Turing's World is a self-contained introduction to Turing machines, one of the fundamental notions of logic and computer science. The text and accompanying diskette allow the user to design, debug, and run sophisticated Turing machines in a graphical environment on the Macintosh. Turning's World introduces users to the key concpets in computability theory through a sequence of over 100 exercises and projects. Within minutes, users learn to build simple Turing machines using a convenient package of graphical functions. (...) Exercises then progress through a significant portion of elementary computability theory, covering such topics as the Halting problem, the Busy Beaver function, recursive functions, and undecidability. Version 3.0 is an extensive revision and enhancement of earlier releases of the program, allowing the construction of one-way and two-way finite state machines (finite automata), as well as nondeterministic Turing and finite-state machines. Special exercises allow users to explore these alternative machines. (shrink)

Well known for this crucial wartime role in breaking the ENIGMA code, this book chronicles Turing's struggle to build the modern computer. Includes first hand accounts by Turing and the pioneers of computing who worked with him.

Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing, the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Gödel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as (...) computer science. Though less well known than his other work, Turing's 1938 Princeton PhD thesis, "Systems of Logic Based on Ordinals," which includes his notion of an oracle machine, has had a lasting influence on computer science and mathematics. This book presents a facsimile of the original typescript of the thesis along with essays by Andrew Appel and Solomon Feferman that explain its still-unfolding significance. A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal--a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine. Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science. (shrink)

Turing wrote that the “guiding principle” of his investigation into the possibility of intelligent machinery was “The analogy [of machinery that might be made to show intelligent behavior] with the human brain.” [10] In his discussion of the investigations that Turing said were guided by this analogy, however, he employs a more far-reaching analogy: he eventually expands the analogy from the human brain out to “the human community as a whole.” Along the way, he takes note of an obvious fact (...) in the bigger scheme of things regarding human intelligence: grownups were once children; this leads him to imagine what a machine analogue of childhood might be. In this paper, I’ll discuss Turing’s child-machine, what he said about different ways of educating it, and what impact the “bringing up” of a child-machine has on its ability to behave in ways that might be taken for intelligent. I’ll also discuss how some of the various games he suggested humans might play with machines are related to this approach. (shrink)

When it comes to the question of what kind of moral claim an intelligent or autonomous machine might have, one way to answer this is by way of comparison with humans: Is there a fundamental difference between humans and other entities? If so, on what basis, and what are the implications for science and ethics? This question is inherently imprecise, however, because it presupposes that we can readily determine what it means for two types of entities to be sufficiently different—what (...) I will refer to as being “discontinuous”. In this paper, I will sketch a formal characterization of what it means for types of entities to be unique with regard to each other. This expands upon Bruce Mazlish’s initial formulation of what he terms a continuity between humans and machines, Alan Turing’s epistemological approach to the question of machine intelligence, and Sigmund Freud’s notion of scientific revolutions dealing blows to the self-esteem of mankind. I will discuss on what basis we should regard entities as (dis-)continuous, the corresponding moral and scientific implications, as well as an important difference between what I term downgrading and upgrading continuities—a dramatic difference in how two previously discontinuous types of entities might become continuous. All of this will be phrased in terms of which scientific levels of explanation we need to presuppose, in principle or in practice, when we seek to explain a given type of entity. The ultimate purpose is to provide a framework that defines which questions we need to ask if we argue that two types of entities ought (not) to be explained (hence treated) in the same manner, as well as what it takes to reconsider scientific and ethical hierarchies imposed on the natural and artificial world. (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)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

Turing’s paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure comprehensively hosting causality at the physical level and beyond. On the other, it can give an insight into the way in which higher order information arises and leads to loss of computational control—while demonstrating how the control can be re-established, in special circumstances, via suitable type (...) reductions. We examine the classical computational framework more closely than is usual, drawing out lessons for the wider application of information–theoretical approaches to characterizing the real world. The problem which arises across a range of contexts is the characterizing of the balance of power between the complexity of informational structure and the means available to bring this information back into the computational fold. We proceed via appropriate mathematical modelling to a more coherent view of the computational structure of information, relevant to a wide spectrum of areas of investigation. (shrink)

The day was particularly appropriate. There was a great deal of publicity for the 50th anniversary of the world's first working modern computer, which ran at Manchester on 21 June 1948. And at 10.30pm the night before, 22 June 1998, the House of Commons had voted by a large majority to change the law so that homosexual and heterosexual acts would alike be governed by an 'age of consent' of 16. It was recognised by all sides that the issue (...) at stake was that of equality. (Note added later: the law was indeed finally changed on 30 November 2000, despite intense opposition from christian leaders in the unelected House of Lords.). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Ad Infinitum: The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back InTony E. JacksonAd Infinitum: The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back In, by Brian Rotman; xii & 203 pp. Stanford: Stanford University Press, 1993, $39.50 cloth, $12.95 paper.Brian Rotman’s book attempts to pull mathematics—the last, most solid home of metaphysical thought—off its absolutist (...) foundations and into the roiling current of postmodernism. His argument works in what has by now become a rather standard manner. He shows how the grounding ideas of mathematics rest upon the insupportable assumption of a realm of absolute truth, “truth perfect, truth outside time, outside space, outside the material world of matter, energy, and entropy, truth somehow able to be discovered or apprehended but never invented” (p. 19). Rotman questions this Platonic grounding in a number of ways.First, with the help of linguistics (de Saussure, Benveniste, Wittgenstein, and Peirce) he provides a semiotic analysis of mathematical arguments as real-world linguistic events. Proof he looks at “not as a form of inference engaged in preserving eternal truths, but as an historically conditioned species of rhetoric” (p. 35). Secondly, Rotman argues that one idea has most secured the fortress-like solidity of mathematical Platonism: the idea of ad infinitum or infinite iteration. Mathematics takes off from the givenness of the natural numbers, our everyday 1, 2, 3 and so on to infinity. It is the “so on to infinity” that frames up the house of arithmetic as it has stood since Euclid. (Euclidean in this context is equivalent to Platonic in the context of philosophy.) But what is the referent of these numbers? Ultimately they are said to be the symbolic representation of the action of counting, which is the “mathematical ur-cognition” (p. 51). And yet we cannot imagine any material human situation in which counting could actually go on to infinity. Therefore, some impossible, supernatural agent (the ghost of the title) must be projected in order for this conception to exist. Similarly, Rotman argues against the assumption of “perfect repeatability” because this notion depends upon some purely imaginary, ideal identity of object or action that could possibly be repeated. Ad infinitum requires and assumes both a disembodied, metaphysical counter and a disembodied, metaphysical counted.So Rotman sets about putting the body back into this symbolic system by rejecting the idea of ad infinitum. Instead of assuming the existence of counting activities that run, in principle, out to infinity without changing their nature, Rotman posits what he calls “realizable iteration,” that is, counting that admits the necessary dissolution or change of whatever repetitive function you begin with. Put simply, he includes the fact that quantitative accumulation or repetition always eventually brings about a qualitative change: at some point in everyday life a difference of degree of the “same” thing or action inevitably becomes a difference of kind. The natural numbers and the ad infinitum [End Page 390] remain blissfully sheltered from this inescapable real world truth, but Rotman reformulates mathematics so as to include this truth in the very idea of numbering.In fact (and he acknowledges this) he theorizes counting after the model of chaos or complexity theory in the physical sciences. Realizable iteration, he says, will bring in the “qualitative differences” forgotten or repressed (the language of Nietzsche, Heidegger, and Freud runs throughout) in the metaphysical version of number, and “display [the qualitative differences] as the emergent effects of repetition” (p. 131). Even numerical addition, for instance, which had been the most linearly continuous of activities, now will be linear beginning from some initial point, but will become increasingly nonlinear as quantitative increase tends towards some point of qualitative discontinuity with what had come before. The point of discontinuity, though certain to appear, is by its nature not specifiable beyond a horizon of probability, and in this, Rotman aligns his arithmetic not only with chaos theory, but also with quantum mechanics. In an appendix, Rotman elaborates at least to a degree the “pre-elements” of his non-Euclidean arithmetic.As chaos theory does not destroy everyday cause... (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930''s. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn''t be computed by a Turing machine but could be computed by means of some other kind of effective (...) procedure. Since that time, however, other interpretations of the thesis have appeared in the literature. In this paper I identify three domains of application which have been claimed for the thesis: (1) the number theoretic functions; (2) all functions; (3) mental and/or physical phenomena. Subsequently, I provide an analysis of our intuitive concept of a procedure which, unlike Turing''s, is based upon ordinary, everyday procedures such as recipes, directions and methods; I call them mundane procedures. I argue that mundane procedures can be said to be effective in the same sense in which Turing machine procedures can be said to be effective. I also argue that mundane procedures differ from Turing machine procedures in a fundamental way, viz., the former, but not the latter, generate causal processes. I apply my analysis to all three of the above mentioned interpretations of the Church-Turing thesis, arguing that the thesis is (i) clearly false under interpretation (3), (ii) false in at least some possible worlds (perhaps even in the actual world) under interpretation (2), and (iii) very much open to question under interpretation (1). (shrink)

This volume celebrates the various facets of Alan Turing (1912–1954), the British mathematician and computing pioneer, widely considered as the father of computer science. It is aimed at the general reader, with additional notes and references for those who wish to explore the life and work of Turing more deeply. -/- The book is divided into eight parts, covering different aspects of Turing’s life and work. -/- Part I presents various biographical aspects of Turing, some from a personal point of (...) view. -/- Part II presents Turing’s universal machine (now known as a Turing machine), which provides a theoretical framework for reasoning about computation. His 1936 paper on this subject is widely seen as providing the starting point for the field of theoretical computer science. -/- Part III presents Turing’s working on codebreaking during World War II. While the War was a disastrous interlude for many, for Turing it provided a nationally important outlet for his creative genius. It is not an overstatement to say that without Turing, the War would probably have lasted longer, and may even have been lost by the Allies. The sensitive nature of Turning’s wartime work meant that much of this has been revealed only relatively recently. -/- Part IV presents Turing’s post-War work on computing, both at the National Physical Laboratory and at the University of Manchester. He made contributions to both hardware design, through the ACE computer at the NPL, and software, especially at Manchester. Part V covers Turing’s contribution to machine intelligence (now known as Artificial Intelligence or AI). Although Turing did not coin the term, he can be considered a founder of this field which is still active today, authoring a seminal paper in 1950. -/- Part VI covers morphogenesis, Turing’s last major scientific contribution, on the generation of seemingly random patterns in biology and on the mathematics behind such patterns. Interest in this area has increased rapidly in recent times in the field of bioinformatics, with Turing’s 1952 paper on this subject being frequently cited. -/- Part VII presents some of Turing’s mathematical influences and achievements. Turing was remarkably free of external influences, with few co-authors – Max Newman was an exception and acted as a mathematical mentor in both Cambridge and Manchester. -/- Part VIII considers Turing in a wider context, including his influence and legacy to science and in the public consciousness. -/- Reflecting Turing’s wide influence, the book includes contributions by authors from a wide variety of backgrounds. Contemporaries provide reminiscences, while there are perspectives by philosophers, mathematicians, computer scientists, historians of science, and museum curators. Some of the contributors gave presentations at Turing Centenary meetings in 2012 in Bletchley Park, King’s College Cambridge, and Oxford University, and several of the chapters in this volume are based on those presentations – some through transcription of the original talks, especially for Turing’s contemporaries, now aged in their 90s. Sadly, some contributors died before the publication of this book, hence its dedication to them. -/- For those interested in personal recollections, Chapters 2, 3, 11, 12, 16, 17, and 36 will be of interest. For philosophical aspects of Turing’s work, see Chapters 6, 7, 26–31, and 41. Mathematical perspectives can be found in Chapters 35 and 37–39. Historical perspectives can be found in Chapters 4, 8, 9, 10, 13–15, 18, 19, 21–25, 34, and 40. With respect to Turing’s body of work, the treatment in Parts II–VI is broadly chronological. We have attempted to be comprehensive with respect to all the important aspects of Turing’s achievements, and the book can be read cover to cover, or the chapters can be tackled individually if desired. There are cross-references between chapters where appropriate, and some chapters will inevitably overlap. -/- We hope that you enjoy this volume as part of your library and that you will dip into it whenever you wish to enter the multifaceted world of Alan Turing. (shrink)

Discussions of artificial intelligence have been shaped by two brilliant thought-experiments: Turing’s Imitation Test for thinking systems and Searle’s Chinese Room Argument. In many ways, debates about large language models (LLMs) struggle to move beyond these original, opposing thought-experiments. So, in this paper, I ask whether we can move debate forward by exploring the features Sceptics about LLM abilities take to ground meaning. Section 1 sketches the options, while Sections 2 and 3 explore the common requirement for a robust relation (...) between linguistic signs and external objects. Section 3 argues that concerns about worldly connections can be met and thus that outputs of LLMs should be viewed as genuinely meaningful. Section 4 then turns to the argument that the kind of derived meaning explored in Section 3 is insufficient for LLMs to count as meaningful systems per se, looking at the claim that they must possess original intentionality before we admit them to the space of meaning. I argue that this demand is not a prerequisite for meaning per se, but rather for some kinds of agency or conscious understanding. I suggest that the demands for original intentionality are not currently met by LLMs and that we should be cautious about whether we want them to be. (shrink)

This personal, yet scientific, letter to Alan Turing, reflects on Turing's personality in order to better understand his scientific quest. It then focuses on the impact of his work today. By joining human attitude and particular scientific method, Turing is able to “immerse himself” into the phenomena on which he works. This peculiar blend justifies the epistolary style. Turing makes himself a “human computer”, he lives the dramatic quest for an undetectable imitation of a man, a woman, a machine. (...) He makes us see the continuous deformations of a material action/reaction/diffusion dynamics of hardware with no software. Each of these investigations opens the way to new scientific paths with major consequences for contemporary live and for knowledge. The uses and the effects of these investigations will be discussed: the passage from classical AI to today's neural nets, the relevance of non-linearity in biological dynamics, but also their abuses, such as the myth of a computational world, from a Turing-machine like universe to an encoded homunculus in the DNA. It is shown that these latter ideas, which are sometimes even made in Turing's name, contradict his views. (shrink)

We begin with the history of the discovery of computability in the 1930’s, the roles of Gödel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines . To whom did Gödel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine and Post’s development of it in [1944, §11], [1948], and finally Kleene-Post [1954] into its present form. A number of topics arose from Turing functionals including continuous functionals on (...) Cantor space and online computations. Almost all the results in theoretical computability use relative reducibility and o-machines rather than a-machines and most computing processes in the real world are potentially online or interactive. Therefore, we argue that Turing o-machines, relative computability, and online computing are the most important concepts in the subject, more so than Turing a-machines and standard computable functions since they are special cases of the former and are presented first only for pedagogical clarity to beginning students. At the end in §10–§13 we consider three displacements in computability theory, and the historical reasons they occurred. Several brief conclusions are drawn in §14. (shrink)

The Turing Test, originally proposed as a simple operational definition of intelligence, has now been with us for exactly half a century. It is safe to say that no other single article in computer science, and few other articles in science in general, have generated so much discussion. The present article chronicles the comments and controversy surrounding Turing's classic article from its publication to the present. The changing perception of the Turing Test over the last fifty years has paralleled (...) the changing attitudes in the scientific community towards artificial intelligence: from the unbridled optimism of 1960's to the current realization of the immense difficulties that still lie ahead. I conclude with the prediction that the Turing Test will remain important, not only as a landmark in the history of the development of intelligent machines, but also with real relevance to future generations of people living in a world in which the cognitive capacities of machines will be vastly greater than they are now. (shrink)

Recent advances in neuroscience lead to a wider realm for philosophy to include the science of the Darwinian-evolved computational brain, our inner world producing organ, a non-recursive super- Turing machine combining 100B synapsing-neuron DNA-computers based on the genetic code. The whole system is a logos machine offering a world map for global context, essential for our intentional grasp of opportunities. We start from the observable contrast between the chaotic universe vs. our orderly inner world, the noumenal cosmos. (...) So far, philosophy has been rehearsing our thoughts, our human-internal world, a grand painting of the outer world, how we comprehend subjectively our experience, worked up by the logos machine, but now we seek a wider horizon, how humans understand the world thanks to Darwinian evolution to adapt in response to the metaphysical gap, the chasm between the human animal and its environment, shaping the organism so it can deal with its variable world. This new horizon embraces global context coded in neural structures that support the noumenal cosmos, our inner mental world, for us as denizens of the outer environment. Kant’s inner and outer senses are fundamental ingredients of scientific philosophy. Several sections devoted to Heidegger, his lizard debunked, but his version of the metaphysical gap & his doctrine of the logos praised. Rorty and others of the behaviorist school discussed also. (shrink)

An investigation of the concept of “surveyability” as traced through the thought of Hilbert, Wittgenstein, and Turing. The communicability and reproducibility of proof, with certainty, are seen as earmarked by the “surveyability” of symbols, sequences, and structures of proof in all these thinkers. Hilbert initiated the idea within his metamathematics, Wittgenstein took up a kind of game formalism in the 1920s and early 1930s in response. Turing carried Hilbert’s conception of the “surveyability” of proof in metamathematics through into his analysis (...) of what a formal system (what a step in a computation) is in “On computable numbers, with an application to the Entscheidungsproblem” (1936). Wittgenstein’s 1939 investigations of the significance of surveyability to the concept of “proof “in Principia Mathematica were influenced, both by Turing’s remarkable everyday analysis of the Hilbertian idea, and by conversations with Turing. Although Turing does not use the word “surveyability” explicitly, it is clear that the Hilbertian idea plays a recurrent role in his work, refracted through his engagement with Wittgenstein’s idea of a “language-game”. This is evinced in some of his later writings, where the “reform” of mathematical notation for the sake of human surveyability (1944/45) may be seen to draw out the Hilbertian idea. For Turing, as for Wittgenstein, the need for “surveyability” earmarks the evolving culture of humans located in an evolving social and scientific world, just as it had for Hilbert. (shrink)

Alan M. Turing, pioneer of computing and WWII codebreaker, is one of the most important and influential thinkers of the twentieth century. In this volume for the first time his key writings are made available to a broad, non-specialist readership. They make fascinating reading both in their own right and for their historic significance: contemporary computational theory, cognitive science, artificial intelligence, and artificial life all spring from this ground-breaking work, which is also rich in philosophical and logical insight. An introduction (...) by leading Turing expert Jack Copeland provides the background and guides the reader through the selection. About Alan Turing Alan Turing FRS OBE, (1912-1954) studied mathematics at King's College, Cambridge. He was elected a Fellow of King's in March 1935, at the age of only 22. In the same year he invented the abstract computing machines - now known simply as Turing machines - on which all subsequent stored-program digital computers are modelled. During 1936-1938 Turing continued his studies, now at Princeton University. He completed a PhD in mathematical logic, analysing the notion of 'intuition' in mathematics and introducing the idea of oracular computation, now fundamental in mathematical recursion theory. An 'oracle' is an abstract device able to solve mathematical problems too difficult for the universal Turing machine. In the summer of 1938 Turing returned to his Fellowship at King's. When WWII started in 1939 he joined the wartime headquarters of the Government Code and Cypher School (GC&CS) at Bletchley Park, Buckinghamshire. Building on earlier work by Polish cryptanalysts, Turing contributed crucially to the design of electro-mechanical machines ('bombes') used to decipher Enigma, the code by means of which the German armed forces sought to protect their radio communications. Turing's work on the version of Enigma used by the German navy was vital to the battle for supremacy in the North Atlantic. He also contributed to the attack on the cyphers known as 'Fish'. Based on binary teleprinter code, Fish was used during the latter part of the war in preference to morse-based Enigma for the encryption of high-level signals, for example messages from Hitler and other members of the German High Command. It is estimated that the work of GC&CS shortened the war in Europe by at least two years. Turing received the Order of the British Empire for the part he played. In 1945, the war over, Turing was recruited to the National Physical Laboratory (NPL) in London, his brief to design and develop an electronic computer - a concrete form of the universal Turing machine. Turing's report setting out his design for the Automatic Computing Engine (ACE) was the first relatively complete specification of an electronic stored-program general-purpose digital computer. Delays beyond Turing's control resulted in NPL's losing the race to build the world's first working electronic stored-program digital computer - an honour that went to the Royal Society Computing Machine Laboratory at Manchester University, in June 1948. Discouraged by the delays at NPL, Turing took up the Deputy Directorship of the Royal Society Computing Machine Laboratory in that year. Turing was a founding father of modern cognitive science and a leading early exponent of the hypothesis that the human brain is in large part a digital computing machine, theorising that the cortex at birth is an 'unorganised machine' which through 'training' becomes organised 'into a universal machine or something like it'. He also pioneered Artificial Intelligence. Turing spent the rest of his short career at Manchester University, being appointed to a specially created Readership in the Theory of Computing in May 1953. He was elected a Fellow of the Royal Society of London in March 1951 (a high honour). (shrink)

A true Turing machine (TM) requires an infinitely long paper tape. Thus a TM can be housed in the infinite world of Newtonian spacetime (the spacetime of common sense), but not necessarily in our world, because our world-at least according to our best spacetime theory, general relativity-may be finite. All the same, one can argue for the "existence" of a TM on the basis that there is no such housing problem in some other relativistic worlds that are (...) similar ("close") to our world. But curiously enough-and this is the main point of this paper-some of these close worlds have a special spacetime structure that allows TMs to perform certain Turing unsolvable tasks. For example, in one kind of spacetime a TM can be used to solve first-order predicate logic and the halting problem. And in a more complicated spacetime, TMs can be used to decide arithmetic. These new computers serve to show that Church's thesis is a thoroughly contingent claim. Moreover, since these new computers share the fundamental properties of a TM in ordinary operation (e.g. intuitive, finitely programmed, limited in computational capability), a computability theory based on these non-Turing computers is no less worthy of investigation than orthodox computability theory. Some ideas about this new mathematical theory are given. (shrink)

The Turing Test is routinely understood as a behaviourist test for machine intelligence. Diane Proudfoot has argued for an alternative interpretation. According to Proudfoot, Turing’s claim that intelligence is what he calls ‘an emotional concept’ indicates that he conceived of intelligence in response-dependence terms. As she puts it: ‘Turing’s criterion for “thinking” is…: x is intelligent if in the actual world, in an unrestricted computer-imitates-human game, x appears intelligent to an average interrogator’. The role of the famous test is (...) thus to provide the conditions in which to examine the average interrogator’s responses. I shall argue that Proudfoot’s analysis falls short. The philosophical literature contains two main models of response-dependence, what I shall call the transparency model and the reference-fixing model. Proudfoot resists the thought that Turing might have endorsed one of these models to the exclusion of the other. But the details of her own analysis indicate that she is, in fact, committed to the claim that Turing’s account of intelligence is grounded in a transparency model, rather than a reference-fixing one. By contrast, I shall argue that while Turing did indeed conceive of intelligence in response-dependence terms, his account is grounded in a reference-fixing model, rather than a transparency one. This is fortunate, because, as an account of intelligence, the transparency model is arguably problematic in a way that the reference-fixing model isn’t. (shrink)

In his short life, Alan Turing (1912-1954) made foundational contributions to philosophy, mathematics, biology, artificial intelligence, and computer science. He, as much as anyone, invented the digital electronic computer. From September, 1939 much of his work on computation was war-driven and brutally practical. He developed high speed computing devices needed to decipher German Enigma Machine messages to and from U-boats, countering the most serious threat by far to Britain's survival during World War Two. Yet few people have an image (...) of him. (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)

Robert French has argued that a disembodied computer is incapable of passing a Turing Test that includes subcognitive questions. Subcognitive questions are designed to probe the network of cultural and perceptual associations that humans naturally develop as we live, embodied and embedded in the world. In this paper, I show how it is possible for a disembodied computer to answer subcognitive questions appropriately, contrary to French’s claim. My approach to answering subcognitive questions is to use statistical information extracted from (...) a very large collection of text. In particular, I show how it is possible to answer a sample of subcognitive questions taken from French, by issuing queries to a search engine that indexes about 350 million Web pages. This simple algorithm may shed light on the nature of human cognition, but the scope of this paper is limited to demonstrating that French is mistaken: a disembodied computer can answer subcognitive questions. (shrink)

This book provides a novel framework for understanding and revising labor markets and education policies in an era of machine learning. It posits that while learning and knowing both require thinking, learning is fundamentally different than knowing because it results in cognitive processes that change over time. Learning, in contrast to knowing, requires time and agency. Therefore, “learning algorithms”—that enable machines to modify their actions based on real-world experiences—are a fundamentally new form of artificial intelligence that have potential to (...) be even more disruptive to labor markets than prior introductions of digital technology. To explore the difference between knowing and learning, Turing’s “Imitation Game,”—that he proposed as a test for machine thinking—is expanded to include time dependence. The arguments presented in the book introduce three novel concepts: (1) Comparative learning advantage: This is a concept analogous to comparative labor advantage but arises from the disparate times required to learn new knowledge bases/skillsets. It is argued that in the future, comparative learning advantages between humans and machines will determine their division of labor. (2) Two dimensions of job performance—expertise and interpersonal: Job tasks can be sorted into two broad categories. Tasks that require expertise have stable endpoints, which makes these tasks inherently repetitive and subject to automation. Tasks that are interpersonal are highly context-dependent and lack stable endpoints, which makes these tasks inherently non-routine. Humans compared to machines have a comparative learning advantage along the interpersonal dimension, which is increasing in value economically. (3) The Learning Game is a time-dependent version of Turing’s “Imitation Game.” It is more than a thought experiment. The “Learning Game” provides a mathematical framework with quantitative criteria for training and assessing comparative learning advantages. The book is highly interdisciplinary—presenting philosophical arguments in economics, artificial intelligence, and education. It also provides data, mathematical analysis, and testable criteria that researchers in these fields will find of practical use. The book calls for a rethinking of how labor markets operate and how the education system should prepare students for future jobs. It concludes with a list of counterintuitive recommendations for future education and labor policies that all stakeholders—employers, employees, educators, students, and political leaders—should heed. (shrink)

This paper commences from the critical observation that the Turing Test (TT) might not be best read as providing a definition or a genuine test of intelligence by proxy of a simulation of conversational behaviour. Firstly, the idea of a machine producing likenesses of this kind served a different purpose in Turing, namely providing a demonstrative simulation to elucidate the force and scope of his computational method, whose primary theoretical import lies within the realm of mathematics rather than cognitive modelling. (...) Secondly, it is argued that a certain bias in Turing’s computational reasoning towards formalism and methodological individualism contributed to systematically unwarranted interpretations of the role of the TT as a simulation of cognitive processes. On the basis of the conceptual distinction in biology between structural homology vs. functional analogy, a view towards alternate versions of the TT is presented that could function as investigative simulations into the emergence of communicative patterns oriented towards shared goals. Unlike the original TT, the purpose of these alternate versions would be co-ordinative rather than deceptive. On this level, genuine functional analogies between human and machine behaviour could arise in quasi-evolutionary fashion. (shrink)

Darwin’s theory of evolution by natural selection unifies the world of physics with the world of meaning and purpose by proposing a deeply counterintuitive ‘‘inversion of reasoning’’ (according to a 19th century critic): ‘‘to make a perfect and beautiful machine, it is not requisite to know how to make it’’ [MacKenzie RB (1868) (Nisbet & Co., London)]. Turing proposed a similar inversion: to be a perfect and beautiful computing machine, it is not requisite to know what arithmetic is. (...) Together, these ideas help to explain how we human intelligences came to be able to discern the reasons for all of the adaptations of life, including our own. (shrink)

Some of the papers in this special issue distribute cognition between what is going on inside individual cognizers' heads and their outside worlds; others distribute cognition among different individual cognizers. Turing's criterion for cognition was individual, autonomous input/output capacity. It is not clear that distributed cognition could pass the Turing Test.

The dilemma confronted by Buridan’s Ass leads into a problem about nil-preference situations, to which there is a solution in the literature that is inspired by Alan Turing: we have evolved with a computational module in our brains that comes into play in such situations by picking a random action among the alternatives that detennines the subject’s choice. We relate these Buridan’s Ass situations to a larger, theoretically interesting category in which there is no alternative that is decisively superior to (...) others with respect to expected utility, and we try to show how our emotional makeup figures in a rational response, particularly as informed by symbolic utilitythat we draw down from our culture’s shared understandings. The category is theoretically interesting because it contains moral dilemmas, as well as hard cases in which an imponant choice must be made without an option that has clearly superior expected utility. We argue that our Emotional Response Model is preferable to Turing’s Randomizer for this category, as well as more illuminating about nil-preference situations or close approximations thereto. (shrink)

Though less well known than his other work, Turings 1938 Princeton Thesis, this title which includes his notion of an oracle machine, has had a lasting influence on computer science and mathematics. It presents a facsimile of the original typescript of the thesis along with essays by Appel and Feferman that explain its still-unfolding significance.

This short paper is meant to be an introduction to the ‘Letter to Alan Turing’ that follows it. It summarizes some basic ideas in information theory and very informally hints at their mathematical properties. In order to introduce Turing’s two main theoretical contributions, in Theory of Computation and in Morphogenesis, the fundamental divide between discrete vs. continuous structures in mathematics is presented, as it is also a divide in his scientific life. The reader who is familiar with these notions, and (...) is convinced that they are relevant in the mathematical understanding of phenomena, may skip this introduction and go directly to the Letter. (shrink)

The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the (...) mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field. (shrink)

The classical view of computing positions computation as a closed-box transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a function-based transformation of an input to an output. Specifically, communication with the outside world happens during the computation, not before or after it. This approach radically changes our understanding of what is computation and how it is modeled. The acceptance of interaction as a (...) new paradigm is hindered by the Strong Church–Turing Thesis (SCT), the widespread belief that Turing Machines (TMs) capture all computation, so models of computation more expressive than TMs are impossible. In this paper, we show that SCT reinterprets the original Church–Turing Thesis (CTT) in a way that Turing never intended; its commonly assumed equivalence to the original is a myth. We identify and analyze the historical reasons for the widespread belief in SCT. Only by accepting that it is false can we begin to adopt interaction as an alternative paradigm of computation. We present Persistent Turing Machines (PTMs), that extend TMs to capture sequential interaction. PTMs allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of TMs and of the CTT. The paradigm shift to interaction provides an alternative understanding of the nature of computing that better reflects the services provided by today’s computing technology. (shrink)

The unprecedented opportunities for experiments in complexity presented by the first modern computers in the late 1940's raised hopes in early computer scientists (eg. John von Neumann and Alan Turing) that the ability to think, our greatest asset in our dealings with the world, might soon be understood well enough to be duplicated. Success in such an endeavor would extend mankind's mind in the same way that the development of energy machinery extended his muscles.

Modern computational natural philosophy conceptualizes the universe in terms of information and computation, establishing a framework for the study of cognition and intelligence. Despite some critiques, this computational perspective has significantly influenced our understanding of the natural world, leading to the development of AI systems like ChatGPT based on deep neural networks. Advancements in this domain have been facilitated by interdisciplinary research, integrating knowledge from multiple fields to simulate complex systems. Large Language Models (LLMs), such as ChatGPT, represent this (...) approach's capabilities, utilizing reinforcement learning with human feedback (RLHF). Current research initiatives aim to integrate neural networks with symbolic computing, introducing a new generation of hybrid computational models. (shrink)

The ancient dualism of a sensible and an intelligible world important in Neoplatonic and medieval philosophy, down to Descartes and Kant, would seem to be supplanted today by a scientific view of mind-in-nature. Here, we revive the old dualism in a modified form, and describe mind as a symbolic language, founded in linguistic recursive computation according to the Church-Turing thesis, constituting a world L that serves the human organism as a map of the Universe U. This methodological distinction (...) of L vs. U helps to understand how and why structures of phenomena come to be opposed to their nature in human thought, a central topic in Heideggerian philosophy. U is uncountable according to Georg Cantor’s set theory but Language L, based on the recursive function system, is countable, and anchored in a Gray Area within U of observable phenomena, typically symbols, prelinguistic structures, genetic-historical records of their origins. Symbols, the phenomena most familiar to mathematicians, are capable of being addressed in L-processing. The Gray Area is the human Environment E, where we can live comfortably, that we manipulate to create our niche within hostile U, with L offering overall competence of the species to survive. The human being is seen in the light of his or her linguistic recursively computational mind. Nature U, by contrast, is the unfathomable abyss of being, infinite labyrinth of darkness, impenetrable and hostile to man. The U-man, biological organism, is a stranger in L-man, the mind-controlled rational person, as expounded by Saint Paul. Noumena can now be seen to reside in L, and are not fully supported by phenomena. Kant’s noumenal cause is the mental L-image of only partly phenomenal causation. Mathematics occurs naturally in pre-linguistic phenomena, including natural laws, which give rise to pure mathematical structures in the world of L. Mathematical foundation within philosophy is reversed to where natural mathematics in the Gray Area of pre-linguistic phenomena can be seen to be a prerequisite for intellectual discourse. Lesser, nonverbal versions of L based on images are shared with animals. (shrink)

Our aim is to propose several itineraries which follow the scenario having as a first step Euclid’s Fifth Postulate; as a second step the Bolyai–Lobachevsky’s non-Euclidean geometries and as a third step Einstein’s relativity theory. The role of Euclid’s fifth postulate is successively assumed by Archimedes’ axiom; Zermelo’s choice axiom; Cantor’s continuum hypothesis; von Neumann’s foundation axiom for set theory; Church–Turing thesis and Turing’s computability; the validity of classical logic under the form of the principles of identity, non-contradiction and excluded (...) middle; the macroscopic universe assumed as form of existence, including the fact that humans are macroscopic beings; human language and semiosis assumed as the unique way to approach understanding and communication; the old Greek ideal of order, symmetry and harmony, adopted for representation of the world in Art and Poetry. The common denominator of these starting steps is the fact that they all correspond to the sensorial-empirical perception of the world. We point out the strong interaction among the steps corresponding, in these different itineraries, to Einstein’s relativity theory. We infer in this way the need of a transdisciplinary approach. (shrink)

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- Axiomatic set (...) theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

Ṣūfī poetry or dīvān poetry, both of our poems have a universal appeal and a classical value just as the poetry of many nations’. Poets of both groups enhanced the consciousness level of every people one by one and created a virtuous society by taking power from the potential that existed in Turkish society already. If it is needed to mention a difference between those two poetries, it could be that dīvān poetry is a static one and sūfī poetry is (...) more dynamic than the dīvān poetry. While dīvān poetry has a unique World of poetry, sūfī poetry has a feeling of universe shaped by different meanings and concepts. Although dīvān poetry is based on intelligence and pleasure, there is intuitive inspiration and wāridāt in ṣūfī poetry. These become poetic sometimes thanks to similes (tashbīḥ), metaphors (majāz) and allusions (talmīḥ), become verse and gradually turns into a poetic theme (mażmūn). Until today, since the first that comes to mind, when it is said poetic themes (mażmūn), is dīvān poetry, poetic themes of ṣūfī poetry were overshadowed. However, ṣūfī poetry is quite interesting and rich in terms of its poetic themes. Although they are called as just terms, each of the terms of ṣūfī literature is a mystical state (ḥāl), mystical station (maqām) or experience. For this reason, many concepts of ṣūfī poetry, which we call term, could be a poetic theme by taking part in a reference (talmīḥ), simile (tashbīḥ) and metaphor (majāz). The aim of this article, which we can sample under three titles and with the couplets of our three ṣūfī poets, is to draw attention to the existence of a rich mażmūn staff independent of the dīvān poetry of Turkish ṣūfī poetry. The subject of mażmūns belonging to Turkish ṣūfī poetry should be considered important since it has not been mentioned much until today. Summary: The poet designs and presents it to his readers on behalf of his readers that he cannot even express what he has imagined but sometimes cannot imagine. Each poem has a specific interlocutor and addresses the emotions of its interlocutor. Each poem is adorned with thought and imagination; yet poems can be understood by annotation accompanied by rhetoric. Ṣūfī poetry and dīvān poetry, both poems have universal appeal and classical value like many poems of many nations. Ṣūfī poet and dīvān poet are the architects of an indestructible art existence that ruled for centuries and their works are masterpieces of their own fields. Both types of poetry, along with literary heritage, are a guide ship that leads the whole cultural heritage to the next generations, leading art branches. Both poets of both groups have strengthened from the potential that existed in Turkish society and raised the level of consciousness of each individual separately and created a virtuous society.Dīvān poetry is as beautiful as the aesthetics of genius; ṣūfī poetry is so beautiful with the sincere sincerity of heathat born to the heart. Without making use of the rich mażmūn staff of the lodge poetry, it would be an exclusivism and a status quo to make the mażmūn exclusive only to dīvān poetry or to search for the mażmūns of dīvān poetry in the lodge poetry and to interpret these mażmūns with the interpretation habits of dīvān poetry.The most obvious difference between ṣūfī poetry and dīvān poetry corresponds to the fact that the terms of Sufism are composed of similes (tashbīḥ), metaphors (majāz); sūfī poetry is a composition of poet’s personally experienced feelings or situations such as waḥdat, ḥalvet, manifest and observant concepts. While the depictions of the dīvān poet, spring and winter depictions, arouse enthusiasm for the reader for a while, the indifference of the ṣūfī poet to this world, the constant heartbeat of Jamāl and the heart beating with the excitement of vuslat (meeting) infiltrates the reader to the eternal world.If there is a difference between the two poems, it is that the poetry of the dīvān is static and the ṣūfī poetry is more dynamic than the poetry of the dīvān. As well as dīvān poetry has a unique world of poetry; ṣūfī poetry has a universe of emotions shaped by different meanings and notions. Even if the beauty they depict is materially and morally different, the poet of both groups wishes to go beyond the beauty that they mean and find themselves there. They do this sometimes with similes (tashbīḥ) and metaphors (majāz) and istiʿāre, succeed with various kinds of poetic themes. In dīvān poetry, there is pleasure and imagination based on intelligence, and inspiration and success based on intuition in ṣūfī poetry. These become poetic sometimes thanks to similes (tashbīḥ), metaphors (majāz) and allusions (talmīḥ), become verse and gradually turns into a poetic theme. Dīvān poetry is an endless effort of abstract imagination, while ṣūfī poetry is an inexhaustible enthusiasm for concrete ecstasy. Although both have found what they are looking for outside of daily life, dīvān poet finds pleasure, yet sūfī poet finds inspire.When it comes to mażmūn, the first thing that comes to mind is Divan literature and it’s figure of beautiful woman’s length (cypress), hair (zulf), eye (daffodil), eyebrow (bow), eyelash (arrow) and skin (silver) and these mażmūns are stereotyped and widespread though each of them is known only to artisans. On the other hand, ṣūfī poetry has many other poetic themes such as Jamāl (grace), Jalāl (distress), ār and nāmūs (mālāmat), gūy (rıza), chevgān (ḳażā) and istignā (reluctant).To date, the first thing that comes to mind when it is said to be mażmūn is dīvān poetry, it has left behind the maẓmūns specific to ṣūfī poetry. However, ṣūfī poetry is also quite rich in terms of its mażmūns. Although they are called as just terms, each of the terms of ṣūfī literature is a mystical state (ḥāl), mystical station (maqām) or experience. For this reason, many concepts of ṣūfī poetry, which we call term, could be a poetic theme by taking part in a reference (talmīḥ), simile (tashbīḥ) and metaphor (majāz).The fact that the dīvān poets used the mażmūn as a word and the use of the term started with the Tanzimat made it inevitable that different definitions of the mażmūn emerged in time. For this reason, many researchers now understand and define the term of mażmūn differently. However, all the academics who researched or wrote about the mażmūn brought to mind only the poem of the dīvān poem, and they selected the examples of the mażmūn from the couplets of the poem of the dīvān poem. In our opinion, the missing part of the subject of mażmūn is here. Nevertheless, our main aim in this article is not to discuss what is or is not a mażmūn, but to try to express that the ṣūfī poem is as rich as the dīvān poem. Whereas the mażmūn has no single generally accepted definition; if so, we can say that ṣūfī poetry may have its own unique mażmūn, we can exemplify it with couplets of ṣūfī poetry. (shrink)