Gödel's Mathematical Intuition and Platonism

This paper has two objectives. The first is to present an interpretation of Gödel's concept of mathematical intuition and defend it against other interpretations like Charles Parsons'. The second objective is to show the necessity of realism for Gödel's mathematical intuition. The first section seeks to show what mathematical intuition is and how it works, focusing on Gödel's works and unpublished texts. Consequently, from this section, I will show that, for Gödel, the concept of mathematical intuition emerges and develops parallel to his platonistic ontological commitment. Gödel's Platonism and mathematical intuition involve not only an ontological dimension but also an epistemological dimension. In the second section, I will discuss Parsons' 1995 paper "Platonism and mathematical intuition in Kurt Gödel's thought", in which he argues for a separation between mathematical intuition and Gödel's Platonism. What I will show is that this separation is not possible in Gödel since according to the recent publications of his philosophical notebooks and works prior to 1964, mathematical intuition was already implicit and, contrary to what Parsons argues, it is not something that arises before and independently of Gödel's realism

Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part

The Contemporary Encyclopedic Novel

This dissertation will define the contemporary American encyclopedic novel and the significant role that irony plays in shaping meaning. The dissertation constructs a model of the encyclopedic novel based upon the history of the encyclopedia – from Denis Diderot\u27s Enlightenment influenced Encyclopédie – and Northrop Frye\u27s conception of the encyclopedic form. It claims (1) that the contemporary encyclopedic novel continues in the cycle of modal progression toward mythic integration that Frye proposes in Anatomy of Criticism; and (2) that the encyclopedic novel utilizes different forms of irony to challenge authoritative discourse and elevate marginal discourse. The first chapter defines the encyclopedic novel by examining the history of the encyclopedia and existing criticism on the encyclopedic text in literature. It draws on theorists such as Denis Diderot and Richard Yeo to define an “encyclopedic project” that adopts a dialogic rhetorical style and seeks to democratize access to information. This chapter also defines the encyclopedic novel as a generic form that combines other forms into a unified whole and utilizes irony as a tool for integration. The second and third chapters form a thematic pairing that shows the self-reflexive progression of the encyclopedic novel from individualistic to humanistic focus. The second chapter argues that Thomas Pynchon\u27s Gravity\u27s Rainbow is an “anarchistic encyclopedic novel” that promotes associational thinking – in the form of paranoia, open forms, and horizontal transmission of knowledge. Gravity\u27s Rainbow adopts a disintegrative irony to empower the oppressed individual against industry-state collusion in the post-WWII era. The third chapter argues that David Foster Wallace\u27s Infinite Jest seeks to reinvent irony as an integrative force and redirect Pynchon\u27s individualistic anarchism toward an inclusive humanism. The fourth chapter demonstrates a break from both of the preceding chapter and argues that Leon Forrest\u27s Divine Days adopts a syndetic model of composition that further works to incorporate forms and integrate irony. Using Northrop Frye\u27s “interpenetration,” I argue that Divine Days integrates competing traditions and discourses by demonstrating their mutual-necessity. In the concluding chapter, I examine “meta-encyclopedic” by Jorge Luis Borges and Roberto Bolaño as an extension of the dissertation

Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined

Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it