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.

What counts as a computation and how it relates to cognitive function are important questions for scientists interested in understanding how the mind thinks. This paper argues that pragmatic aspects of explanation ultimately determine how we answer those questions by examining what is needed to make rigorous the notion of computation used in the (cognitive) sciences. It (1) outlines the connection between the Church-Turing Thesis and computational theories of physical systems, (2) differentiates merely satisfying a computational function from true computation, (...) and finally (3) relates how we determine a true computation to the functional methodology in cognitive science. All of the discussion will be directed toward showing that the only way to connect formal notions of computation to empirical theory will be in virtue of the pragmatic aspects of explanation. (shrink)

We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that θ does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7]. We apply the result to show that Φ′ does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new (...) characterization of effectively simple sets and show that simple sets are not btt-cuppable. (shrink)

Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.

In this paper we will discuss the active part played by certain diagonal arguments in the genesis of computability theory. 1 In some cases it is enough to assume the enumerability of Y while in others the effective enumerability is a substantial demand. These enigmatical words by Kleene were our point of departure: When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the λ–definable functions. But, quickly realizing that the diagonalization cannot (...) be done effectively, I became overnight a supporter of the thesis. (1981, p. 59) The title of our paper alludes to this very work, a task on which Kleene claims to have set out after hearing such a remarkable statement from Church, who was his teacher at the time. There are quite a few points made in this extract that may be surprising. First, it talks about a proof by diagonalization in order to test—in fact to try to falsify—a hypothesis that is not strictly formal. Second, it states that such a proof or diagonal construction fails. Third, it seems to use the failure as a support for the thesis. Finally, the episode we have just described took place at a time, autumn 1933, in which many of the results that characterize Computability Theory had not yet materialized. The aim of this paper is to show that Church and Kleene discovered a way to block a very particular instance of a diagonal construction: one that is closely related to the content of Church's thesis. We will start by analysing the logical structure of a diagonal construction. Then we will introduce the historical context in order to analyse the reasons that might have led Kleene to think that the failure of this very specific diagonal proof could support the thesis. This is a joint paper. We have both attempted to add a small piece to an amazing historical jigsaw puzzle at a juncture we feel to be appropiate. In the paper by Manzano 1997 the aforementioned words by Kleene were quoted, and since then several logicians, Enrique Alonso first and foremost, have questioned her on this issue. Here we both submit our reply. (1999, pp. 249--273). (shrink)

A colleague’s personal recollections of Jean van Heijenoort’s contributions to the editing of volumes I–III of Gödel’s Collected Works and of his interactions with the other editors.

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both (...) Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake. (shrink)

The purpose of this paper is threefold. First, I visit the Fogelin–Geach-dispute, criticizeMiller''s interpretation of the Geachian notationN(x:N(fx)) and conclude that Fogelin''s argumentagainst the expressive completeness of the Tractariansystem of logic is unacceptable and that the adoptionof the Geachian notation N(x:fx) would not violate TLP5.32. Second, I prove that a system of quantificationtheory with finite domains and with N as the solefundamental operation is expressively complete. Lastly, I argue that the Tractarian system is apredicate-eliminated many-sorted theory (withoutidentity) with finite domains (...) and with N as the solefundamental operation, and thus is expressivelycomplete. (shrink)

Physicalism is the default position in science and in the philosophy of mind, but it should not be, I argue, because of two errors. By its epistemological error, physicalism gives physics priority over the evidence of first person experience. Only what I experience in first person is certain, so observation is prior to any theory. Physics itself is based on observation, avoiding the epistemological error, and then physics can progress, even changing its own ontology. However, physicalism imposes the ontology of (...) physics on every science, and in physics everything is causal. By its ontological error, physicalism tries to explain causally what is intentional. And it happens that causality and intentionality are mutually exclusive, showing that the ontology of physics is insufficient wherever intentions are present. This ontological insufficiency prevents that physicalism can repeat the success of physics with any science where intentions play a rôle, and thus it is blocking the advance of both the social sciences and the philosophy of mind. To overcome this obstacle, I propose to go back to the essentials: we should consider again the transcendental problem raised by Descartes and its solutions by Hume and Kant. On top of this subjectivist solution, we should take advantage of Darwin and Turing, and we should extend our ontology beyond causality to include intentionality, and here my proposal is problem solving. Then you could join our Post-Kantian subjectivism and say with me: The world is not a huge machine, as physicalism proposes, but an enigmatic problem. (shrink)

Christopher G. Timpson provides the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. He argues for an ontologically deflationary account of the nature of quantum information, which is grounded in a revisionary analysis of the concepts of information.

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

The Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna a?

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)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

Alonzo Church (1903–1995) was a renowned mathematical logician, philosophical logician, philosopher, teacher and editor. He was one of the founders of the discipline of mathematical logic as it developed after Cantor, Frege and Russell. He was also one of the principal founders of the Association for Symbolic Logic and the Journal of Symbolic Logic. The list of his students, mathematical and philosophical, is striking as it contains the names of renowned logicians and philosophers. In this article, we focus primarily on (...) Church’s philosophical contributions. (For an account of his life and academic history see the Introduction to The Collected Works of Alonzo Church (2019).) However, we also discuss his mathematical results when it is desirable to do so in order to pursue a philosophical issue. (shrink)

Combining physics, mathematics and computer science, quantum computing and its sister discipline of quantum information have developed in the past few decades from visionary ideas to two of the most fascinating areas of quantum theory. General interest and excitement in quantum computing was initially triggered by Peter Shor (1994) who showed how a quantum algorithm could exponentially “speed-up” classical computation and factor large numbers into primes far more efficiently than any (known) classical algorithm. Shor’s algorithm was soon followed by several (...) other algorithms that aimed to solve combinatorial and algebraic problems, and in the years since theoretical study of quantum systems serving as computational devices has achieved tremendous progress. Common belief has it that the implementation of Shor’s algorithm on a large scale quantum computer would have devastating consequences for current cryptography protocols which rely on the premise that all known classical worst-case algorithms for factoring take time exponential in the length of their input (see, e.g., Preskill 2005). Consequently, experimentalists around the world are engaged in attempts to tackle the technological difficulties that prevent the realisation of a large scale quantum computer. But regardless whether these technological problems can be overcome (Unruh 1995; Ekert and Jozsa 1996; Haroche and Raimond 1996), it is noteworthy that no proof exists yet for the general superiority of quantum computers over their classical counterparts. -/- The philosophical interest in quantum computing is manifold. From a social-historical perspective, quantum computing is a domain where experimentalists find themselves ahead of their fellow theorists. Indeed, quantum mysteries such as entanglement and nonlocality were historically considered a philosophical quibble, until physicists discovered that these mysteries might be harnessed to devise new efficient algorithms. But while the technology for harnessing the power of 50–100 qubits (the basic unit of information in the quantum computer) is now within reach (Preskill 2018), only a handful of quantum algorithms exist, and the question of whether these can truly outperform any conceivable classical alternative is still open. From a more philosophical perspective, advances in quantum computing may yield foundational benefits. For example, it may turn out that the technological capabilities that allow us to isolate quantum systems by shielding them from the effects of decoherence for a period of time long enough to manipulate them will also allow us to make progress in some fundamental problems in the foundations of quantum theory itself. Indeed, the development and the implementation of efficient quantum algorithms may help us understand better the border between classical and quantum physics (Cuffaro 2017, 2018a; cf. Pitowsky 1994, 100), and perhaps even illuminate fundamental concepts such as measurement and causality. Finally, the idea that abstract mathematical concepts such as computability and complexity may not only be translated into physics, but also re-written by physics bears directly on the autonomous character of computer science and the status of its theoretical entities—the so-called “computational kinds”. As such it is also relevant to the long-standing philosophical debate on the relationship between mathematics and the physical world. (shrink)

1. Logic, determinism and free will. 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; my concern here is to limit attention to two arguments from logic. To begin with, 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. Surely no such argument could really establish determinism, but one is hard pressed to explain where it goes wrong. One now classic dismantling of it has been given by Gilbert Ryle, in the chapter ‘What was to be’ of his fine book, Dilemmas (Ryle 1954). We leave it to the interested reader to pursue that and the subsequent literature. (shrink)

We highlight the limitations of formal methods by exhibiting two results in recursive function theory: that there is no effective means of finding a program that satisfies a given formal specification; or checking that a program meets a specification. We also exhibit a ‘simple’ MAS which has all the power of a Turing machine. We then argue that any ‘pure design’ methodology will face insurmountable difficulties in today’s open and complex MAS. Rather we suggest a methodology based on the classic (...) experimental method – that is ‘scientific foundations’ for the construction and control of complex MAS. (shrink)

Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.

Este ensayo ofrece un análisis crítico del último argumento que el matemático y filósofo Roger Penrose ofrece a favor de la tesis según la cual hay habilidades de la mente humana que nunca podrán ser igualadas por ingenio mecánico alguno. Al mismo tiempo se ofrece una descripción general de los últimos episodios del eterno enfrentamiento entre mentalismo y mecanicismo y se concluye con una sugerencia acerca de los puntos en los que cabe esperar nuevas situaciones de tensión entre estos dos (...) grandes paradigmas. Este trabajo pretende, además, mostrar la vigencia de un tipo de filosofía elaborada a partir de las herramientas de la ciencia moderna cuyos problemas son, no obstante, los del pensamiento filosófico tradicional. (shrink)

In this article it is argued that the relation between the socalled Gutenberg galaxis of print culture and the Turing galaxis of digital media is not one of opposition and substitution, but rather one of co-evolution and integration. Or more precisely: that the Gutenberg galaxis on the one hand can be inscribed into the Turing galaxis, which on the other hand is textual in character since it is based on linear and serially processed representations manifested in a binary alphabet. In (...) continuation of this text and hypertext is described as notions of related but different sorts of complex systems and it is argued that both texts and hypertexts are best thought of as sequentially organised. The first part of the paper contains a description of the basic properties of computers and digital representation answering the question: what are the properties common to all kinds of use of computers – whether used for one purpose or another? The second part is concerned with the relationship between printed and electronic text. The third part addresses the idea of hypertext as systems which exploits modal shifts between node-mode and link-mode as a significant part of the semantics of the system. (shrink)

This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...) a retraction, has been proved in Section 11.4. The last section gives a laconic answer to the question posed in the title of the paper. (shrink)

Almost everything Turing wrote is now accessible on-line in some form, much of it in the Turing Digital Archive, which makes available scanned versions of the physical papers held in the archive at King's College, Cambridge University. See..

The article discusses the limitations of psycho-physical parallellism and the implications of a pscycho-physical interaction paradigm considering the notion Downward Causation. The focus is on the notion of levels in nature and their interrelations, and it argues that the notion of rule-based systems should be considered a subcategory of rule-generating systems partly based on redundancy functions rather than rules.