In a paper published in 1975, Robert Jeroslow introduced the concept of an experimental logic as a generalization of ordinary formal systems such that theoremhood is a (or in practice ) rather than . These systems can be viewed as (rather crude) representations of axiomatic theories evolving stepwise over time. Similar ideas can be found in papers by Putnam (1965) and McCarthy and Shapiro (1987). The topic of the present article is a discussion of a suggestion by Allen Hazen, that (...) these experimental logics might provide an illuminating way of representing “the human mathematical mind”. This is done in the context of the well-known Lucas-Penrose thesis. Though we agree that Jeroslow's model has some merit in this context, and that the Lucas-Penrose arguments certainly are less than persuasive, some semi-technical doubts are raised concerning the alleged impact of experimental logics on the question of knowable self-consistency. (shrink)

Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing sophisticated (...) results from Mathematical Logic into play cannot furnish insights which would otherwise be unavailable. Lastly, we conclude that there are simply too many weighty details left unfilled in Hofstadter’s proposal. These really need to be fleshed out before we can even hope to say that our understanding of classical mind-body problems has been advanced through metamathematical parallels with Gödel’s work. (shrink)

Disertační práce se zabývá problematikou připisování myšlení jiným entitám, a to pomocí imitační hry navržené v roce 1950 britským filosofem Alanem Turingem. Jeho kritérium, známé v dějinách filosofie jako Turingův test, je podrobeno detailní analýze. Práce popisuje nejen původní námitky samotného Turinga, ale především pozdější diskuse v druhé polovině 20. století. Největší pozornost je věnována těmto kritikám: Lucasova matematická námitka využívající Gödelovu větu o neúplnosti, Searlův argument čínského pokoje konstatující nedostatečnost syntaxe pro sémantiku, Blockův návrh na použití brutální síly pro (...) řešení imitační hry, Frenchova teorie subkognitivních informací a Michieho skepticismus ohledně možnosti umělého vědomí. Závěr práce zachycuje současný stav recepce Turingova testu a představuje pokusy o jeho praktickou realizaci, například v každoroční soutěži o Loebnerovu cenu. Autor práce zastává názor, že ani po více než šedesáti letech od uveřejnění Turingova paradigmatického eseje stále neexistují žádné vážné důvody pro zamítnutí jeho tvrzení. Tradiční komputační funkcionalismus možná není ideální teorií vysvětlující činnost myslí a jako slibnější se může jevit vývoj v neurálních vědách, ale Turingův test je přesto užitečným a snad i jediným nástrojem pro detekci inteligence u lidmi vytvořených strojů. (shrink)

Why is nature amenable to mathematical description? This question has received attention in the philosophy of science but rarely from a phenomenological perspective. Nevertheless Husserl’s late essay “The Origin of Geometry,” which has received some critical scholarly attention in recent years, contains the beginning of a striking answer. This answer proceeds from Husserl’s main claim in that essay, which he also makes in the Crisis of the European Sciences, that the original meaning of science has been covered over or “sedimented” (...) by concepts that obscure the true intentional core of scientific meaning. In the first three sections of this paper I develop Husserl’s central insights about mathematics in light of two contemporary critiques of his project of “reactivation” of the original, sedimented meaning of science. In the latter two sections, I argue that accepting Husserl’s account of the original meaning and development of science offers a promising explanation of why nature is amenable to mathematics. This explanation hinges on a conception of the objects and methods of mathematics and the mathematized physical sciences as accomplishments, that is, as constituted contents of consciousness. (shrink)

Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.

This article is an inquiry into how the relationship between the principle of non-contradiction and the limits of thought has been understood by thinkers as diverse as Hegel, Heidegger, Levinas, and Graham Priest. While Heidegger and Levinas focus on the question of temporality and Priest takes a formal approach, all these philosophers effectively maintain that the principle of non-contradiction imposes a restriction on thought that disables it from adequately accounting for its own limits and thus what lies beyond those limits, (...) the implication being that the violation of the principle is necessary for such an accounting to take place. However, the ultimate argument here is that, contrary to Priest’s interpretation, Hegel’s philosophy can be convincingly read as supporting the idea that the mind’s ability to go beyond any particular limit of thought can actually be said to involve an adherence to a normative demand to locate and dispel the contradictions that emerge through the very setting of determinative limits. This is a non-formal consistency that evinces a “logic” that is unknowingly followed by the Heideggerian and Levinasian phenomenological philosophies of transcendence. (shrink)

While art and science still functioned side-by-side during the Renaissance, their methods and perspectives diverged during the nineteenth century, creating a still enduring separation between the "two cultures". Recently, artists and scientists again collaborate more frequently, as promoted most radically by the ArtScience movement. This approach aims at a true synthesis between the intuitive, imaginative methods of art and the rational, rule-governed methods of science. To prepare the grounds for a theoretical synthesis, this paper surveys the fundamental commonalities and differences (...) between science and art. Science and art are united in their creative investigation, where coherence, pattern or meaning play a vital role in the development of concepts, while relying on concrete representations to experiment with the resulting insights. On the other hand, according to the standard conception, science seeks an understanding that is universal, objective and unambiguous, while art focuses on unique, subjective and open-ended experiences. Both offer prospect and coherence, mystery and complexity, albeit with science preferring the former and art, the latter. The paper concludes with some examples of artscience works that combine all these aspects. (shrink)

SUMMARY Major theories of philosophical psychology and philosophy of mind are examined on the basis of the fundamental questions of ontology, metaphysics, epistemology, semantics and logic. The result is the choice between language of eliminative reductionism and dualism, neither of which answers properly the relation between mind and body. In the search for a non–dualistic and non–reductive language, Wittgenstein’s notion of language–games as the representative links between language and the world is considered together with Peirce’s semeiosis of cognition. The result (...) is a redefined notion of cognition as the goal–oriented ability to instantiate strategies from the rules of the language–games. Cognition is considered within the four-dimensional languagegames: rules/strategies, primitive/complete language-games, family-resemblances/forms of life, and non-temporal continuum as the language-games’ modality. Cognition is seen as a continuum process of non–individualized elements that unites perception, volition, emotion, will and thought against the commonly accepted understandings of cognition in exclusive terms of perception and thinking. The alternative to the mind/body problem is the continuity of cognition where rules and strategies, syntax and semantics, are considered inseparable within language–games. This continuity of cognition is explained in terms of virtual (univocal) identity, replacing the old philosophical paradigm of the mind/body problem. -/- SOMMARIO Le teorie maggiori della psicologia filosofica e filosofia della mente sono esaminate a partire dalle questioni fondamentali di ontologia, metafisica, epistemologia, semantica e logica. Il risultato è la scelta tra un linguaggio riduzionista eliminativo e un dualismo che però non confronta in modo adeguato la relazione tra mente e corpo. Nella ricerca per un linguaggio non-dualistico e non-riduttivo, viene considerata la nozione del gioco linguistico di Wittgenstein come un legame rappresentativo tra il linguaggio e il mondo insieme con la semiosi cognitiva di Peirce. Il risultato è la nozione ridefinita della cognizione come abilità goal-orientata di creare le strategie dalle regole dei giochi linguistici. La cognizione è considerata entro le quattro dimensioni dei giochi linguistici: regole/strategie, giochi linguistici primitivi/completi, somiglianze di famiglia/forme di vita, e la continuità atemporale come la modalità dei giochi linguistici. La cognizione diventa un processo continuo degli elementi non-individualizzati che tende ad unificare la percezione, volizione, emozione, la volontà e il pensiero contro un concetto generalmente accettato di vedere la cognizione nei termini esclusivamente di percezione e di pensiero. L’alternativa al problema della mente/corpo è la continuità della cognizione, dove le regole e le strategie, la sintassi e la semantica, sono considerate inseparabili nei giochi linguistici. Questa continuità della cognizione è spiegata nei termini dell’identità virtuale (univoca) sostituendo un paradigma filosofico previo del problema della mente/corpo. (shrink)

A response to a recent critique by Cem Bozşahin of the theory of syntactic semantics as it applies to Helen Keller, and some applications of the theory to the philosophy of computer science.

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.

What do you get when you cross a fallacy with a good argument? A fugu, that is, a valid argument that tempts you to reach its conclusion invalidly. You have yielded to the temptation more than you realize. If you are a teacher, you may have served many fugus. They arise systematically through several mechanisms. Fugus are interesting intermediate cases that shed light on the following issues: bare evidentialism, false pleasure, philosophy of education, and the ethics of argument. Normally, a (...) fugu will not yield knowledge from known premises. But if the reasoning is only slightly fallacious, they do yield knowledge. These mild fugus show that we can gain knowledge by invalid reasoning. This is a conservative resource for historians. They want to credit discoveries to Euclid rather than those who made minor corrections to his proofs, such as David Hilbert. We also benefit from this practice of grandfathering in old standards of knowledge attribution. For we can compete spiritedly for priority. We do not need to worry that credit will instead go to future scholars who will make the minor amendments needed to raise present proofs to a future standard of demonstration. (shrink)

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her ideas. (...) But we also present some further problems with her view, concerning the role of proof for mathematical knowledge. (shrink)

In this reply to James H. Fetzer’s “Minds and Machines: Limits to Simulations of Thought and Action”, I argue that computationalism should not be the view that (human) cognition is computation, but that it should be the view that cognition (simpliciter) is computable. It follows that computationalism can be true even if (human) cognition is not the result of computations in the brain. I also argue that, if semiotic systems are systems that interpret signs, then both humans and computers are (...) semiotic systems. Finally, I suggest that minds can be considered as virtual machines implemented in certain semiotic systems, primarily the brain, but also AI computers. In doing so, I take issue with Fetzer’s arguments to the contrary. (shrink)

Goal-directed problem solving as originally advocated by Herbert Simon’s means-ends analysis model has primarily shaped the course of design research on artificially intelligent systems for problem-solving. We contend that there is a definite disregard of a key phase within the overall design process that in fact logically precedes the actual problem solving phase. While systems designers have traditionally been obsessed with goal-directed problem solving, the basic determinants of the ultimate desired goal state still remain to be fully understood or categorically (...) defined. We propose a rational framework built on a set of logically inter-connected conjectures to specifically recognize this neglected phase in the overall design process of intelligent systems for practical problem-solving applications. (shrink)

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.

ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.

Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions (...) related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that—if one is willing to assume a thesis due to Donald Knuth—the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent—including a not-necessarily infallible human agent—satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Self-comprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

We show that the name “Lucas-Penrose thesis” encompasses several different theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interesting versions of the much weaker thesis known as “Gö- del disjunction”. Our main conclusions are that (...) pure mathematical theorems cannot decide alone any question which is not purely mathematical, and that an argument that cannot be fully formalized cannot be taken as a mathematical proof. (shrink)

Lucas-Penrose type arguments have been the focus of many papers in the literature. In the present paper we attempt to evaluate the consequences of Gödel’s incompleteness theorems for the philosophy of the mind. We argue that the best answer to this question was given by Gödel already in 1951 when he realized that either our intellectual capability is not representable by a Turing Machine, or we can never know with mathematical certainty what such a machine is. But his considerations became (...) known only in recent times when many scholars were already aware of Benacerraf’s and Chihara’s analyses on the consequences of Gödel’s incompleteness theorems for the philosophy of the mind. Benacerraf and Chihara, in fact, discussing Lucas’ paper, arrived at the same conclusions as Gödel in the sixties, but in a more formal way. After Penrose’s provocative arguments, Shapiro again shed light on the question. In our paper, after a broad and simple presentation of the contributions to the debate made by different authors, we show how to present Gödel’s argument in a rigorous way, highlighting the necessary philosophical premises of Gödel’s argument and more in general of Gödelian arguments. (shrink)

In this paper I discuss possible ways of measuring the power of arithmetical theories, and the possiblity of making an explication in Carnap's sense of this concept. Chaitin formulates several suggestions how to construct measures, and these suggestions are reviewed together with some new and old critical arguments. I also briefly review a measure I have designed together with some shortcomings of this measure. The conclusion of the paper is that it is not possible to formulate an explication of the (...) concept. (shrink)