EDGAR MORIN'S PARADIGM OF COMPLEXITY AND GÖDEL'S INCOMPLETENESS THEOREM

This article shows that in two respects, Gödel's incompleteness theorem strongly supports the arguments of Edgar Morin's complexity paradigm. First, from the viewpoint of the content of Gödel's theorem, the latter justifies the basic view of complexity paradigm according to which knowledge is a dynamic, unfinished process, and develops by way of self-criticism and self-transcendence. Second, from the viewpoint of the proof procedure of Gödel's theorem, the latter confirms the complexity paradigm's circular line of inference through which is formed the all-round knowledge of a concrete object.     

Completeness: from Gödel to Henkin

This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödel's and Tarski's crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity in the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödel's proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkin's work concerning the generalization of his completeness proof to any logic from his initial work in type theory.     

Gödel,Kant, and the Path of a Science

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz and Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallels between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added.     

Gödel on Concepts

This article is an attempt to present Gödel's discussion on concepts, from 1944 to the late 1970s, in particular relation to the thought of Frege and Russell. The discussion takes its point of departure from Gödel's claim in notes on Bernay's review of ‘Russell's mathematical logic’. It then retraces the historical background of the notion of intension which both Russell and Gödel use, and offers some grounds for claiming that Gödel consistently considered logic as a free-type theory of concepts, called intensions, considered as the denotations of predicate names.     

Kurt Gödel's Anticipation of the Turing Machine: A Vitalistic Approach

In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of a computing machine which closely resembles Turing's groundbreaking conception. After introducing the notebooks on quantum mechanics, this article describes Gödel's vitalistic Weltbild and the ideas leading to the development of his computing machine. It investigates a notion of lawlike sequence which closely resembles Turing's concept of a computable number and which Gödel himself calls ‘problematic’, and compares it to the opposed concept of randomness, drawing upon the notion of program-size complexity. Finally, Gödel's machine is implemented in a dialect of the Lisp programing language.     

Gödel,Carnap and the Fregean Heritage

Synthese - Thorough a detailed analysis of version III of Gödel's Is mathematics syntax of language?, we propose a new interpretation of Gödel's criticism against the...     

Gödel And The Intuition Of Concepts

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition.     

Gödelian time-travel and anthropic cosmology

This paper looks at Kurt Gödel's causally‐pathological cosmological models (derived from general relativity), in the light of anthropic explanations. If a Gödelian world is a possible world, could anthropic reasoning shed any light on whether or not our world is Gödelian? This paper argues that while there are some good anthropic reasons why our world ought to be Gödelian, too many observations suggest that our world can’t possibly be Gödelian in fact. If Gödel's world is a possible one, anthropic teleology alone cannot explain why it isn’t the world we inhabit. Furthermore, if our world were Gödelian, anthropic arguments against the existence of extraterrestrial intelligences would imply a bleak human future. En route, some general objections to relativistic causal pathologies are addressed and some anthropic arguments to the effect that Gödelian worlds couldn’t sustain life are also addressed and dismissed.     

Gödel's and Other Paradoxes

Francesco Berto has recently written “The Gödel Paradox and Wittgenstein's Reasons,” about a paradox first formulated by Graham Priest in 1971. The major reason for disagreeing with Berto's conclusions concerns his elucidation of Wittgenstein's understanding of Gödel's theorems. Seemingly, Wittgenstein was some kind of proto‐paraconsistentist. Priest himself has also, though in a different way, tried to tar Wittgenstein with the same brush. But the resolution of other paradoxes is intimately linked with the resolution of the Gödel Paradox, and with understanding Wittgenstein's views of Gödel's theorems. So this paper discusses some other paradoxes before looking at Wittgenstein's relevant views.     

(Non)Issues of Infinite Regress in Modeling Motor Behavior

In the past, infinite regress criticisms that have been raised about models of motor behavior have been reserved for executive-type models (e.g., Beek & Meijer, 1988). On the basis of Gödel's (1930/1986) proof that an algorithm cannot prove its own validity, the authors reason that executive- as well as self-organized-type explanatory models of motor behavior have infinite regress difficulties. The conclusion offered in the present article is that judgments on a model's theoretical importance should be based not on issues of infinite regress but on other relevant characteristics, such as its propensity for falsification (Popper, 1959).     

L'idée de la logique formelle dans les appendices VI à X du volume 12 des Husserliana (1970)

Gödel always claimed that he did not know Skolem's Helsinki lecture when writing his dissertation. Some questions and doubts have been raised about this claim, in particular on the basis of a library slip showing that he had requested Skolem's paper in 1928. It is shown that this library slip does not constitute evidence against Gödel's claim, and that, on the contrary, the library slip and other archive material actually corroborate what Gödel said.     

Satisfying Predicates: Kleene's Proof of the Hilbert–Bernays Theorem

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.     

Gödel's Argument for Cantorian Cardinality

On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's argument, showing that it fails in two important ways: (i) Its premises are not sufficiently compelling to discredit countervailing intuitions and pragmatic considerations, nor pluralism, and (ii) its final inference, from the superiority of Cantor's theory as applied to sets of changeable physical objects to the unique acceptability of that theory for all sets, is irredeemably invalid.     

Remarks on the development of computability

The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. It is suggested in Section 3 that a central item was the problem of generalizing Gödel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of computability often called ‘argument by Church's thesis’.     

Platonism and metaphor in the texts of mathematics: Gödel and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of ‘grasping’ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes ‘grasping’ more as theoretical activity than as a kind of inner mental ‘seeing’.

     

Some Limitations to the Psychological Orientation in Semantic Theory

The psychological orientation treats semantics as a matter of idealized computation over symbolic structures, and semantic relations like denotation as relations between linguistic expressions and these structures. I argue that results similar to Gödel’s incompleteness theorems and Tarski’s theorem on truth create foundational difficulties for this view of semantics.     

An Intuitionistic Completeness Theorem for Classical Predicate Logic

This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel’s completeness theorem for classical predicate logic.     

The experiential foundations of mathematical knowledge

A view of the sources of mathematical knowledge is sketched which emphasizes the close connections between mathematical and empirical knowledge. A platonistic interpretation of mathematical discourse is adopted throughout. Two skeptical views are discussed and rejected. One of these, due to Maturana, is supposed to be based on biological considerations. The other, due to Dummett, is derived from a Wittgensteinian position in the philosophy of language. The paper ends with an elaboration of Gödel's analogy between the mathematician and the physicist.     

Logical Foundations and Kant's Principles of Formal Logic

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system.