formulation the Thesis appears to be an empirical hypothesis, subject to physical falsification. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, analog, quantum, and retrocausal computation. The conclusions are that these models reduce to supertasks, i.e. infinite computation, and that even supertasks are no solution for recursive incomputability. This yields that the realization (...) of hypercomputing devices is implausible, and that Thesis P is not essentially different from the standard Church-Turing Thesis. (shrink)

Sumário Mostramos que, em virtude dos limites teóricos da computação, nem toda a ciência formulada com carácter preditivo pode ser simulada. Em particular, evidencia- se que a Fisica Clássica, nomeadamente a Físíca Newtoniana, padece deste mal, encerrando processos de Zenão.

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)

Alan Turing anticipated many areas of current research incomputer and cognitive science. This article outlines his contributionsto Artificial Intelligence, connectionism, hypercomputation, andArtificial Life, and also describes Turing's pioneering role in thedevelopment of electronic stored-program digital computers. It locatesthe origins of Artificial Intelligence in postwar Britain. It examinesthe intellectual connections between the work of Turing and ofWittgenstein in respect of their views on cognition, on machineintelligence, and on the relation between provability and truth. Wecriticise widespread and influential misunderstandings of theChurch–Turing thesis (...) and of the halting theorem. We also explore theidea of hypercomputation, outlining a number of notional machines thatcompute the uncomputable. (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.

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.

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)

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)

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