Alan Turing and the Mathematical Objection
Gualtiero Piccinini2003, Minds and Machines
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
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.
Related papers
Turing's Approaches to Computability, Mathematical Reasoning and Intelligence
2013
In this paper a distinction is made between Turing's approach to computability, on the one hand, and his approach to mathematical reasoning and intelligence, on the other hand. Unlike Church's approach to computability, which is top-down being based on the axiomatic method, Turing's approach to computability is bottom-up, being based on an analysis of the actions of a human computer. It is argued that, for this reason, Turing's approach to computability is convincing. On the other hand, his approach to mathematical reasoning and intelligence is not equally convincing, because it is based on the assumption that intelligent processes are basically mechanical processes, which however from time to time may require some decision by an external operator, based on intuition. This contrasts with the fact that intelligent processes can be better accounted for in rational terms, specifically, in terms of non-deductive inferences, rather than in term of inscrutable intuition.
On mind & Turing’s machines
Natural Computing, 2007
Turing's notion of human computability is exactly right not only for obtaining a negative solution of Hilbert's Entscheidungsproblem that is conclusive, but also for achieving a precise characterization of formal systems that is needed for the general formulation of the incompleteness theorems. The broad intellectual context reaches back to Leibniz and requires a focus on mechanical procedures; these procedures are to be carried out by human computers without invoking higher cognitive capacities. The question whether there are strictly broader notions of effectiveness has of course been asked for both cognitive and physical processes. I address this question not in any general way, but rather by focusing on aspects of mathematical reasoning that transcend mechanical procedures. Section 1 discusses Go¨del's perspective on mechanical computability as articulated in his [193?], where he drew a dramatic conclusion from the undecidability of certain Diophantine propositions, namely, that mathematicians cannot be replaced by machines. That theme is taken up in the Gibbs Lecture of 1951; Go¨del argues there in greater detail that the human mind infinitely surpasses the powers of any finite machine. An analysis of the argument is presented in Section 2 under the heading Beyond calculation. Section 3 is entitled Beyond discipline and gives Turing's view of intelligent machinery; it is devoted to the seemingly sharp conflict between Go¨del's and Turing's views on mind. Their deeper disagreement really concerns the nature of machines, and I'll end with some brief remarks on (supra-) mechanical devices in Section 4.
A note on Turing's 1936
arXiv:1308.0497v3, 2023
Turing historical article of 1936 is the result of a special endeavor focused around the factuality of a general process for algorithmic computation. The Turing machine as a universal feasibility test for computing procedures is applied up to closely examining what are considered to be the limits of computation itself. In this regard he claims to have defined a number which is not computable, arguing that there can be no machine computing the diagonal on the enumeration of the computable sequences. This article closely examines the original 1936 argument, displaying how it cannot be considered a demontration, and that there is indeed no evidence of such a defined number that is not computable.
Turing's Error-revised
Many important lines of argumentation have been presented during the last decades claiming that machines cannot think like people. Yet, it has been possible to construct devices and information systems, which replace people in tasks which have previously been occupied by people as the tasks require intelligence. The long and versatile discourse over, what machine intelligence is, suggests that there is something unclear in the foundations of the discourse itself. Therefore, we critically studied the foundations of used theory languages. By looking critically some of the main arguments of machine thinking, one can find unifying factors. Most of them are based on the fact that computers cannot perform sense-making selections without human support and supervision. This calls attention to mathematics and computation itself as a representational constructing language and as a theory language in analysing human mentality. It is possible to notice that selections must be based on relevance, i.e., on why some elements of sets belong to one class and others do not. Since there is no mathematical justification to such selection, it is possible to say that relevance and related concepts are beyond the power of expression of mathematics and computation. Consequently, Turing erroneously assumed that mathematics and formal language is equivalent with natural languages. He missed the fact that mathematics cannot express relevance, and for this reason, mathematical representations cannot be complete models of the human mind. Preface This paper is of a programmatic nature. We fully acknowledge the enormous achievements of modern science, engineering and design by calling on the most advanced machine models, as we always did in the past and will continue into the future. We will not primarily discuss the physical limitations (Markov 2014), but instead focus on the conceptual constraints, the intuitive assumptions of underlying theories, resp. their foundations (Saariluoma 1997). We try to understand and find an answer to the fundamental question: Is the human mind capable of understanding itself beyond computability? We question the mainstream assumption that everything is (or at least should be) 'computational' (Chatelin 2012, Chalmers 1996, Sun, Wilson, and Lynch 2016). We argue that the foundations of such an assumption are (still) not fully justifiable. Therefore, we imitate Kant's (1781/1922) famous 'Copernican Revolution' from a kind of Wittgensteinian (1921/1974) perspective and ask whether the properties of the theory of language itself used in discourse can explain why the problems have proven to be so hard. In other words, we ask whether formal theory languages (i.e., logic, mathematics and computation) are powerful enough to express problems of human thinking and represent thoughts. Since many of the foundational issues concentrate on one theoretical construct, Turing machines (TM), we must once again consider whether people 'think like machines'.
Turing and Computationalism
Philosophia: International Journal of Philosophy (2014) 15(1): 50-62
Due to his significant role in the development of computer technology and the discipline of artificial intelligence, Alan Turing has supposedly subscribed to the theory of mind that has been greatly inspired by the power of the said technology which has ev entually become the dominant framework for current researches in artificial intelligence and cognitive science, namely, computationalism or the computational theory of mind. In this essay, I challenge this supposition. In particular, I will try to show tha t there is no evidence in Turing's two seminal works that supports such a supposition. His 1936 paper is all about the notion of computation or computability as it applies to mathematical functions and not to the nature or workings of intelligence. On the other hand, while his 1950 work is about intelligence, it is, however, particularly concerned with the problem of whether intelligence can be attributed to computing machines and not of whether computationality can be attributed to human intelligence or to intelligence in general.
Thinking Like a Machine: Alan Turing, Computation and the Praxeological Foundations of AI
Science & Technology Studies, 2023
As part of ongoing research bridging ethnomethodology and computer science, in this article we offer an alternate reading of Alan Turing's 1936 paper, "On Computable Numbers". Following through Turing's machinic respecification of computation, we hope to contribute to a deflationary position on AI by showing that the activities attributed to AIs are achieved in the course of methodic hands-on work with computational systems and not in isolation by them. Turing's major innovation was a demonstration that mathematical and logical operations could be broken down into elementary, mechanically executable operations, devoid of intellectual content. Drawing out lessons from a re-enactment of Turing's methods as a means of reflecting on contemporary artificial intelligence (AI), including the way those methods disappear into the technology, we will suggest the interesting question raised in "On Computable Numbers" is less about the possibilities of designing machines that "can think" (cf. Turing, 1950), but the practical work we do, and which is made possible, when we ourselves set out to think like machines.
JF 2016 Introduction to Turing 100: Philosophical Explorations of the Legacy of Alan Turing
An overview of essays in this volume, with an emphasis on the philosophical legacy of Turing's work, specifically the ways in which his work bridges not only the gap between the sciences and the humanities, but also foundational and practical aspects of science and everyday life. Three sections of the volume are outlined, framing the overarching structure of Turing's intellectual development: I) Turing on the foundations of mathematics, incompleteness, the the limits of analysis; II) Turing's Universal Machine, implying the ubiquity of computational processes in our world, exemplified by applications in the early history of voice encryption, the history of computer music, the frontiers of computation, and the topic of emergence; III) Turing's work on machines and mind, including his famed "Turing test" as a societal mechanism, the nature of perception as cognition, his views on freedom of the will and the integration of human and machine intelligence, and the developing idea of social algorithms.
Cogito ergo sum non machina! About Gödel’s First Incompleteness Theorem and Turing machines.
2007
Abstract The aim of this paper is to argue about the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm that represent the human capacity of recognizing mathematical truths. More specifically, based on a direct argument from Gödel's First Incompleteness Theorem, we discuss the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm to the human capacity of recognition of first-order arithmetical truths and so of mathematical truths in general.
Turing-, Human- and Physical Computability: An Unasked Question
Minds and Machines, 2008
Godel on Turing on Computability
Church's Thesis after 70 years, 2006
Toward assessing some of the arguments of "AI's first philosopher," Alan Turing
Toward assessing some of the arguments of "AI's first philosopher," Alan Turing Patrick S. O'Donnell (2022) "Alan Turing did not fit easily with any of the intellectual movements of his time, aesthetic, technocratic or marxist. In the 1950s, commentators struggled to find discreet words to categorise him: as 'a scientific Shelley,' as possessing great 'moral integrity.' Until the 1970s, the reality of his life was unmentionable. He is still hard to place within twentieth-century thought. He exalted the science that according to existentialists had robbed life of meaning. The most original figure, the most insistent on personal freedom, he held originality and will to be susceptible to mechanisation. The mind of Alan Turing continues to be an enigma. But it is an enigma to which the twenty-first century seems increasingly drawn. The year of his centenary, 2012, witnessed numerous conferences, publications, and cultural events in his honor. Some reasons for this explosion of interest are obvious. One is that the question of the Turkle, Sherry. Alone Together: Why We Expect More from Technology and Less from Each Other (Basic Books, 2011).
The Genius of Alan Turing: The Computing Classical Model
International Journal of Machine Learning and Computing, 2013
This paper aims to examine the basis of Calculus and computus from first philosophical principles, having a focus on the internal representations and acts of spontaneity, proper of genius that the concept of creativity is affiliate with. Our guiding author is Alan Turing and we will enquire closely the computing classical model. The paper explores the traditions of computing and philosophy, theorizing about the question of bio-machine hybrids in relation with imagination, the form of representation most free from nature. The first section is called calculus et computus. It examines the developments associated with the notions of algorithm, function and rule. In the second section the faculty of imagining is addressed through the abbreviated table, hoping to identify the boundaries both theoretical and practical of the computing classical model, following the seminal paper on computable numbers with Application to the Entscheidungs Problem (1936). We show how much hybridization of ideas fostered by both traditions was to find a place in the imaginary of artificial intelligence. Flanked by intuitions and concepts, imagination, the synthesis of reproduction, is capable of discerning about cosmos through bios and computus, so powerfully as if it sketched ideas in images, as the Turing machine clearly exemplifies.
Alan Turing’s “Computing Machinery and Intelligence”. Anachronistic review.
TOPOI, 32 (2):293-299 (2013)
It is the anachronistic review of the classical paper of Turing (as published this year), for discussing current issues and contradictions in AI and Cognitive Science.
Turing versus Gödel on Computability and the Mind (with B.J. Copeland)
Computability: Turing, Gödel, Church, and Beyond (eds. B. Jack Copeland, Carl Posy, Oron Shagrir), MIT Press, 2013
Nowadays the work of Alan Turing and Kurt Gödel rightly takes center stage in discussions of the relationships between computability and the mind. After decades of comparative neglect, Turing's 1936 paper "On Computable Numbers" is now regarded as the foundation stone of computability theory, and it is the fons et origo of the concept of computability employed in modern theoretical computer science. Moreover, Turing's 1950 essay, "Computing Machinery and Intelligence," sparked a rich literature on the mind-machine issue. Gödel's 1931
An Interpretation of the Church-Turing Thesis
The Church-Turing thesis is given a provable interpretation based on the idea that a computation by an idealized human agent must be a logically definable finite mathematical object. The argument is preserved under a large variation in the expressive power of the underlying logical language, thus providing a possible explanation of why the notion of effective computability is so robust.
Turing versus Godel on computability and the mind
2013
This chapter contains sections titled: 1.1 Godel on Turing's “Philosophical Error”, 1.2 Two Approaches to the Analysis of Computability, 1.3 Godel and Turing on the Mind, 1.4 Conclusion, Acknowledgments, Notes, References
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.