This thesis addresses an issue in the philosophy of Mathematics which is little discussed, and indeed little recognised. This issue is the phenomenon of a ‘change of setting’. Changes of setting are events which involve a change in a scientific framework which is fruitful for answering questions which were, under an old framework, intractable. The formulation of the new setting usually involves a conceptual re-orientation to the subject matter. In the natural sciences, such re-orientations are arguably unremarkable, inasmuch as it is possible that within the former setting for one’s thinking one was merely in error, and under the new orientation one is merely getting closer to the truth of the matter. However, when the subject matter is pure mathematics, a problem arises in that mathematical truth is (in appearance) timelessly immutable. The conceptions that had been settled upon previously seem not the sort of thing that could be vitiated. Yet within a change of setting that is just what seems to happen. Changes of setting, in particular in their effects on the truth of individual propositions, pose a problem for how to understand mathematical truth. Thus this thesis aims to give a philosophical analysis of the phenomenon of changes of setting, in the spirit of the investigations performed in Wilson (1992) and Manders (1987) and (1989). It does so in three stages, each of which occupies a chapter of the thesis: 1. An analysis of the relationship between mathematical truth and settingchanges, in terms of how the former must be viewed to allow for the latter. This results in a conception of truth in the mathematical sciences which gives a large role to the notion that a mathematical setting must ‘explain itself’ in terms of the problems it is intended to address. 2. In light of (1), I begin an analysis of the change of setting engendered in mathematical logic by Gottlob Frege. In particular, this chapter will address the question of whether Frege’s innovation constitutes a change of setting, by asking the question of whether he is seeking to answer questions which were present in the frameworks which preceded his innovations. I argue that the answer is yes, in that he is addressing the Kantian question of whether alternative systems of arithmetic are possible. This question arises because it had been shown earlier in the 19th century that Kant’s conclusion, that Euclid’s is the only possible description of space, was incorrect. 3. I conclude with an in-depth look at a specific aspect of the logical system constructed in Frege’s Grundgesetze der Arithmetik. The purpose of this chapter is to find evidence for the conclusions of chapter two in Frege’s technical work (as opposed to the philosophical). This is necessitated by chapter one’s conclusions regarding the epistemic interdependence of formal systems and informal views of those frameworks. The overall goal is to give a contemporary account of the possibility of setting-changes; it will turn out that an epistemic grasp of a mathematical system requires that one understand it within a broader, somewhat historical context.

[Abridged abstract] The goal is to give structural proof-theoretical analyses of logical constants, and thereby provide a criterion for what a logical constant is. Another goal is to illustrate the thesis that structural assumptions of logic are basic and that alternative logics (later called substructural logics) differ from each other only in their structural assumptions, and not in their assumptions about logical constants.

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture.

No description available.

The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.

No description available.

The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H¹(k(x), μ_n). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.

FARIAS, Pablo Mayckon Silva. Um estudo sobre as origens da Lógica Matemática e os limites da sua aplicabilidade à formalização da Matemática. 2007. 110 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2007. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-12T14:54:53Z No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-20T13:48:23Z (GMT) No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) / Made available in DSpace on 2016-07-20T13:48:23Z (GMT). No. of bitstreams: 1 2007_dis_pmsfarias.pdf: 859405 bytes, checksum: 9d580356cce3820f228499085b2e3cde (MD5) Previous issue date: 2007 / This work is a study about the origins of Mathematical Logic and the limits of its applicability to the formal development of Mathematics. Firstly, Dedekind’s arithmetical theory is presented, which was the first theory to provide a precise definition for natural numbers and to demonstrate relying on it all facts commonly known about them. Peano’s axiomatization for Arithmetic is also presented, which in a sense simplified Dedekind’s theory. Then, Frege’s Begriffsschrift is presented, the formal language from which modern Logic originated, and in it are represented Frege’s basic definitions concerning the notion of number. Afterwards, a summary of important topics on the foundations of Mathematics from the first three decades of the twentieth century is presented, beginning with the paradoxes in Set Theory and ending with Hilbert’s formalist doctrine. At last, are presented, in general terms, Gödel’s incompleteness. theorems and Turing’s computability concept, which provided precise answers to the two most important points in Hilbert’s program, to wit, a direct proof of consistency for Arithmetic and the decision problem, respectively. Keywords: 1. Mathematical Logic 2. Foundations of Mathematics 3. Gödel’s incompleteness theorems / Este trabalho é um estudo sobre as origens da Lógica Matemática e os limites da sua aplicabilidade ao desenvolvimento formal da Matemática. Primeiramente, é apresentada a teoria aritmética de Dedekind, a primeira teoria a fornecer uma definição precisa para os números naturais e com base nela demonstrar todos os fatos comumente conhecidos a seu respeito. É também apresentada a axiomatização da Aritmética feita por Peano, que de certa forma simplificou a teoria de Dedekind. Em seguida, é apresentada a ome{german}{Begriffsschrift} de Frege, a linguagem formal que deu origem à Lógica moderna, e nela são representadas as definições básicas de Frege a respeito da noção de número. Posteriormente, é apresentado um resumo de questões importantes em fundamentos da Matemática durante as primeiras três décadas do século XX, iniciando com os paradoxos na Teoria dos Conjuntos e terminando com a doutrina formalista de Hilbert. Por fim, são apresentados, em linhas gerais, os teoremas de incompletude de Gödel e o conceito de computabilidade de Turing, que apresentaram respostas precisas às duas mais importantes questões do programa de Hilbert, a saber, uma prova direta de consistência para a Aritmética e o problema da decisão, respectivamente.

Submitted by Claudinei Pracidelli (cpracide@ipen.br) on 2015-01-21T10:14:57Z No. of bitstreams: 0 / Made available in DSpace on 2015-01-21T10:14:57Z (GMT). No. of bitstreams: 0 / Dissertação (Mestrado em Tecnologia Nuclear) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

Orientador: Walter Alexandre Carnielli / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas / Made available in DSpace on 2018-08-17T17:02:46Z (GMT). No. of bitstreams: 1 Araujo_Anderson_D.pdf: 1286485 bytes, checksum: 1e51db7a5721f4affeaf8f512d23269e (MD5) Previous issue date: 2011 / Resumo: Esta tese propõe uma nova abordagem da computabilidade de Turing clássica, denominada abordagem modelo-teórica. De acordo com essa abordagem, estruturas e teorias são associadas às máquinas de Turing a fim de investigar as características de suas computações. Uma abordagem modelo-teórica da computabilidade de Turing através da lógica de primeira ordem é desenvolvida, e resultados de correspondência, correção, representação e completude entre máquinas, estruturas e teorias de Turing são demonstrados. Nessa direção, os resultados obtidos a respeito de propriedades tais como estabilidade, absoluticidade, universalidade e logicidade enfatizam as potencialidades da computabilidade modelo-teórica de primeira ordem. Demonstra-se que a lógica subjacente às teorias de Turing é uma lógica minimal intuicio-nista, sendo capaz, inclusive, de internalizar um operador de negação clássico. As técnicas formuladas nesta tese permitem, sobretudo, investigar a computabilidade de Turing em modelos não-padrão da aritmética. Nesse contexto, uma nova perspectiva acerca do fenômeno de Tennenbaum e uma avaliação crítica da abordagem de Dershowitz e Gurevich da tese de Church-Turing sào apresentadas. Como conseqüência, postula-se um princípio de interna-lidade aritmética na computabilidade, segundo o qual o próprio conceito de computação é relativo ao modelo aritmético em que as máquinas de Turing operam. Assim, a tese unifica as caracterizações modelo-aritméticas do problema P versus NP existentes na literatura, revelando, por fim, uma barreira modelo-aritmética para a possibilidade de solução desse problema central em complexidade computacional no que diz respeito a certos métodos. Em sua totalidade, a tese sustenta que características cruciais do conceito de computação podem ser vislumbradas a partir da dualidade entre finitude e infinitude presente na distinção entre números naturais padrão e não-padrão / Abstract: This PhD thesis proposes a new approach to classical Turing computability, called a model-theoretic approach. In that approach, structures and theories are associated to Turing machines in order to study the characteristics of their computations. A model-theoretic approach to Turing computability through first-order logic is developed, and first results about correspondence, soundness, representation and completeness among Turing machines, structures and theories are proved. In this line, the results about properties as stability, absoluteness, universality and logicality emphasize the importance of the model-theoretic standpoint. It is shown that the underlying logic of Turing theories is a minimal intuicionistic logic, being able to internalize a classical negation operator. The techniques obtained in the present dissertation permit us to examine the Turing computability over nonstandard models of arithmetic as well. In this context, a new perspective about Tennenbaum's phenomenon and a critical evaluation of Dershowitz and Gurevich's account on Church-Turing's thesis are given. As a consequence, an arithmetic internality principle is postulated, according to which the concept of computation itself is relative to the arithmetic model that Turing machines operate. In this way, the dissertation unifies the existing model-arithmetic characterizations of the P versus NP problem, leading, as a by-product, to a model-arithmetic barrier to the solvability of that central problem in computational complexity with respect to certain techniques. As a whole, the dissertation sustains that crucial characteristics of the concept of computation may be understood from the duality between finiteness and infiniteness inherent within the distinction between standard and nonstandard natural numbers / Doutorado / Filosofia / Doutor em Filosofia