Made available in DSpace on 2016-04-28T14:16:18Z (GMT). No. of bitstreams: 1 Danilo Gustavo Bispo.pdf: 2512902 bytes, checksum: 2261f415993066c8892733480af9c1c9 (MD5) Previous issue date: 2013-04-15 / In this paper I present initially in order to contextualize the influences involved in the emergence of the theory of Alan Turing computability on a history of some issues that mobilized mathematicians in the early twentieth century. In chapter 1, an overview will be exposed to the emergence of ideology Formalist designed by mathematician David Hilbert in the early twentieth century. The aim was to base the formalism elementary mathematics from the method and axiomatic theories eliminating contradictions and paradoxes. Although Hilbert has not obtained full success in your program, it will be demonstrated how their ideas influenced the development of the theory of computation Turing. The theory proposes that Turing is a decision procedure, a method that analyzes any arbitrary formula of logic and determines whether it is likely or not. Turing proves that there can be no general decision. For that will be used as a primary source document On Computable Numbers, with an application to the Entscheidungsproblem. In Chapter 2, you will see the main sections of the document Turing exploring some of its concepts. The project will be completed with a critique of this classic text in the history of mathematics based on historiographical proposals presented in the first chapter / Neste texto apresento inicialmente com o intuito de contextualizar as influências envolvidas no surgimento da teoria de Alan Turing sobre computabilidade um histórico de algum problemas que mobilizaram os matemáticos no início do século XX. No capítulo 1, será exposto um panorama do surgimento da ideologia formalista concebida pelo matemático David Hilbert no início do século XX. O objetivo do formalismo era de fundamentar a matemática elementar a partir do método e axiomático, eliminando das teorias suas contradições e paradoxos. Embora Hilbert não tenha obtido pleno êxito em seu programa, será demonstrado como suas concepções influenciaram o desenvolvimento da teoria da computação de Turing. A teoria que Turing propõe é um procedimento de decisão, um método que analisa qualquer fórmula arbitrária da lógica e determina se ela é provável ou não. Turing prova que nenhuma decisão geral pode existir. Para tanto será utilizado como fonte primária o documento On computable numbers, with an application to the Entscheidungsproblem. No capítulo 2, será apresentado as principais seções do documento de Turing explorando alguns de seus conceitos. O projeto será finalizado com uma crítica a este texto clássico da história da matemática com base nas propostas historiográficas apresentadas no primeiro capítulo

The central question of this thesis is: what is mathematics about? The answer arrived at by the thesis is an unsettling and unsatisfying one. By examining two of the most promising contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of giving us a conclusive answer to our question. The conclusion is arrived at by a combination of historical and conceptual analysis. It begins with the historical fact that, since the middle of the nineteenth century, mathematics has undergone a radical transformation. This transformation occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier images of geometry understood it as the science of space. In the wake of the emergence of multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to the description of physical space, the old picture of Euclidean geometry as the sole correct description of physical space was no longer tenable. The first chapter of the dissertation provides an historical account of some of the forces which led to the destabilization of the traditional picture of geometry. The second chapter examines the debate between Gottlob Frege and David Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting that Hilbert’s views are ultimately unsatisfying. The third chapter continues to probe the work of Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an impasse: the traditional understanding of mathematics cannot hold, but neither can the two most promising modern accounts. The fourth and final chapter of the thesis investigates mathematical structuralism—a more recent development in the philosophy of mathematics—in order to see whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued that the contemporary debate between ‘assertoric’ structuralists and ‘algebraic’ structuralists recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that neither of the two most promising contemporary accounts can offer us a satisfying philosophical answer to the question ‘what is mathematics about?’.

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<p>This sentence is whatever truth is <em>not</em>. The subject of this master’s thesis is the power, influence, and solvability of the liar paradox. This paradox can be constructed through the application of a standard conception of truth and rules of inference are applied to sentences such as the first sentence of this abstract. The liar has been a powerful problem of philosophy for thousands of years, from its ancient origin (examined in Chapter One) to a particularly intensive period in the twentieth century featuring many ingenious but ultimately unsuccessful solutions from brilliant logicians, mathematicians and philosophers (examined in Chapter Two, Chapter Three, and Chapter Four). Most of these solutions were unsuccessful because of a recurring problem known as the liar’s revenge; whatever truth is <em>not</em> includes, as it turns out, not <em>just</em> falsity, but also meaninglessness, ungroundedness, gappyness, and so on. The aim of this master’s thesis is to prove that we should not consign ourselves to the admission that the liar is and always will just be a paradox, and thus unsolvable. Rather, I argue that the liar <em>is</em> solvable; I propose and defend a novel solution which is examined in detail in the latter half of Chapter Two, and throughout Chapter Three. The alternative solution I examine and endorse (in Chapter Four) is not my own, owing its origin and energetic support to Graham Priest. I argue, however, for a more qualified version of Priest’s solution. I show that, even if we accept a very select few true contradictions, it should <em>not</em> be assumed that inconsistency inevitably spreads throughout other sets of sentences used to describe everyday phenomena such as motion, change, and vague predicates in the empirical world.</p> / Master of Arts (MA)

A philosophic account of quantum physics. The thesis is divided into two parts. Part I is dedicated to laying the groundwork of quantum physics, and explaining some of the primary difficulties. Subjects of interest will include the principle of locality, the quantum uncertainty principle, and Einstein's criterion for reality. Quantum dilemmas discussed include the double-slit experiment, observations of spin and polarization, EPR, and Bell's theorem. The first part will argue that mathematical-physical descriptions of the world fall short of explaining the experimental observations of quantum phenomenon. The problem, as will be argued, is framework of the physical descriptive schema. Part I includes in-depth discussions of mathematical principles. Part II will discuss the Copenhagen interpretation as put forth by its founders. The Copenhagen interpretation will be expressed as a paradox: The classical physical language cannot describe quantum phenomenon completely and with certainty, yet this language is the only possible method of articulating the physical world. The paradox of Copenhagen will segway into Kant's critique of metaphysics. Kant's understanding of causality, things-in-themselves, and a priori synthetic metaphysics. The thesis will end with a conclusion of the quantum paradox by juxtaposing anti-materialist Martin Heidegger with quantum founder Werner Heisenberg. Our conclusion will be primarily a discussion of how we understand the world, and specifically how our understanding of the world creates potential for truth.