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João Vitor Schmidt on Frege's Definition of the Ancestral Relation

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João Vitor Schmidt on Frege's Definition of the Ancestral Relation Universidade Estadual de Campinas Instituto de Filosofia e Ciências Humanas João Vitor Schmidt On Frege’s definition of the Ancestral Relation: logical and philosophical considerations Sobre a definição fregeana da Relação Ancestral: considerações lógicas e filosóficas CAMPINAS 2017 João Vitor Schmidt On Frege’s definition of the Ancestral Relation: logical and philosophical considerations Sobre a definição fregeana da Relação Ancestral: considerações lógicas e filosóficas Dissertação apresentada ao Instituto de Filosofia e Ciências Humanas da Universidade Estadual de Campinas como parte dos req- uisitos exigidos para a obtenção do título de Mestre em Filosofia. Dissertation presented to the Institute of Phi- losophy and Human Sciences of the University of Campinas in partial fulfillment of the re- quirements for the degree of Master in the area of Philosophy. Orientador: Marco Antonio Caron Ruffino Este exemplar corresponde à versão final da dissertação defendida pelo aluno João Vitor Schmidt, e orientada pelo Prof. Dr. Marco Antonio Caron Ruffino. Campinas 2017 Agência(s) de fomento e nº(s) de processo(s): CNPq, 131481/2015-0 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Filosofia e Ciências Humanas Paulo Roberto de Oliveira - CRB 8/6272 Schmidt, João Vitor, 1987- Sch52o SchOn Frege's definition of the ancestral relation : logical and philosophical considerations / João Vitor Schmidt. – Campinas, SP : [s.n.], 2017. SchOrientador: Marco Antonio Caron Ruffino. SchDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. Sch1. Frege, Gottlob, 1848-1925. 2. Kant, Immanuel, 1724-1804. 3. Filosofia contemporânea. 4. Lógica simbólica e Matemática. 5. Matemática - Filosofia. I. Ruffino, Marco Antonio Caron,1963-. II. Universidade Estadual de Campinas. Instituto de Filosofia e Ciências Humanas. III. Título. Informações para Biblioteca Digital Título em outro idioma: Sobre a definição fregeana da relação ancestral : considerações lógicas e filosóficas Palavras-chave em inglês: Contemporary philosophy Symbolic and mathematical logic Mathematics - Philosophy Área de concentração: Filosofia Titulação: Mestre em Filosofia Banca examinadora: Marco Antonio Caron Ruffino [Orientador] Dirk Greimann Marcelo Esteban Coniglio Data de defesa: 12-09-2017 Programa de Pós-Graduação: Filosofia Powered by TCPDF (www.tcpdf.org) Universidade Estadual de Campinas Instituto de Filosofia e Ciências Humanas A Comissão Julgadora dos trabalhos de Defesa de Dissertação de Mestrado composta pelos professores Doutores a seguir descritos, em sessão pública realizada em 12 de Setembro de 2017, considerou o candidato João Vitor Schmidt aprovado. Prof. Dr. Marco Antonio Caron Ruffino Prof. Dr. Dirk Greimann Prof. Dr. Marcelo Esteban Coniglio A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processo de vida acadêmica do aluno. Aos meus ancestrais: Orlando Schmidt Eugênia Regis (in memorian) Gino Freddo Ivany Augustina Moretto (in memorian) Agradecimentos A todos aqueles que, direta ou indiretamente, tornaram este trabalho possível. Ao CNPq, pela concessão da bolsa. Aos meus familiares, Eliane, Jair, Gina e Marcos Schmidt, condições sem as quais este trabalho não seria possível. A todos os colegas da UNICAMP, pelas amizades e incontáveis aprendizados, em espe- cial Bruno Mendonça, Edson Bezerra, Henrique Almeida, Igor Camargo, Rodrigo Costa e Vincenzo Ciccarelli. Aos professores Dirk Greimann, Emiliano Boccardi, Giorgio Venturi, Marcelo Coniglio e Walter Carnielli pela disposição em ler e contribuir com a redação do texto final. Em especial a meu orientador Marco Ruffino, pela aceitação e acolhimento em Campinas e pela rica orientação e amizade dispostas ao longo da pesquisa. Finalmente à Aniely Cristina Mussoi, companheira e amiga incansável, que com todo amor e incentivo fez deste trabalho tanto dela quanto meu. Resumo Neste trabalho, examinamos a famosa definição fregeana da Relação Ancestral em seus aspectos lógicos e filosóficos. O logicismo de Frege é o tema central: demonstrar as bases lógicas da aritmética. Isto é feito ao mostrar-se a natureza lógica dos números e a natureza lógica do raciocínio matemático. Ao fazê-lo, faz-se importante mostrar como os números nat- urais constituem uma série ordenada. Esta é a tarefa da definição do Ancestral. Com isso em mente, focaremos nos seguintes tópicos. Na introdução, o problema sobre séries será discu- tido, seguido de uma introdução do oponente mais famoso de Frege sobre isso: Kant. Isto será feito no Capítulo 1. Entender a filosofia de Kant é o passo inicial para as motivações filosófi- cas de Frege. Elas serão então discutidas no Capítulo 2, juntamente às suas reinterpretações da terminologia kantiana. Terminaremos este capítulo discutindo a lógica conceitográfica de Frege. No capítulo 3, introduzimos um sistema simples para lógica de segunda ordem, necessário para avaliar a definição de Frege e fiel o bastante às suas motivações filosóficas. Então, o Ancestral será finalmente e detalhadamente introduzido e discutido. Além disso, provamos os teoremas necessários para o logicismo de Frege, similarmente ao modo como o próprio antecipou na Begriffsschrift e Die Grundlagen der Arithmetik. Isso culminará no que hoje é conhecido como o Teorema de Frege. Finalmente, no capítulo 4, discutimos um dos problemas da definição de Frege: sua suposta circularidade. Mais precisamente, ar- gumentamos contra essa conclusão, acrescentando que o Ancestral, embora impredicativo, não é prejudicial como esperado, dadas as motivações filosóficas e lógicas de Frege perante ela. Palavras-chave: Frege, Definição Ancestral, Logicismo, Kant, Teorema de Frege. Abstract In this work, we examine Frege’s famous definition of the Ancestral Relation both in its philosophical and logical aspects. Frege’s logicism is the main theme on both: to show the logical grounds of arithmetic. This is done by showing the logical nature of numbers and the logical nature of mathematical reasoning. In doing so, it’s important to show how the natural numbers constitute an ordered series. This is the task of the Ancestral. With that in mind, we focus on the following topics. In the Introduction, the problem about series is discussed, followed by an assessment of Frege’s most famous opponent regarding it: Kant. This is done in chapter 1. Understanding Kant’s philosophy is the starting point for Frege’s own philosophical motivations. In Chapter 2, we avaluate them, alongside Frege’s own inter- pretations of the kantian notions. We finish the chapter by introducing Frege’s concept-script logic. In Chapter 3 we introduce a simple system for second-order logic, necessary to evaluate Frege’s definition and faithful enough to his philosophical motivations. Then, the Ancestral is finally and thoroughly introduced and discussed. Most importantly, we prove the theorems necessary for Frege’s logicism, similarly as himself envisaged in the Begriffsschrift and Die Grundlagen der Arithmetik. This culminates in what is nowadays known as Frege’s Theo- rem. Finally, in chapter 4, we discuss one of the problems of Frege’s definition: its alleged circularity. More precisely, we argue against this conclusion, adding that the Ancestral, al- though impredicative, is not as harmful as supposed, given Frege’s philosophical and logical motivations for it. Keywords: Frege, Ancestral definition, Logicism, Kant, Frege’s Theorem. List of Abbreviations 1. Frege’s Works BS - Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, translated by William T. Bynum as Conceptual notation: A formula lan- guage of pure thought modelled upon the formula language of arithmetic, (FREGE, 1992) GLA - Die Grundlagen der Arithmetik, translated by John Austin as The Foundations of Arithmetic, (FREGE, 1953) GGA - Grundgesetze der Arithmetik, translated by Philip Ebert and Marcus Rossberg as The Basic Laws of Arithmetic, (FREGE, 2013) PMC - Wissenschaftlicher Britfwechsel, translated by Felix Klein as Philosophical and Mathematical Correspondence, (FREGE, 1980) PW - Nachgelassene Schriften und Wissenschaftlicher Briefwechsel, translated by Pe- ter Long and Roger White as Posthumous Writings, (FREGE, 1979) CP - Kleine Schriften, translated by Max Black, V. H. Dudman et. al., edited by Brian McGuinness as Collected Papers on Mathematics, Logic, and Philosophy, (FREGE, 1984) 2. Kant’s Works CPR - Kritik der Reinen Vernunft, translated and edited by Paul Guyer and Allen Wood as Critique of Pure Reason, (KANT, 1998) JL - Immanuel Kant’s Logik, ein Handbuch zu Vorlesungen, translated by J. Michael Young as Immanuel Kant’s Logic - A manual for lectures in (KANT, 1992). Contents Introduction 12 1 Kant’s Proposal 21 1.1 Kant’s Critical Philosophy . 21 1.2 Aesthetic and Intuitions . 25 1.3 Logic and Concepts . 31 1.4 The place of Time in human thinking . 36 1.5 Time, Numbers and Arithmetic . 39 1.6 Conclusion of Kant’s proposal . 46 2 Frege’s Proposal 48 2.1 Philosophical Motivations . 49 2.1.1 The denial of Kantian Intuitions . 52 2.1.2 Fregean Analyticity . 56 2.1.3 Fregean Definitions, Conceptual-Contents . 59 2.1.4 Logic and Informativity . 67 2.2 The Logic of the Concept-script . 70 3 Frege’s Ancestral 79 3.1 Concept-script logic reconsidered . 79 3.2 The Ancestral of a relation . 85 3.3 Important Theorems . 93 3.3.1 Induction . 94 3.3.2 Transitivity . 96 3.3.3 Trichotomy . 98 3.3.4 (IP) and (CP), once again . 100 3.4 The route to Frege’s Theorem . 104 4 The impredicativity of Frege’s Ancestral 114 4.1 Kerry and Angelelli . 115 4.1.1 The Circularity of (FA) . 117 4.2 The circularity revisited . 119 4.2.1 Frege’s Reduction . 120 4.2.2 Is there a circularity? . 123 4.2.3 Some further points . 127 4.3 Conclusion . 129 Concluding Remarks 132 A Arithmetic in GLA 136 Bibliography 149 Introduction This work have one major theme: Frege’s famous definition of the Ancestral of a Relation. But the approach here taken is twofold: to discuss such definition in the philosophical per- spective, and to show its accomplishments logicwise. Both perspectives are mutually related. The quest for the foundations of arithmetic is the starting point for both, and the problem of series is one of the main fields of discussion.
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