Machine learning, and more broadly artificial intelligence (AI), is a fascinating technology and can be considered as the closest approximation to the Cartesian “thinking thing” that humans have ever created. Just as the industrial revolution required a new ethos, the age of intelligent machines will create its own, challenging the established moral, economic, and political presuppositions. This paper discusses the relationship between AI and society; it presents several thought experiments to explore the complexity of the relationship and highlights the insufficiency (...) of the current normative paradigm in addressing technological expansion. I argue that many of the externalities, such as security risks, loss of privacy, and economic instability will result from trying to fit the emerging technologies into the existing frame of efficiency and utility, by redefining the notions of human value, identity, autonomy, purity, and truth, among others. The age of the intelligent machine is elevating alienation to new levels, treating the individual as mere patterns in data—its primary commodity. I further argue that while the possibility of unintended consequences, due to the potential misuse of AI is ever present, the intelligent machine per se is unlikely to engage in a zero-sum game for power on its own initiative. I question whether singularity is at all attainable and argue that technology will forever remain a proxy for human interests. I conclude by posing questions for charting the path forward. Through this analysis, I aim to provide a more nuanced understanding of the complex relationship between the AI and humans. (shrink)

The word algorithm has become the default descriptor for anything vaguely computational to the extent that it appears synonymous with computing itself. It functions in this respect as the master signifier under which a spectrum of sense is subsumed, less a well-defined and stable expression than the vehicle through which innumerable concerns are projected. Commenting on this nebulous quality, Massimo Mazzotti has dubbed the term “a site of semantic confusion.” Yet, rather than “engaging in a taxonomic exercise to norm the (...) usage of the word,” Mazzotti proposes that a more generative approach would “consider its flexible, ill-defined, and often inconsistent meanings as a resource: a messy map of our increasingly algorithmic life”—that is, he attempts to take “the omnipresent figure of the algorithm as an object that refracts collective expectations and anxieties.” Similarly, this study has little fondness for semantic discipline. Unlike Mazzotti, however, my focus here is primarily historical: How did the algorithm come to be such an “omnipresent figure”? What was at stake in aligning the computational with the algorithmic? (shrink)

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

Since the beginning of the twenty-first century there has been an increasing awareness that software rep- resents a blind spot in new media theory. The growing interest in software also influences the argument in this paper, which sets out from the assumption that Alan M. Turing's concept of the universal machine, the first theoretical description of a computer program, is a kind of bachelor machine. Previous writings based on a similar hypothesis have focused either on a comparison of the universal (...) machine and the bachelor machine in terms of the similarities of their structural features, or they have taken the bachelor machine as a metaphor for a man or a computer. Unlike them, this paper stresses the importance of the con- text as a key to interpreting the universal Turing machine as a bachelor machine and, potentially, as a self-portrait. (shrink)

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

Modern computing is generally taken to consist primarily of symbol manipulation. But symbols are abstract, and computers are physical. How can a physical device manipulate abstract symbols? Neither Church nor Turing considered this question. My answer is that the bit, as a hardware-implemented abstract data type, serves as a bridge between materiality and abstraction. Computing also relies on three other primitive—but more straightforward—abstractions: Sequentiality, State, and Transition. These physically-implemented abstractions define the borderline between hardware and software and between physicality and (...) abstraction. At a deeper level, asking how a physical device can interact with abstract symbols is the wrong question. The relationship between symbols and physical devices begins with the realization that human beings already know what it means to manipulate symbols. We build and program computers to do what we understand to be symbol manipulation. To understand what that means, consider a light switch. A light switch doesn’t turn a light on or off. Those are abstractions. Light switches don’t operate with abstractions. We build light switches, so that when flipped, the world is changed in such a way that we understand the light to be on or off. Similarly, we build computers to perform operations that we understand as manipulating symbols. (shrink)

In this paper, I argue that computationalism is a progressive research tradition. Its metaphysical assumptions are that nervous systems are computational, and that information processing is necessary for cognition to occur. First, the primary reasons why information processing should explain cognition are reviewed. Then I argue that early formulations of these reasons are outdated. However, by relying on the mechanistic account of physical computation, they can be recast in a compelling way. Next, I contrast two computational models of working memory (...) to show how modeling has progressed over the years. The methodological assumptions of new modeling work are best understood in the mechanistic framework, which is evidenced by the way in which models are empirically validated. Moreover, the methodological and theoretical progress in computational neuroscience vindicates the new mechanistic approach to explanation, which, at the same time, justifies the best practices of computational modeling. Overall, computational modeling is deservedly successful in cognitive science. Its successes are related to deep conceptual connections between cognition and computation. Computationalism is not only here to stay, it becomes stronger every year. (shrink)

The notion of computation has changed the world more than any previous expressions of knowledge. However, as know-how in its particular algorithmic embodiment, computation is closed to meaning. Therefore, computer-based data processing can only mimic life’s creative aspects, without being creative itself. AI’s current record of accomplishments shows that it automates tasks associated with intelligence, without being intelligent itself. Mistaking the abstract for the concrete has led to the religion of “everything is an output of computation”—even the humankind that conceived (...) the computer. The hypostatized role of computers explains the increased dependence on them. The convergence machine called deep learning is only the most recent form through which the deterministic theology of the machine claims more than what it actually is: extremely effective data processing. A proper understanding of complexity, as well as the need to distinguish between the reactive nature of the artificial and the anticipatory nature of the living are suggested as practical responses to the challenges posed by machine theology. (shrink)

In this paper, I review the objections against the claim that brains are computers, or, to be precise, information-processing mechanisms. By showing that practically all the popular objections are either based on uncharitable interpretation of the claim, or simply wrong, I argue that the claim is likely to be true, relevant to contemporary cognitive (neuro)science, and non-trivial.

Traditionally, computational theory (CT) and dynamical systems theory (DST) have presented themselves as opposed and incompatible paradigms in cognitive science. There have been some efforts to reconcile these paradigms, mainly, by assimilating DST to CT at the expenses of its anti-representationalist commitments. In this paper, building on Piccinini’s mechanistic account of computation and the notion of functional closure, we explore an alternative conciliatory strategy. We try to assimilate CT to DST by dropping its representationalist commitments, and by inviting CT to (...) recognize the functionally closed nature of some computational systems. (shrink)

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more (...) mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

This article presents three extracts from the introductory course in mathematical logic that Gödel gave at the University of Notre Dame in 1939. The lectures include a few digressions, which give insight into Gödel's views on logic prior to his philosophical papers of the 1940s. The first extract is Gödel's first lecture. It gives the flavour of Gödel's leisurely style in this course. It also includes a curious definition of logic and a discussion of implication in logic and natural language. (...) The second extract is a discussion on undecidability and on Leibniz. The third extract concerns the paradoxes and Russell's theory of types. (shrink)

Nykyaikaisen logiikan keskeisenä tutkimuskohteena ovat erilaiset formalisoidut teoriat. Erityisesti vuosisadan vaihteen aikoihin matematiikan perusteiden tutkimuksessa ilmaantuneiden hämmentävien paradoksien (Russell 1902, 1903) jälkeen (ks. kuitenkin jo Frege 1879, Dedekind 1888, Peano 1889; vrt. Wang 1957) keskeiset matemaattiset teoriat on pyritty tällaisten vaikeuksien välttämiseksi uudelleen muotoilemaan täsmällisesti keinotekoisessa symbolikielessä, jonka lauseenmuodostussäännöt on täsmällisesti ja yksikäsitteisesti määrätty. Edelleen teoriat on pyritty aksiomatisoimaan, ts. on pyritty antamaan joukko peruslauseita, joista kaikki muut - tai ainakin mahdollisimman monet - teorian todet lauseet voidaan loogisesti johtaa tarkoin (...) määrättyjen päättelysääntöjen mukaisesti (erityisesti Hilbert 1904, 1918, 1927; ks. myös Russell 1908, Zermelo 1908; vrt. Kleene 1952, §§ 14−15). (shrink)

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)

Arguments to the effect that Church's thesis is intrinsically unprovable because proof cannot relate an informal, intuitive concept to a mathematically defined one are unconvincing, since other 'theses' of this kind have indeed been proved, and Church's thesis has been proved in one direction. However, though evidence for the truth of the thesis in the other direction is overwhelming, it does not yet amount to proof.

Informatics is generally understood as a “new technology” and is therewith discussed according to technological aspects such as speed, data retrieval, information control and so on. Its widespread use from home appliances to enterprises and universities is not the result of a clear-cut analysis of its inner possibilities but is rather dependent on all sorts of ideological promises of unlimited progress. We will discuss the theoretical definition of informatics proposed in 1936 by Alan Turing in order to show that it (...) should be taken as final and complete. This definition has no relation to the technology because Turing defines computers as doing the work of solving problems with numbers. This formal definition implies nonetheless a relation to the non-formalized elements around informatics, which we shall discuss through the Greek notion of téchne. (shrink)

This paper revisits the often debated question Can machines think? It is argued that the usual identification of machines with the notion of algorithm has been both counter-intuitive and counter-productive. This is based on the fact that the notion of algorithm just requires an algorithm to contain a finite but arbitrary number of rules. It is argued that intuitively people tend to think of an algorithm to have a rather limited number of rules. The paper will further propose a modification (...) of the above mentioned explication of the notion of machines by quantifying the length of an algorithm. Based on that it appears possible to reconcile the opposing views on the topic, which people have been arguing about for more than half a century. (shrink)

Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses (...) of action(- types) from the particular courses of action(-tokens) that actually instantiate them and the causal processes and/or interpretations that ultimately make them effective. On my analysis, effectiveness is not just a matter of logical form; `content' matters. The analysis I provide has the advantage of applying to ordinary, everyday procedures such as recipes and methods, as well as the more refined procedures of mathematics and computer science. It also has the virtue of making better sense of the physical possibilities for hypercomputation than the received view and its extensions, e.g. Turing's o-machines, accelerating machines. (shrink)

There are two main ways to teach a course online: synchronously or asynchronously. In an asynchronous course, students can log on at their convenience and do the course work. In a synchronous course, there is a requirement that all students be online at specific times, to allow for a shared course environment. In this article, the author discusses the strengths and weaknesses of synchronous online learning for the teaching of undergraduate philosophy courses. The author discusses specific strategies and technologies he (...) uses in the teaching of online philosophy courses. In particular, the author discusses how he uses videoconferencing to create a classroom-like environment in an online class. (shrink)

Computational complexity theory is a branch of computer science dedicated to classifying computational problems in terms of their difficulty. While computability theory tells us what we can compute in principle, complexity theory informs us regarding our practical limits. In this chapter I argue that the science of \emph{quantum computing} illuminates complexity theory by emphasising that its fundamental concepts are not model-independent, but that this does not, as some suggest, force us to radically revise the foundations of the theory. For model-independence (...) never has been essential to those foundations. The fundamental aim of complexity theory is to describe what is achievable in practice under various models of computation for our various practical purposes. Reflecting on quantum computing illuminates complexity theory by reminding us of this, too often under-emphasised, fact. (shrink)

Gottfried Wilhelm Leibniz is the self-proclaimed inventor of the binary system and is considered as such by most historians of mathematics and/or mathematicians. Really though, we owe the groundwork of today’s computing not to Leibniz but to the Englishman Thomas Harriot and the Spaniard Juan Caramuel de Lobkowitz, whom Leibniz plagiarized. This plagiarism has been identified on the basis of several facts: Caramuel’s work on the binary system is earlier than Leibniz’s, Leibniz was acquainted—both directly and indirectly—with Caramuel’s work and (...) Leibniz had a natural tendency to plagiarize scientific works. (shrink)

Since the cognitive revolution, it has become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this paper, we distinguish information processing from computation and examine some of their mutual relations, shedding light on the role each can play in a theory of cognition. We recommend that theorists of cognition be explicit and careful (...) in choosing notions of computation and information and connecting them together.Keywords: Computation; Information processing; Computationalism; Computational theory of mind; Cognitivism. (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)

I.It is probably no controversial claim to state that there has been a major change in the communicative landscape during the last 10 to 20 years, due to technological innovations that have created utterly new types of digital communicative media. In the following, I apply an analysis, rooted in Kant's analysis of the antinomies of reason in Critique of Pure Reason, by which I argue that we can see a dogmatic strain in the digital media. Kant arrives at the discussion (...) of the antinomies in book two, “The Dialectical Inferences of Pure Reason.”1 The aim of the transcendental dialectic…. (shrink)

In this thesis I present a virtual laboratory which implements five different models for controlling animats: a rule-based system, a behaviour-based system, a concept-based system, a neural network, and a Braitenberg architecture. Through different experiments, I compare the performance of the models and conclude that there is no best model, since different models are better for different things in different contexts. The models I chose, although quite simple, represent different approaches for studying cognition. Using the results as an empirical philosophical (...) aid, I note that there is no best approach for studying cognition, since different approaches have all advantages and disadvantages, because they study different aspects of cognition from different contexts. This has implications for current debates on proper approaches for cognition: all approaches are a bit proper, but none will be proper enough. I draw remarks on the notion of cognition abstracting from all the approaches used to study it, and propose a simple classification for different types of cognition. (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)

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of (...) basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. (shrink)

Wittgenstein’s “machines-as-symbols” are considered with respect to their historical sources and their symbolic and logical nature. Among these sources and precursors, along with Leonardo’s drawings of machines, there are illustrated “machine books”, a kind of book published in the period from the 16th to the 18th centuries which consist of pictures and descriptions of a variety of mechanical devices. Most probably, these books were one of Wittgenstein’s inspirations for his view of machines as components of language-games. The picture of homo (...) volans in Vrančić’s machine book possessed by Wittgenstein is taken as an example. In particular, homo volans is shown to contain patterns of logical laws and rules and to be abstractly interpretable as a logical symbol. A machine, taken as a symbol, is shown to be a precondition of a meaningful “overview” of a mechanical work that exceeds the limits of decidability, and to possess causal features if causality is understood teleologically and in a deeper sense of a “binding” life. (shrink)

Kripke’s argument against functionalism extended to physical computers poses a deep philosophical problem for understanding the standard view of what computers are. The problem puts into jeopardy the definition in the standard view that computers are physical machines for performing physical computations. Indeed, it is entirely possible that, unless this philosophical problem is resolved, we will never have a good understanding of computers and may never know just what they are.

In the centenary year of Turing’s birth, a lot of good things are sure to be written about him. But it is hard to find something new to write about Turing. This is the biggest merit of this article: it shows how von Neumann’s architecture of the modern computer is a serendipitous consequence of the universal Turing machine, built to solve a logical problem.

Toisaalta ennennäkemätön äärettömien joukko-opillisten menetelmien hyödyntäminen sekä toisaalta epäilyt niiden hyväksyttävyydestä ja halu oikeuttaa niiden käyttö ovat ratkaisevasti muovanneet vuosisatamme matematiikkaa ja logiikkaa. Tämän kehityksen vaikutus nykyajan filosofiaan on myös ollut valtaisa; merkittävää osaa siitä ei voi edes ymmärtää tuntematta sen yhteyttä tähän matematiikan ja logiikan vallankumoukseen. Lähestymistapoja, jotka tavalla tai toisella hyväksyvät äärettömän matematiikan ja perinteisten logiikan sääntöjen (erityisesti kolmannen poissuljetun lain) soveltamisen myös sen piirissä, on tullut tavaksi kutsua klassiseksi matematiikaksi ja logiikaksi erotuksena nämä hylkäävistä radikaaleista intuitionistisista ja (...) konstruktivistisista opeista. (shrink)

Church's thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing's work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church's thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing's work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic (...) entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular. (shrink)

Over the past decades, due to the course towards digitalization of all areas of life, interest in modeling and creating intelligent systems has increased significantly. However, there are now a stagnation in the industry, a lack of attention to analog and bionic approaches as alternatives to digital, numerous speculations on “neuro” issues for commercial and other purposes, and an increase in social and environmental risks. The article provides an overview of the development of artificial intelligence (AI) conceptions toward increasing the (...) human likeness of machines: from the key ideas of A. Turing and J. von Neumann, who initiated the digitalization of society, to discussions about the definition of AI and the emergence of conceptions of strong and weak AI. Special attention is paid to the approach of A. Sloman, to ideas about the architecture and design of complex artificial systems are considered, which make it possible to “emotionally” expand the idea of weak/strong AI. In the article's section on the necessity and possibility of incorporating emotions into the architecture of AI, the authors reveal the goals and methodological limitations for creating an emotional artificial agent. In addition, the article briefly presents the main principles of the authors' architectural approach to the creation of emotional intellectual systems on the example of the cognitive-affective model of architecture, which allow modeling the impact of emotions on the cognitive processes involved in decision-making processes. The described architectural approach to modeling intelligent systems can be used as a conceptual basis for discussing and formulating a strategy for the development of neurocomputing, philosophy of artificial intelligence, and experimental philosophy, for developing innovative research programs, formulating and solving theoretical and methodological problems. (shrink)

This essay discusses Peirce’s appeal to logical machines as an argument against psychologism. It also contends that some of Peirce’s anti-psychologistic remarks on logic contain interesting premonitions arising from his perception of the asymmetry of proof complexity in monadic and relational logical calculi that were only given full formulation and explication in the early twentieth century through Church’s Theorem and Hilbert’s broad-ranging Entscheidungsproblem.

In the beginning was the word, or grunt, or groan, or signal of some sort. This, however, hardly qualifies as an information revolution, at least in any standard technological sense. Nature is replete with meaningful signs, and we must imagine that our early ancestors noticed natural patterns that helped to determine when to sow and when to reap, which animal tracks to follow, what to eat, and so forth. Spoken words at first must have been meaningful in some similar sense. (...) But in time the word became flesh (corpus) and dwelt among us, as "inscription" (literally, to put into writing) inaugurated the dawn of human history. This did not happen instantly. One place to enter the story is with clay tokens to represent trade transactions that in time became accounting tablets and, then, the world's first literature (Enmerkar and the Lord of Aratta, The Epic of Gilgamesh, etc.) and codes of law (The Codes of Ur-Nammu, Lipit-Ishtar, Hammurabi, and so forth.) This event happened around the north shore of the Persian Gulf sometime in the 4th millennium BCE and was enshrouded in mystery as the role of the scribe trained in the art of inscribing and deciphering signs belonged to the priest (Deibert 1997). With the sanction of religion, writing gave birth to "civility" (literally, life in the city) and defined the line between "history" and "pre-history," the latter being a term designating everything that happened before. There is little doubt that the invention of writing was significant and that it deserves recognition as the first revolution in the history of information. Life as we live it today would have been impossible otherwise. Innovations in writing technologies happened with significant effects, but at various points in the history of information, changes in technology were so dramatic that they reshaped the course of human history in radical ways. The revolution in printing is well-studied; the invention of the printing press and movable type (c.. (shrink)

The foundations of AI design discourse are worth analyzing. Here, attention is paid to the nature of theory languages used in designing new AI technologies because the limits of these languages can clarify some fundamental questions in the development of AI. We discuss three types of theory language used in designing AI products: formal, computational, and natural. Formal languages, such as mathematics, logic, and programming languages, have fixed meanings and no actual-world semantics. They are context- and practically content-free. Computational languages (...) use terms referring to the actual world, i.e., to entities, events, and thoughts. Thus, computational languages have actual-world references and semantics. They are thus no longer context- or content-free. However, computational languages always have fixed meanings and, for this reason, limited domains of reference. Finally, unlike formal and computational languages, natural languages are creative, dynamic, and productive. Consequently, they can refer to an unlimited number of objects and their attributes in an unlimited number of domains. The differences between the three theory languages enable us to reflect on the traditional problems of strong and weak AI. (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)

Turing was an exceptional mathematician with a peculiar and fascinating personality and yet he remains largely unknown. In fact, he might be considered the father of the von Neumann architecture computer and the pioneer of Artificial Intelligence. And all thanks to his machines; both those that Church called “Turing machines” and the a-, c-, o-, unorganized- and p-machines, which gave rise to evolutionary computations and genetic programming as well as connectionism and learning. This paper looks at all of these and (...) at why he is such an often overlooked and misunderstood figure. (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)

This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...) remarks on _Lebensformen_ is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the _Entscheidungsproblem_“, as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of _Lebensform_ in the PI is then given in terms of a logical “regression” to _Lebensform_ as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?”. (shrink)

There are simple, purely syntactic axiomatic proof systems for both the logical truths and the logical falsehoods of propositional logic. However, to date no such system has been developed for the logical contingencies, that is, formulas that are both satisfiable and falsifiable. This paper formalizes the purely syntactic axiomatic proof systems for the logical contingencies and proves its soundness as well as completeness.

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

A critical aspect of much human behaviour is the generation and regulation of sequential activities. Such behaviour is seen in both naturalistic settings such as routine action and language production and laboratory tasks such as serial recall and many reaction time experiments. There are a variety of computational mechanisms that may support the generation and regulation of sequential behaviours, ranging from those underlying Turing machines to those employed by recurrent connectionist networks. This paper surveys a range of such mechanisms, together (...) with a range of empirical phenomena related to human sequential behaviour. It is argued that the empirical phenomena pose difficulties for most sequencing mechanisms, but that converging evidence from behavioural flexibility, error data arising from when the system is stressed or when it is damaged following brain injury, and between-trial effects in reaction time tasks, point to a hybrid symbolic activation-based mechanism for the generation and regulation of sequential behaviour. Some implications of this view for the nature of mental computation are highlighted. (shrink)

A. M. Turing has bequeathed us a conceptulary including 'Turing, or Turing-Church, thesis', 'Turing machine', 'universal Turing machine', 'Turing test' and 'Turing structures', plus other unnamed achievements. These include a proof that any formal language adequate to express arithmetic contains undecidable formulas, as well as achievements in computer science, artificial intelligence, mathematics, biology, and cognitive science. Here it is argued that these achievements hang together and have prospered well in the 50 years since Turing's death.

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an n such (...) that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)