De Houdbaarheid van Kurt Gödels Wiskundige Intuïtie

Wiskunde heeft filosofen al eeuwenlang beziggehouden en nog steeds bestaat er een levendig onderzoek naar dit onderwerp binnen de filosofie. De ontologie van de wiskunde is een belangrijk onderdeel van deze filosofie van de wiskunde. Binnen de ontologie van de wiskunde vindt men vragen als: ‘wat is een getal?’, ‘over wat of welke dingen gaat wiskunde?’ en ‘gaat wiskunde over dingen?’ Een mogelijk antwoord op deze vragen wordt gegeven door het platonisme. Het platonisme stelt dat wiskunde gaat over objecten die abstract en onafhankelijk van het menselijk denken zijn. Bovendien zijn het deze objecten die de wiskunde waar maken. Eén van de grootste problemen waar een wiskundig platonist mee te maken heeft, is dat hij niet kan verklaren hoe het kan dat wij kennis van deze abstracte objecten hebben. Voor dit epistemologische probleem moet de platonist een oplossing zien te formuleren. Een mogelijke oplossing voor dit probleem vinden we bij de logicus Kurt Gödel. Het is deze oplossing, Gödels notie van ‘wiskundige intuïtie’ (voorts: wiskundige intuïtie), die het onderwerp is van dit paper. Dit paper zal zich richten op de houdbaarheid van wiskundige intuïtie zoals ingezet door Kurt Gödel, namelijk als een epistemologisch vermogen om ons inzicht in wiskunde en haar axioma’s mee te verschaffen. Dit paper bestaat uit vijf onderdelen. Ten eerste zal ik uiteenzetten wat wiskundige intuïtie bij Gödel inhoudt en hoe deze positie zich verhoudt tot het wiskundig platonisme. Gödel achtte deze stroming binnen de ontologie van de wiskunde zeer nauw verbonden met wiskundige intuïtie. Ten tweede zal ik twee kritieken op Gödels opvatting over wiskundige intuïtie behandelen. De auteurs van deze twee kritieken, Paul Benacerraf en Charles Chihara, zetten argumenten in die in hun ogen niet alleen de plausibiliteit maar ook de houdbaarheid van wiskundige intuïtie aantasten. Hoewel er andere kritieken mogelijk zijn, wil ik me in dit paper tot deze twee kritieken beperken. Ten derde zal Gödels eigen argument voor wiskundige intuïtie uiteen worden gezet. Dit argument sluit aan op de praktijk van het ontdekken van nieuwe axioma’s in de wiskunde, geïllustreerd aan de hand van de continuümhypothese. Ten vierde beschouw ik een mogelijk filosofisch gevolg van Gödels eerste incompleetheidsstelling. Dit gevolg heeft te maken met de aard van wiskundige kennis. In dit verband zal ik tevens de kenleer van Immanuel Kant en David Hume bespreken. Ten slotte beschouw ik een argument voor wiskundige intuïtie op basis van de functie die wiskundige afbeeldingen zouden moeten innemen in de wiskundige praktijk. Een aantal van de argumenten die ik doorheen dit paper zal geven zijn directe argumenten voor wiskundige intuïtie. Het doel van dit paper is echter niet de lezer te overtuigen van de correctheid of het bestaan van wiskundige intuïtie. Ik wil niet pleiten voor de geldi

Causal loops: logically consistent correlations, time travel, and computation

Causal loops are loops in cause-effect chains: An effect can be the cause of that effect's cause. We show that causal loops can be unproblematic, and explore them from different points of view. This thesis is motivated by quantum theory, general relativity, and quantum gravity. By accepting all of quantum theory one can ask whether the possibility to take superpositions extends to causal structures. Then again, quantum theory comes with conceptual problems: Can we overcome these problems by dropping causality? General relativity is consistent with space-time geometries that allow for time-travel: What happens to systems traveling along closed time-like curves, are there reasons to rule out the existence of closed time-like curves in nature? Finally, a candidate for a theory of quantum gravity is quantum theory with a different, relaxed space-time geometry. Motivated by these questions, we explore the classical world of the non-causal. This world is non-empty; and what can happen in such a world is sometimes weird, but not too crazy. What is weird is that in these worlds, a party (or event) can be in the future and in the past of some other party (time travel). What is not too crazy is that this theoretical possibility does not lead to any contradiction. Moreover, one can identify logical consistency with the existence of a unique fixed point in a cause-effect chain. This can be understood as follows: No fixed point is the same as having a contradiction (too stiff), multiple fixed points, then again, is the same as having an unspecified system (too loose). This leads to a series of results in that field: Characterization of classical non-causal correlations, closed time- like curves that do not restrict the actions of experimenters, and a self-referential model of computation. We study the computational power of this model and use it to upper bound the computational power of closed time-like curves. Time travel has ever since been term weird, what we show here, however, is that time travel is not too crazy: It is not possible to solve hard problems by traveling through time. Finally, we apply our results on causal loops to other fields: an analysis with Kolmogorov complexity, local and classical simulation of PR-box correlations with closed time-like curves, and a short note on self-referentiality in language

On the Inherent Incompleteness of Scientific Theories

We examine the question of whether scientific theories can ever be complete. For two closely related reasons, we will argue that they cannot. The first reason is the inability to determine what are “valid empirical observations”, a result that is based on a self-reference Gödel/Tarski-like proof. The second reason is the existence of “meta-empirical” evidence of the inherent incompleteness of observations. These reasons, along with theoretical incompleteness, are intimately connected to the notion of belief and to theses within the philosophy of science: the Quine-Duhem (and underdetermination) thesis and the observational/theoretical distinction failure. Some puzzling aspects of the philosophical theses will become clearer in light of these connections. Other results that follow are: no absolute measure of the informational content of empirical data, no absolute measure of the entropy of physical systems, and no complete computer simulation of the natural world are possible. The connections with the mathematical theorems of Gödel and Tarski reveal the existence of other connections between scientific and mathematical incompleteness: computational irreducibility, complexity, infinity, arbitrariness and self-reference. Finally, suggestions will be offered of where a more rigorous (or formal) “proof” of scientific incompleteness can be found

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

A Redemption of Meaning in Three Novels by Italo Calvino

In this thesis I present three readings of Italo Calvino’s later novels: Invisible Cities (1972), If on a winter’s night a traveler (1979), and Mr. Palomar (1983). My primary aim is to defend Calvino against dominant scholarly interpretations that position him as a Postmodern nihilist, a literary trickster interested solely in toying with the mechanics of language. My analysis of Calvino’s work re-envisions him as a special breed of Postmodernist concerned with humanity’s ability to create spaces for meaning in spite of an indifferent cosmos. Drawing from psychoanalytic theory, cognitive science, analytic philosophy, and phenomenology, I synthesize my own critical lens to demonstrate Calvino’s ecstatic faith in human creativity. I claim that Calvino’s later novels contain a fundamentally ethical message: they call on us to live our lives with the intensity and vigilance of the artist, to see the world as a landscape for the collective life of our minds, to endure “the inferno of the living” through acts of creation

Discovering the Harmony of Reason and Faith in the Symphony of Eternal Creation

Tensions between the domain of reason and the domain of faith have been one of the most controversial issues in the history of our civilization for over three hundred years. They have contributed to many divisions, conflicts, and even wars. Contributions that have sought to reconcile the two domains have largely used the cultural approach in trying to solve this problem. The approach used in this essay views faith and reason from the perspective of cognitive operations. It shows that viewed from this perspective, faith and reason emerge as two aspects of the process of creation of new levels of organization that takes place in the human mind. The essay correlates faith and reason with such cognitive operations as equilibration and the production of disequilibrium. This approach shows that there is no fundamental ontological contradiction between faith and reason, and that cooperation between them is not only possible but is actually essential for sustaining our mental work and the survival of our civilization

Discovering the Harmony of Reason and Faith in the Symphony of Eternal Creation

Tensions between the domain of reason and the domain of faith have been one of the most controversial issues in the history of our civilization for over three hundred years. They have contributed to many divisions, conflicts, and even wars. Contributions that have sought to reconcile the two domains have largely used the cultural approach in trying to solve this problem. The approach used in this essay views faith and reason from the perspective of cognitive operations. It shows that viewed from this perspective, faith and reason emerge as two aspects of the process of creation of new levels of organization that takes place in the human mind. The essay correlates faith and reason with such cognitive operations as equilibration and the production of disequilibrium. This approach shows that there is no fundamental ontological contradiction between faith and reason, and that cooperation between them is not only possible but is actually essential for sustaining our mental work and the survival of our civilization

Speaking in circles: completeness in Kant's metaphysics and mathematics

This dissertation presents and responds to the following problem. For Kant a field of enquiry can be a science only if it is systematic. Most sciences achieve systematicity through having a unified content and method. Physics, for example, has a unified content, as it is the science of matter in motion, and a unified method because all claims in physics must be verified through empirical testing. In order for metaphysics to be a science it also must be systematic. However, metaphysics cannot have a unified content or method because metaphysicians lack a positive conception of what its content and method are. On Kant's account, metaphysicians can say with certainty what metaphysics does not study and what methods it cannot use, but never how it should proceed. Without unified content and method systematicity can only be guaranteed by some either means, namely, completeness. Without completeness metaphysics cannot have systematicity and every science must be systematic. Completeness can only be achieved if we severely limit the scope of metaphysics so that it contains only the conditions for the possibility of experience. This dissertation defends the claims made about the centrality of completeness in understanding Kant's conception of metaphysics as a science in two ways. First, the first two chapters point to a substantial body of textual evidence that supports the idea that Kant was directly concerned about the notion of completeness and links it to his conception of metaphysics as a science. Chapters 3 and 4 consider some possible objections to thinking that metaphysics as a science can be complete, giving special consideration to Gödel's incompleteness theorem. Chapter 5 explains why, if this position is as clear as this dissertation has argued, previous scholars have failed to acknowledge it. Giving a full answer to this question requires considering the general neglect of the "Doctrine of Method" section of Kant's primary theoretical text, The Critique of Pure Reason. The Doctrine of Method contains many of the passages which most directly support my thesis. Chapter 6 explains why scholars have ignored this important passage and argues that they should not continue to do so