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I revisit my 1993 paper on Putnam and mathematical realism focusing on the indispensability argument and how it has fared over the years. This argument starts from the claim that mathematics is an indispensable part of science and draws the conclusion, from holistic considerations about confirmation, that the ontology of science includes abstract objects as well as the physical entities science deals with. [ABSTRACT FROM AUTHOR]

In contrast to dominant approaches to human reason involving essentially a logical and instrumental conception of rationality easily modeled by artificial intelligence mechanisms, I argue that the specific capacities of the human mind are meta-analytic in nature, understood as irreducible to the analytic or the logical, or else the computational. Firstly, the assumption of a meta-analytical level of rationality is derived from key insights developed in various branches of the social sciences. This meta-analytical level is then inferred from Gödel's incompleteness theorem. On these bases, and with the help of psychological approaches to consciousness, I argue that human rationality may be characterized as the capacity to make meaningful use of signs. [ABSTRACT FROM AUTHOR]

1936 was a watershed year for computability. Debates among Gödel, Church and others over the correct analysis of the intuitive concept "human effectively computable", an analysis at the heart of the Incompleteness Theorems, the Entscheidungsproblem, the question of what a finite computation is, and most urgently—for Gödel—the generality of the Incompleteness Theorems, were definitively set to rest with the appearance, in that year, of the Turing Machine. The question I explore here is, do the mathematical facts exhaust what is to be said about the thinking behind the "confluence of ideas in 1936"? I will argue for a cultural role in Gödel's, and, by extension, the larger logical community's absorption of Turing's 1936 model. As scaffolding I employ a conceptual framework due to the critic Leo Marx of the technological sublime; I also make use of the distinction within the technological sublime due to Caroline Jones, between its iconic and performative modes—a distinction operating within the conceptual art of the 1960s, but serving the history of computability equally well. [ABSTRACT FROM AUTHOR]

Alexander Pruss has given a quick argument against the claim that consistency is possibility using Gödel's second incompleteness theorem. The argument does not distinguish metalanguage claims of consistency from object-language ones, rendering it unsound. [ABSTRACT FROM AUTHOR]

The paper presents an interpretation of Platonism, the seeds of which can be found in the writings of Gödel and Wittgenstein. Although it is widely accepted that Wittgenstein is an anti-Platonist the author points to some striking affinities between Gödel’s and Wittgenstein’s accounts of mathematical concepts and the role of feeling and intuition in mathematics. A version of Platonism emerging from these considerations combines realism with respect to concepts with a view of concepts as accessible to feeling and able to guide our behavior through feeling.

Despite his unreserved appreciation of Turing's analysis for being a "precise and unquestionably adequate definition" of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a "philosophical error" in Turing's argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing's work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing's response to the "mathematical objection" based on Gödel's incompleteness theorem and Gödel's own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no "error" on Turing's side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together. [ABSTRACT FROM AUTHOR]

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Recent discussions of how axioms are extrinsically justified have appealed to abductive considerations: on such accounts, axioms are adopted on the basis that they constitute the best explanation of some mathematical data, or phenomena. In the first part of this paper, I set out a potential problem caused by the appeal made to the notion of mathematical explanation and suggest that it can be remedied once it is noted that all the justificatory work is done by appeal to the theoretical virtues. In the second part of the paper, I appeal to the theoretical virtues account of axiom justification to provide an argument that judgements of theoretical virtuousness, and therefore of extrinsic justification, are subjective in a substantive sense. This tells against a recent claim by Penelope Maddy that such justification is "wholly objective". [ABSTRACT FROM AUTHOR]

1936 was a watershed year for computability. Debates among Gödel, Church and others over the correct analysis of the intuitive concept “human effectively computable”, an analysis at the heart of the Incompleteness Theorems, the Entscheidungsproblem, the question of what a finite computation is, and most urgently—for Gödel—the generality of the Incompleteness Theorems, were definitively set to rest with the appearance, in that year, of the Turing Machine. The question I explore here is, do the mathematical facts exhaust what is to be said about the thinking behind the “confluence of ideas in 1936”? I will argue for a cultural role in Gödel’s, and, by extension, the larger logical community’s absorption of Turing’s 1936 model. As scaffolding I employ a conceptual framework due to the critic Leo Marx of the technological sublime; I also make use of the distinction within the technological sublime due to Caroline Jones, between its iconic and performative modes—a distinction operating within the conceptual art of the 1960s, but serving the history of computability equally well.

This article considers Gödel's remarks on Plato's dialogue Euthyphro, which he made in conversation with the proof theorist Sue Toledo in the years 1972–1975. [ABSTRACT FROM AUTHOR]

This paper examines Nick Land’s “numbering practices” for opening up language to modernity’s increasing technological entanglement beyond human comprehension. I begin by examining Land’s attempt to override our linguistic systems with machinic code and binary symbolism in a way which mirrors modernity’s technological future shock. I then consider his appropriation of Cantor’s set theory and Gödel’s incompleteness theorem to plot ever greater degrees of reality’s excess to all anthropic logic. The third section looks at Land’s use of qabbalistic numerology to uncover the absolute contingency of all our most strongly held beliefs, truths and values. The fourth section considers Land’s radicalisation of qabbalism through his own particularly abstract and inhuman notational “gematria.” I conclude by looking at Land’s interest in the computer keyboard’s lock-in to the QWERTY layout for proffering a glimpse of modern technology’s increasingly dehumanising meltdown of our anthropocentric delusions of grandeur.

Mathematical intuition is a widely discussed topic in the philosophy of science. This article studies the idea that Kurt Gödel has about it, one of the greatest mathematicians of all time. When his works are reviewed, it is inevitable to find the use of the word "intuition" in many of his demonstrations, and in some cases it is not clear what we should understand by this notion. Sometimes he uses it to indicate that it is a faculty similar to the perception of physics. On other occasions, it assumes that it is in charge of validating the axioms by providing them with evidence, which implies seeing it as a psychological fact. But it is curious, for example, that it does not appeal to her when it comes to creativity. [ABSTRACT FROM AUTHOR]

It is widely alleged that metaphysical possibility is "absolute" possibility (Kripke in Naming and necessity, Harvard University Press, Cambridge, 1980; Lewis in On the plurality of worlds, Blackwell, Oxford, 1986; van Inwagen in Philos Stud 92:68–84, 1997; Rosen, in: Gendler and Hawthorne (eds) Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker (ed) Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215; Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity "necessity in the highest degree" (1980, p. 99). Van Inwagen claims that if P is metaphysically possible, then it is possible "tout court. Possible simpliciter. Possible period.... possib(le) without qualification (1997, p. 72)." And Stalnaker writes, "we can agree with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, and most others who allow themselves to talk about possible worlds at all, that metaphysical necessity is necessity in the widest sense (2003, p. 203)." What exactly does the thesis that metaphysical possibility is absolute amount to? Is it true? In this article, I argue that, assuming that the thesis is not merely terminological, and lacking in any metaphysical interest, it is an article of faith. I conclude with the suggestion that metaphysical possibility may lack the metaphysical significance that is widely attributed to it. [ABSTRACT FROM AUTHOR]

Critical phenomenology is gaining currency as a progressive philosophy of emancipation, but there is no consensus on what its "criticality" entails. From a Derridean perspective, critique can be said to involve radical self-interrogation; a philosophy that questions its own conditions of possibility or grounds is one that opens itself to its auto-deconstruction. Deconstruction produces undecidability, however, which means that the philosophy in question can no longer account for its political claims or its normative force. This is the predicament in which critical phenomenology, like any other critical theory, will find itself when it takes its critical injunction to heart. [ABSTRACT FROM AUTHOR]

We investigate static cylindrically symmetric solutions of the Weyl and Gödel space–times in the framework of modified gravity. With this aim, we consider the modified higher-order theory of gravity based on nonconformal invariant gravitational waves. From the modified Einstein equations, we derive two exact solutions of the Weyl space–time and find one exact and one numerical solution of the Gödel space–time. In particular, we obtain a family of exact solutions with a constant scalar curvature depending on arbitrary constants for both space–times. It is interesting that the second solution for the Weyl metric has a nonconstant Ricci scalar. We find that the result obtained by solving the higher-order theory of gravity is similar to the result for the Einstein field equations with a cosmological constant. Moreover, we graphically study the role of the metric coefficients in both space–times. [ABSTRACT FROM AUTHOR]

DOI: http://doi.org/10.26333/sts.xxxiv1.12 Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.

Despite his unreserved appreciation of Turing’s analysis for being a “precise and unquestionably adequate definition” of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a “philosophical error” in Turing’s argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing’s work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing’s response to the “mathematical objection” based on Gödel’s incompleteness theorem and Gödel’s own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no “error” on Turing’s side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together.

This review of mathematical art exhibited at Bridges Linz in 2019 focusses on the use of two 20th c. mathematical developments, the theory of tilings and aperiodic order, and of fractals, in the work of Dugan Hammock, Ryan Webb, Lisa Shier, Doug Dunham, Pallavi Nataragan, Peter Stempfli, Saara Lehto, Andrea Dozmati, Hans Kuiper, Bernat Espigulé, Mingjang Chen, Flora Olivier, Kevin Lee and Demian Nahuel Goos Bosco. [ABSTRACT FROM AUTHOR]

Kurt Gödels Ontological Proof von 1970 ist einer der bedeutendsten Beweise der Existenz Gottes unserer Zeit. Die Formulierung in formaler Logik erschwert erstens das Verständnis und scheint zweitens einen unbedingten Wahrheitsbezug zu implizieren. Seit seiner Veröffentlichung bis heute wurde sich auf vielfältige Weise mit dem Beweis auseinandergesetzt und dazu publiziert. Allerdings fehlt bislang eine theologische – insbesondere eine protestantisch-theologische – Auseinandersetzung. Dieser Aufsatz soll eine systematische Übersicht über die bisherige Forschung sowie eine Einführung und Erläuterung des Beweises geben. Der theologische Forschungsbeitrag besteht hierbei vor allem in einer Relativierung einiger mit dem Beweis und seiner Rezeption verbundenen Ansprüche. Summary: Kurt Gödel's Ontological Proof (1970) is probably one of the most important proofs of God's existence of our times. The argument is based on formal logic: Firstly, this makes the argument quite hard to access; secondly, the argument seems to claim universal truth. Since its publication, many papers have been written, dealing with different aspects of this argument. However, a theological – and especially a protestant theological – consideration of the argument is still missing. This paper will give a systematic overview of the research that has already been conducted as well as introduce and explain the formal structure of the argument. The theological contribution can be seen by putting claims about God's ontological status based on Gödels' work into perspective. [ABSTRACT FROM AUTHOR]

This paper deals with the issue of interdisciplinarity in literature, tracing the mathematical influences -- and particularly those of the incompleteness theorems by Kurt Gödel -- on the literary works of Hans Magnus Enzensberger. Our attention is focused on Enzensberger's publications in mathematics, on his poem Hommage à Gödel, and his montage epos The Sinking of the Titanic. This paper submits a new, interdisciplinary interpretation of the poem Hommage à Gödel as a kind of art manifesto, and comes to the conclusion that the reason Enzensberger finds the incompleteness theorems so interesting is that they offer a scientific basis for undecidable problems and paradoxes. He applies Gödel's theorems to literature, where they enable a creative, playful oscillation between two positions or aspects of an undecidable problem and self-reference in literature. [ABSTRACT FROM AUTHOR]

Gödel's first incompleteness theorem is sometimes said to refute mechanism about the mind. §1 contains a discussion of mechanism. We look into its origins, motivations and commitments, both in general and with regard to the human mind, and ask about the place of modern computers and modern cognitive science within the general mechanistic paradigm. In §2 we give a sharp formulation of a mechanistic thesis about the mind in terms of the mathematical notion of computability. We present the argument from Gödel's theorem against mechanism in terms of this formulation and raise two objections, one of which is known but is here given a more precise formulation, and the other is new and based on the discussion in §1. [ABSTRACT FROM AUTHOR]

Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed. [ABSTRACT FROM AUTHOR]

This paper explores the connection between the natural language and a formal language from a particular point of view: self-referential constructions. Such constructions lead to some kind of limits of thought, either in the form of paradoxical constructions (Liar-type or Grellingtype), or in the form of the so called limitative theorems in mathematical logic (e.g. Gödel's theorem). By deriving Gödel's significant results from paradoxical constructions the limitative character of such self-referential constructions is preserved, but they open the ways for a new representation of a great variety of arguments in the field of logic, mathematics and philosophy. [ABSTRACT FROM AUTHOR]

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

A theoretical discussion of units in linguistic theory would be, in a sense, incomplete without a discussion of the systems, whether overt or implied, that the units are associated with. This paper traces conceptualizations of units and their accompanying systems in several disciplines. We identify some important problems with rule-based accounts (Parsons 1937) of social action and discuss the transition to non-rule-based theory afforded by ethnomethodology (e.g. Garfinkel 1963, 1967; Heritage 1984, 2011). We draw direct parallels between these issues and analogous developments in mathematical logic (Gödel 1992) and philosophy of mind (Fodor 1968, 1983; Lucas 1961; Putnam 1960, 1967 etc.), and argue that these stem directly from fundamental properties of a class of all formal systems which permit self-reference. We argue that, since these issues are architectural in nature, linguistic theory which postulates that linguistic units are the outputs of a consistent, self-referential, rule-based formal systems (e.g. Hauser, Chomsky & Fitch 2002) will inevitably run into similar problems. This is further supported by examples from actual language use which, as a class, will elude any theoretical explanation grounded in such a system. [ABSTRACT FROM AUTHOR]

This paper examines Nick Land's "numbering practices" for opening up language to modernity's increasing technological entanglement beyond human comprehension. I begin by examining Land's attempt to override our linguistic systems with machinic code and binary symbolism in a way which mirrors modernity's technological future shock. I then consider his appropriation of Cantor's set theory and Gödel's incompleteness theorem to plot ever greater degrees of reality's excess to all anthropic logic. The third section looks at Land's use of qabbalistic numerology to uncover the absolute contingency of all our most strongly held beliefs, truths and values. The fourth section considers Land's radicalisation of qabbalism through his own particularly abstract and inhuman notational "gematria." I conclude by looking at Land's interest in the computer keyboard's lock-in to the QWERTY layout for proffering a glimpse of modern technology's increasingly dehumanising meltdown of our anthropocentric delusions of grandeur. [ABSTRACT FROM AUTHOR]