Kurt Gödel

Este texto é a tradução da introdução do livro Logical Forms feita por Guido Imaguire e revisada pelo próprio autor (que também fez algumas pequenas alterações no texto e é responsável por sua forma final). Muitas citações de textos originais nas notas de rodapé foram suprimidas na tradução.

Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come delle pietre miliari e come tali richiamati ancora oggi nella letteratura specialistica, investono aspetti fondamentali per la ricerca in campo matematico, quali la relazione tra verità e dimostrabilità o questioni relative alla natura e ai limiti della conoscenza matematica. Questo profilo nasce dal tentativo di correggere l’impressione che la grandezza del personaggio possa essere ricondotta a uno dei suoi molteplici contributi in particolare, come ad esempio il teorema di incompletezza per le teorie assiomatiche al quale egli deve gran parte della sua fama. Al contrario, il volume cerca di offrire una panoramica per quanto possibile esaustiva dell’opera logica e matematica di Gödel, così da valutarne meglio la portata.

This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide a partial vindication of Gödel and show that if a particular solution to the intensional paradoxes is adopted, one can indeed give an argument to the effect that the mind is not a machine.

.....the double bind as constituting an entirely new, non-material, principle of evolution, major conceptual revisions may need to be made in our general understanding of evolution as contained within the “modern evolutionary synthesis”.

KEYWORDS:
behavior, communication, context, contradiction, deutero-learning, double bind, epistemology, evolution, incompleteness theorum, kinesics, learning, logical type theory, metacommunication, metamessages, metatheory, mystification, paradox, relationships, schismogenesis, schizophrenia,  scientific elegance, symmathesy, teleology, temporality, transcontextual, transdisciplinary, trauma

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.

In 2013, the German conservative newspaper Die Welt reported “Scientists Prove Existence of God," As an eye-catching headline as it was, unsurprisingly, there is a rather significant caveat to that claim. When Gödel died in 1978, he left behind a tantalizing theory based on principles of modal logic - that a higher being must exist. The details of the mathematics involved in Gödel's ontological proof are complicated, but in essence he was arguing that, by definition, God is that for which no greater can be conceived. And while God exists in the understanding of the concept, we could conceive of him as greater if he existed in reality. Therefore, he must exist.

Even at the time, the argument was not exactly a new one. For centuries many have tried to use this kind of abstract reasoning to prove the possibility or necessity of the existence of God. But the mathematical model composed by Gödel proposed a proof of the idea. It’s theorems and axioms, assumptions which cannot be proven can be expressed as mathematical equations, and that means they can be proven.

lecturer on alchemy and its modern applications. www.DWHauck.com Go directly to the start of the text. Abstract The alchemists believed that no transformation— whether in the laboratory, in the body, or in the soul—could succeed without the presence of a mysterious ingredient known as the Materia Prima (First Matter). Nothing was more important to alchemist's work than this energetic essence, which they believed could be extracted from any substance and actually rendered tangible and visible. This paper examines the nature and properties of the elusive First Matter. We will look at how the alchemists exposed it, accumulated it, and worked with it to accomplish the regeneration of the substance at hand. We will explore how the concept of the First Matter evolved in both the scientific and esoteric traditions to become a fundamental, although often unrecognized, tenet of both modern Hermeticism and modern physics.

We claim that Kant's doctrine of the "transcendental ideal of pure reason" contains, in an anticipatory sense, a second-order theory of reality (as a second-order property) and of the highest being. Such a theory, as reconstructed in this paper, is a transformation of Kant's metatheoretical regulative and heuristic presuppositions of empirical theories into a hypothetical ontotheology. We show that this metaphysical theory, in distinction to Descartes' and Leibniz's ontotheology, in many aspects resembles Gödel's theoretical conception of the possibility of a supreme being. The proposed second-order modal formalization of the Kantian doctrine of the "transcendental ideal" of the supreme being includes some specific features of Kantian general logic like "categorical", "hypothetical", and exclusive disjunctive propositions as basic forms. (Download: http://www.collegepublications.co.uk/downloads/ifcolog00027.pdf)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.

By way of commenting on the prior literature, it is argued that both the first and the second of Gödel's Incompleteness Theorems have a bearing on the view on mathematics that Husserl presents in Formale und transzendentale Logik, and that this bearing is not small.

Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in the title, with special attention to the method of analysis and synthesis in Greek geometry, fixed point theorems, and Kreisel's contact with Gödel.

This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard of proof that completes a system containing genuine, not contrived, undecidable statements. Gödel's First Incompleteness Theorem is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are disproved in F."

Scientific thought was once the paradigm of unchanging truth. While the natural sciences have broken with this presumption, the formal sciences have been slower to separate themselves from this antique conception. Yet the experience of change in the formal sciences must be recognized as central to the theoretical enterprise. It is at the boundaries of the formal sciences that most of this change occurs. Frege extended the scope of formal thought into foundations, while Cantor extended the scope of formal thought toward the infinite. Gödel synthesized foundationalist rigor with Cantorian insights to redefine the boundaries of mathematics with limitative theorems.

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predica-tivity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-Löf.

Lyotard raccomanda, obtorto collo, non di affidarsi al relativismo, ovviamente, ma di imparare a convivere con esso e dunque a ripensare realtà e supposizioni, in quanto l’essere è tramontato e il divenire ha mostrato le sue carte. La convivenza con il relativismo, aggiunge modestamente l’autore dell’articolo, ha il pregio di stimolare la ricerca e, grazie agli strumenti nuovi (il computer, internet) di porre sotto discussione i progressi scientifici e culturali, di farlo, per lo meno, da un punto di vista della dignità intellettuale e morale che appartiene ad ognuno di noi, altro che Orwell 1984!

Incompleteness specifically refers to the fact that given any system of symbols, there will always be questions that can be formulated in that system that cannot be answered using just the symbols of that system.

An example would be that if a person had a box of objects that person was holding, and an interrogator asked to show the objects in the box which cannot be shown. That would be a paradoxical question with no possible answer.
Those sorts of questions can always be formulated, and this shows that there are some propositions that will be unresolvable no matter what the premises are of a system.
So, if one wanted to judge an axiomatic system—as David Hilbert did at the turn of the last century —by whether it could have an answer for any question one could ask of it, then one will not be able to make that judgement of some system being more thoroughly 'complete' (able to answer questions asked of it) relative to another.

Can, this, then can be applied to law and moral judgements?  If one perceives law or morality as a propositional system then, yes, it would be applicable to moral judgements. Many pragmatists do not perceive law as such a propositional system, though. Some of the quotes I have collected refer to such pragmatists with that different approach.

Morality cannot be approved and everything is subjective intent including law. So, in the same way that in the early 1900s Bertrand Russell’s attempts with mathematics were approved futile in Principia Mathematica. Are the philosophical foundations of law proved futile?

How is Gödel's incompleteness theorem as being a problem for morals? Nobody has "true" premises to go from? Therefore—it can be said—all our conclusions are subjective. Is this accurate?

Resumen
Se explora la naturaleza recursiva del lenguaje, la iteración a la que parece condenado  cuando debe dar cuenta de sí mismo. El carácter autorreferencial del lenguaje, como  némesis, se verifica en las paradojas que reingresan constante en sí mismas, expresiones arquitectónicamente circulares en cuyo movimiento de pliegue y repliegue comparece el infinito. De allí la perplejidad frente al principio de “información faltante”, el teorema de la incompletitud, los sistemas caóticos (ciencias de la complejidad) que parece amenazar nuestra racionalidad. También entran a consideración las exploraciones realizadas por W. S. Burroughs desde la literatura pulp donde el ser humano aparece alienado por el lenguaje. El lenguaje (y las normas gramaticales y sintácticas que le caracterizan) sería un organismo parásito, un virus, que ha elegido nuestras mentes como hábitat. El sujeto se encuentra manipulado y transformado por los procesos de contagio; el lenguaje es un virus que se reproduce con gran facilidad y condiciona cualquier actividad humana, dando cuenta de su intoxicada naturaleza. Se finaliza con la contraposición entre el pensamiento occidental con el pensamiento oriental, en particular con el budismo Zen, entendiendo este último como una visión alternativa del mundo, basada en el flujo y transformación, en lugar de la inmutabilidad y subordinación de los conceptos lógicos griegos.
Abstract
The recursive nature of language is explored, iterating that seems doomed when to give an account of himself. The self-referential nature of language, as nemesis, is verified in constant re-entering the paradoxes themselves architecturally circular motion expressions in which fold and stow appears infinity. Hence the perplexity against the principle of "missing information", the incompleteness theorem, the (complexity science) chaotic systems appears to threaten our rationality. Also suitable for consideration examinations by WS Burroughs from pulp literature where man appears alienated by language. The language (and grammatical and syntactical rules that characterize it) would be a parasitic organism, a virus that has chosen our minds as habitat. The subject is manipulated and transformed by the processes of contagion; language is a virus that is easily reproduced and conditions any human activity, realizing his intoxicated nature. It ends with the contrast between Western thought with Eastern thought, understanding the latter as an alternative worldview, based on the flow and transformation, rather than the immutability and subordination of Greek logical concepts.
Palabras Clave:
Lógica, paradoja, consistencia, recursividad, contradicción, lenguaje, racionalidad, metáfora viral, lecturas parasitarias
Keywords:
Logic, paradox, consistency, resourcefulness, contradiction, language, rationality, metaphor viral, parasitic readings

Guided by a passage in Kreisel, this is a discussion of the relations between the phenomena in the title, with special attention to the method of analysis and synthesis in Greek geometry, fixed point theorems, and Kreisel's contact with Gödel.

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The demonstration of Gödel’s 1st Incompleteness Theorem relies on the recourse to a formula G supposed to assert its own indemonstrable nature.
However, in order to build the formula G, Gödel had no other option than transgress the law of causality, i.e. the very law which precludes any possibility for an effect to act as its own cause.
The present draft, based on the author’s conviction that this law rules the abstract world with the same rigour as it does the material one, provides a detailed analysis of two lines of reasoning which, in the demonstration, fail to achieve such an unachievable result.

Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]).  There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent.  But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently).  Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge.  The purpose of this paper is to determine what Benacerraf’s challenge could be such that this view is warranted.  I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field.  Not only are none of the challenges easier for the pluralist to meet.  None satisfies a key constraint that has been placed on Benacerraf’s challenge.  However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones.  Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding.  If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be.

- Draws extensively on Gödel's archive
- Analyses historical and systematical aspects of Gödel's philosophical project
- Evaluates Gödel's use of phenomenology and challenges recent interpretations

This volume tackles Gödel's two-stage project of first using Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology, and then founding classical mathematics on the metaphysics thus obtained. The author analyses the historical and systematic aspects of that project, and then evaluates it, with an emphasis on the second stage.

The book is organised around Gödel's use of Leibniz, Husserl and Brouwer. Far from considering past philosophers irrelevant to actual systematic concerns, Gödel embraced the use of historical authors to frame his own philosophical perspective. The philosophies of Leibniz and Husserl define his project, while Brouwer's intuitionism is its principal foil: the close affinities between phenomenology and intuitionism set the bar for Gödel's attempt to go far beyond intuitionism.

The four central essays are `Monads and sets', `On the philosophical development of Kurt Gödel', `Gödel and intuitionism', and `Construction and constitution in mathematics'. The first analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. It also provides further support for Gödel's idea that the monadology needs to be reconstructed phenomenologically, by showing that the unsupplemented monadology is not able to found mathematics directly. The second studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his published writings. The third discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation. The fourth addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged, and concludes that it does not.

The remaining essays provide further context.  The essays collected here were written and published over the last decade. Notes have been added to record further thoughts, changes of mind, connections between the essays, and updates of references.

Content Level » Research

Keywords » Edmund Husserl - Gottfried Wilhelm von Leibniz - Kurt Gödel - L.E.J. Brouwer - Platonism - intuitionism - monadology - phenomenology of mathematics - philosophy of mathematics

Related subjects » Epistemology & Philosophy of Science - Mathematics - Philosophical Traditions

This paper introduces an original proposal intended to encode timelines as univocal integers, by the use of arithmetic mapping, exploring inherent properties of prime numbers. A group of algorithms were developed in order to encode drum-set timelines (as well as to retrieve the original rhythms from the codes), either individually or taken together, forming grooves. Additional parameters (dynamic levels and timbre) are also included in the encoding process. Geometrical representation of the grooves, adopting terminology and methodology proposed by Gottfried Toussaint[19], is also provided. Some practical application, addressing analysis and composition, are suggested at the last section of the study.

Ontological arguments like those of Gödel (1995) and Pruss (2009; 2012) rely on premises that initially seem plausible, but on closer scrutiny are not. The premises have modal import that is required for the arguments but is not immediately grasped on inspection, and which ultimately undermines the simpler logical intuitions that make the premises seem plausible. Furthermore, the notion of necessity that they involve goes unspecified, and yet must go beyond standard varieties of logical necessity. This leaves us little reason to believe the premises, while their implausible existential import gives us good reason not to. Gödel (1995) introduced a new class of formal arguments for the existence of God, appealing to a notion of " positive " property and applying modal logic. Gödel's premises were later shown to imply modal collapse, i.e., if they are true, then everything true is necessary (Sobel 1987), and then to be inconsistent (Benzmuller and Paleo 2016). However, Anderson (1990) and Pruss (2009; 2012) give simpler versions that avoid these problems. We will focus here on one of Pruss's formulations, but our observations apply generally. We will see that Pruss's premises are not as innocent as they first appear. Once their modal import is unpacked, and their unclear foundations exposed, they are not very plausible at all.

JOHN CORCORAN, Numerically-qualified omega completeness.


  In GF, the framework of the 1931 Gödel paper, there are two explicit non-logical constants, ‘o’ and ‘s’; the concept “number” is absorbed into the intension of the individual variables by taking the domain D of number theory as the universe of discourse. In the framework of the 1889 Peano booklet, D is taken as a subset of the range of individual variables and, accordingly, the concept “number” takes a distinctive non-logical constant. In a modern version, PF, of the Peano framework, the numbers are treated as individuals having a distinctive quality, say, “being numerical”, indicated by a one-place predicate symbol ‘N’ supplementing ‘o’ and ‘s’.
  This paper discusses semantic omega completeness in GF juxtaposed with its analogue Numerically-qualified omega completeness in PF.

About two hundred years after the Kant’s critique, which was supposed to be devastating since it had declared all ontological arguments of God’s existence impossible, in the middle of the 20th century the argument of St. Anselm got into the centre of attention once again. Besides the reconstructions of the classical train of thought with the help of modal logic, numerous new constructions were born that strove to approximate to the existence of God with the most modern devices of formal logic. All of these constructions, despite the fact that his work constitutes a single typed page, the attempt of Kurt Gödel, one of the greatest mathematicians and logicians of the century, stands out. Today it is a generally accepted opinion that his deduction is formally correct and his five axioms and three definitions constitute a consistent system. Nevertheless, the Ontologischer Beweis as an argument of God’s existence is meaningless unless we succeed in continuing Gödel’s work by adding adequate semantics to its formulas. This work should be done by the simultaneous implementation of two important points of view. On the one hand, it is necessary to give an interpretation of logical symbols that makes it possible to call the deducted being god. On the other hand, it is important to make sure that from the semantics, acceptable for the philosophy of religion, no untenable statements could be drawn to the world that belongs to the god of the argument. This present study enumerates the possible interpretations of the central technical words of the argumentation, “positive property” and “essence,” and arrives at the conclusion that in Gödel’s system these two conditions cannot be fulfilled simultaneously.
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Mintegy kétszáz évvel Kant megsemmisítőnek vélt kritikája után, amely lehetetlennek nyilvánított minden ontológiai istenérvet, a 20. század közepén újra a figyelem középpontjába került Szent Anzelm istenérve. A klasszikus gondolatmenetnek a modális logika eredményei segítségével történő rekonstrukciói mellett számos olyan új konstrukció is született, amely Isten létét a formális logika legkorszerűbb eszköztárával próbálta megközelíteni. Ezek közül is kiemelkedik Kurt Gödelnek, a század egyik legnagyobb matematikusának és logikusának mindössze egy gépelt oldalt kitevő kísérlete. A mára általánossá vált a vélemény szerint a levezetés formailag korrekt, és öt axiómája valamint három definíciója konzisztens rendszert alkot. Ennek ellenére az Ontologischer Beweis mint istenérv semmitmondó, ha nem sikerül Gödel munkáját azzal folytatni, hogy formuláihoz megfelelő szemantikát szerkesztünk. Ezt két fontos szempont egyidejű érvényesítésével kellene elvégezni. Egyrészt a logikai szimbólumok olyan interpretációjára van szükség, amely lehetővé teszi, hogy a levezetett lény istennek legyen nevezhető. Másrészt ügyelni kell arra, hogy a vallásfilozófiailag elfogadható szemantikából ne következzenek az érv istenéhez tartozó világra vonatkozó tarthatatlan állítások. A tanulmány a bizonyítás központi terminusai, a „pozitív tulajdonság”, illetve a „lényegi tulajdonság” lehetséges értelmezéseit veszi számba, és arra a megállapításra jut, hogy Gödel rendszerében e két feltétel egyidejűleg nem elégíthető ki.

In part one (of two parts) I show that any purely physical-scientific account of reality must be deficient. Instead, I present a general-ontological framework that should prove fruitful when discussing or resolving philosophic controversies; indeed, I show that the ...

International audienceWe are pleased to present here a second volume of the electronic edition of Kurt Gödel’s Maxims and Philosophical Remarks, commonly called Max Phil. Since volume X (⟨hal-01459188⟩ of this electronic edition appeared in January 2017 hundreds of Gödel scholars have taken the opportunity to download it. We here present the first complete transcription, from Gabelsberger shorthand to German, of Max Phil IX. A preliminary version of this volume IX, as well as of volumes XI, XII, XIV and XV, was until now only available to the public at the CNRS Library Granger Guillermit.Notebook IX is catalogued as item 030095 in Series III, box 6b, folder 69.A scan of Gödel's manuscript in Gabelsberger is avalaible at https://library.ias.edu/gödelpapers

The paper shows that it is possible to obtain a "slingshot" result in Gödel's theory of positiveness in the presence of the theorem of the necessary existence of God. In the context of the reconstruction of Gödel's original "slingshot" argument on the suppositions of non-Fregean logic, this is a natural result. The "slingshot" result occurs in sufficiently strong non-Fregean theories accepting the necessary existence of some entities. However, this feature of a Gödelian theory may be considered not as a trivialisation, but as an intended consequence of Gödel's ontotheological views. (http://semper.istore.pl/pl/p/Goedels-Ontological-Argument/23076163; https://philpapers.org/rec/KOVGQA)