Translator's summary The translated paper is an extract, published in 1945, of an unpublished thesis, of both historical and technical import, dealing with notions of definability and their relation to invariance under automorphisms. The author develops a metamathematical Galois theory, and discusses and anticipates some aspects of higher-order model theory in an informal but conceptually rich manner.

We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful (...) historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be "Computability Theory" or simply Computability rather than "Recursive Function Theory.". (shrink)

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

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)

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)

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)

§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions ; and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his (...) thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a. (shrink)

This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for (...) Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical
objection” to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it
should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human
mathematicians presumably do. (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-umwelts bridging artificial intelligence with biology in a novel way. (shrink)

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

Should you be targeted by police for a crime that AI predicts you will commit? In this paper, we analyse when, and to what extent, the person-based predictive policing (PP) — using AI technology to identify and handle individuals who are likely to breach the law — could be justifiably employed. We first examine PP’s epistemological limits, and then argue that these defects by no means refrain from its usage; they are worse in humans. Next, based on major AI ethics (...) guidelines (IEEE, EU, and RIKEN, etc.), we refine three basic moral principles specific to person-based PP. We also derive further requirements from case studies, including debates in Chicago, New Orleans, San Francisco, Tokyo, and cities in China. Instead of rejecting PP programs, we analyse what necessary conditions should be met for using the tool to achieve social good. While acknowledging its risks, we conclude that the person-based PP could be beneficial in community policing, especially when merging into a larger governance framework of the social safety net. (shrink)

This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...) paper is in three sections. The first is written in journalistic style:the story of the life of AlonzoChurch is told, including some of the many anecdotes I have collected from different sources. The secondpart is devoted to his work, but is far from being exhaustive. The last part is more original; in it I attempto show that Church’s great discovery was lambda calculus and that his remaining contributions weremainly inspired afterthoughts in the sense that most of his contributions as well as some of his pupils derivefrom that initial achievement. Included are Kleene’s Recursion Theory and the completeness proof ofHenkin. I have added an appendix in which is presented the typed lambda calculus and a proof of theundecidability of first-order logic. (shrink)

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

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)

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations—whatever they are—must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise (...) to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an “assembly language” for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems. (shrink)

How can we harness nature’s power for computations? Our society comprises a collection of individuals, each of whom handles decision-making tasks that are abstracted as computational problems of finding the most profitable option from a set of options that stochastically provide rewards. Society is expected to maximize the total rewards, while the individuals compete for common rewards. Such collective decision making is formulated as the “competitive multi-armed bandit problem.” Herein, we demonstrate an analog computing device that uses numerous fluids in (...) coupled cylinders to efficiently solve CBP for the maximization of social rewards, without paying the conventionally-required huge computational cost. The fluids estimate the reward probabilities of the options for the exploitation of past knowledge, and generate random fluctuations for the exploration of new knowledge for which the utilization of the fluid-derived fluctuations is more advantageous than applying artificial fluctuations. The fluid-derived fluctuations, which require exponentially-many combinatorial efforts when they are emulated using conventional digital computers, would exhibit their maximal computational power when tackling classes of problems that are more complex than CBP. Extending the current configuration of the device would trigger further studies related to harnessing the huge computational power of natural phenomena to solve a wide variety of complex societal problems. (shrink)

We investigate Turing's contributions to computability theory for real numbers and real functions presented in [22, 24, 26]. In particular, it is shown how two fundamental approaches to computable analysis, the so-called ‘Type-2 Theory of Effectivity' (TTE) and the ‘realRAM machine' model, have their foundations in Turing's work, in spite of the two incompatible notions of computability they involve. It is also shown, by contrast, how the modern conceptual tools provided by these two paradigms allow a systematic interpretation of Turing's (...) pioneering work in the subject. (shrink)

This paper considers the ontological significance of invisibility in relation to the question ‘what is a digital image?’ Its argument in a nutshell is that the emphasis on visibility comes at the expense of latency and is symptomatic of the style of thinking that dominated Western philosophy since Plato. This privileging of visible content necessarily binds images to linguistic paradigms of interpretation which promote representation, subjectivity, identity and negation over multiplicity, indeterminacy and affect. Photography is the case in point because (...) until recently critical approaches to photography had one thing in common: they all shared in the implicit and incontrovertible understanding that photographs are a medium that must be approached visually; they took it as a given that photographs are there to be looked at and they all agreed that it is only through the practices of spectatorship that the secrets of the image can be unlocked. Whatever subsequent interpretations followed, the priori- ty of vision in relation to the image remained unperturbed. This undisputed belief in the visibility of the image has such a strong grasp on theory that it imperceptibly bonded together otherwise dissimilar and sometimes contradictory methodol- ogies, preventing them from noticing that which is the most unexplained about images: the precedence of looking itself. This self-evident truth of visibility casts a long shadow on im- age theory because it blocks the possibility of inquiring after everything that is invisible, latent and hidden. (shrink)

It is sometimes suggested that the history of computation in cognitive science is one in which the formal apparatus of Turing-equivalent computation, or effective computability, was exported from mathematical logic to ever wider areas of cognitive science and its environs. This paper, however, indicates some respects in which this suggestion is inaccurate. Computability theory has not been focused exclusively on Turing-equivalent computation. Many essential features of Turing-equivalent computation are not captured in definitions of computation as symbol manipulation. Turing-equivalent computation did (...) not play the role in McCulloch and Pitts’s early cybernetic work that is sometimes attributed to it. Finally, various segments of the neuroscientific community invoke a notion of computation that differs from the Turing-equivalent notion.Keywords: Circular causality; Computation; Cortical maps; Neural networks; Symbol manipulation; Turing-equivalent computation. (shrink)

The aim of this work is to apply information theoretic ideas to the notion of fundamentality. I will argue that if one adopts pancomputationalism as a metaphysics for the universe, then there are higher-level structures which are just as fundamental for computation as anything from microphysics.