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Origins of Generic Point in Algebraic Geometry and Mathematical Practice

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Origins of Generic Point in Algebraic Geometry and Mathematical Practice UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de Filosofia e Ciências Humanas Renato Reis Leme Origens do ponto genérico em geometria algébrica e na prática matemática Origins of generic point in algebraic geometry and mathematical practice CAMPINAS 2020 Renato Reis Leme Origins of generic point in algebraic geometry and mathematical practice Origens do ponto genérico em geometria algébrica e na prática matemática 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 Philosophy and Human Sciences of the Uni- versity of Campinas in partial fulfillment of the requirements for the degree of Master of Philosophy. Supervisor/Orientador: Giorgio Venturi ESTE TRABALHO CORRESPONDE À VER- SÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO RENATO REIS LEME, E ORI- ENTADA PELO PROF. DR. GIORGIO VEN- TURI. Campinas 2020 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Filosofia e Ciências Humanas Cecília Maria Jorge Nicolau - CRB 8/3387 Leme, Renato Reis, 1993- L542o LemOrigins of generic point in algebraic geometry and mathematical practice / Renato Reis Leme. – Campinas, SP : [s.n.], 2020. LemOrientador: Giorgio Venturi. LemDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. Lem1. Lógica simbólica e matemática. 2. Geometria algébrica. 3. Matemática - História. 4. Matemática - Filosofia. I. Venturi, Giorgio, 1983-. II. Universidade Estadual de Campinas. Instituto de Filosofia e Ciências Humanas. III. Título. Informações para Biblioteca Digital Título em outro idioma: Origens do ponto genérico em geometria algébrica e na prática matemática Palavras-chave em inglês: Logic, symbolic and mathematical Algebraic geometry Mathematics - History Mathematics - Philosophy Área de concentração: Filosofia Titulação: Mestre em Filosofia Banca examinadora: Giorgio Venturi [Orientador] Rodrigo de Alvarenga Freire Bruno Ramos Mendonça Data de defesa: 18-12-2020 Programa de Pós-Graduação: Filosofia Identificação e informações acadêmicas do(a) aluno(a) - ORCID do autor: https://orcid.org/0000-0001-8199-1261 - Currículo Lattes do autor: http://buscatextual.cnpq.br/buscatextual/visu 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 Mes- trado, composta pelos Professores Doutores a seguir descritos, em sessão pública realizada em 18/12/2020, considerou o candidato Renato Reis Leme aprovado. Prof Dr Giorgio Venturi Prof Dr Rodrigo de Alvarenga Freire Prof Dr Bruno Ramos Mendonça A Ata de Defesa com as respectivas assinaturas dos membros encontra-se no SIGA/Sistema de Fluxo de Dissertações/Teses e na Secretaria do Pro- grama de Pós-Graduação em Filosofia do Instituto de Filosofia e Ciências Humanas. To The Timeless Agradecimentos Agradeço ao meu orientador Giorgio Venturi pelas aulas, reuniões de orientação e, acima de tudo, pelo exemplo de pesquisador. Agradeço aos colegas do grupo de pesquisa Edson Bezerra, João Vitor Schmidt, Pedro Yago e Santiago Jockwich, e aos colegas do IMECC Fábio Meneghetti e Vinícius Vasconcelos pelas discussões que ajudaram a dissipar dúvidas e desbravar o caminho, por vezes tortuoso, das ideias. Agradeço aos membros da banca pela cuidadosa leitura e valiosas sugestões, sem as quais este trabalho não estaria completo. Agradeço a Jessica Andrade por me fazer acreditar que sempre posso mais, e a minha família pelo apoio incondicional que, nos piores momentos, fez saber que não estou sozinho. Agradeço a todos os funcionários do IFCH pelo profissionalismo e prestatividade tão importantes para o dia a dia do Instituto. Agradeço a todos os professores e funcionários do CLE por preservarem e manterem vivo e atuante este centro de excelência de pesquisa em Lógica. Por fim, agradeço a Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, processo nº 19/01580-7) pelo fundamental apoio financeiro ao longo da condução desta pesquisa. Resumo O presente trabalho explora o fenômeno da genericidade e suas conexões em três eixos centrais: histórico, filosófico e lógico. No capítulo histórico deste trabalho, é apresentado o desenvolvimento da noção de genericidade desde suas origens no pensamento ocidental até a emergência do conceito de ponto genérico em geometria algébrica. No capítulo filosófico, partindo da crítica fregeana aos objetos variáveis na matemática, é avaliada a relação entre genericidade e arbitrariedade. Por fim, no capítulo lógico, mostramos a conexão entre gene- ricidade e possibilidade a partir da semântica topológica de McKinsey-Tarski para o cálculo modal 푆4. Palavras-chave: Lógica simbólica e matemática, Geometria algébrica, Matemática - História, Matemática - Filosofia. Abstract The present work explores genericity phenomena and its connections over three central axis: historical, philosophical and logical. In the historical chapter of this work, the devel- opment of the notion of genericity is presented from its origins in Western thought to the emergence of the concept of generic point in algebraic geometry. In the philosophical chapter, based on the Fregean critique of variable objects in mathematics, the relation between gener- icity and arbitrariness is evaluated. Finally, in the logical chapter, we show the connection between genericity and possibility based on McKinsey-Tarski’s topological semantics for the modal calculus 푆4. Keywords: Logic symbolic and mathematical, Algebraic geometry, Mathematics - His- tory, Mathematics - Philosophy. List of Figures 1.1 Our metaphysical target. 14 1.2 Our six-sided die. 15 2.1 Plato’s square. 20 2.2 An artist drawing with his perspective machinery. “Man drawing a Lute” – By Albrecht Dürer (1530)............................. 22 2.3 A circle of radius equal to 1 illustrating Descartes concept of coordinated system. 23 2.4 Vanishing point at the horizon. 25 2.5 Photo taken in July 1935, during the first official conference of Bourbaki, in Besse-et-Saint-Anastaise. Standing, from left to right: Henri Cartan, René de Possel, Jean Dieudonné, André Weil and Luc Olivier. Sitting, from left to right: a “guinea pig” called Mirles, Claude Chevalley and Szolem Mandelbrojt. 28 2.6 푆푝푒푐(Z). See Mumford [7]. 33 2.7 Mumford’s arithmetic surface........................... 34 3.1 A topological space with generic point. 40 3.2 Begriffsschrift language. 46 3.3 (((퐴 → 퐵) → 퐴) → 퐴) [Peirce’s law] . 47 3.4 A topological representation of 휔휔........................ 51 4.1 Topological spaces. Examples from Mumkres [17]. 57 4.2 Graph representation of generic classes. 61 4.3 Plots of the subspaces generated by the power-set of arbitrary sets with cardi- nality equals to 2, 3, 4, 5 and 6, respectively, all filtered generic sets and they ideal counterpart. 62 Contents 1 Introduction 12 1.0.1 Generic property . 13 2 From Plato to Weil and beyond: Genericity through the history of math- ematics 16 2.1 Introduction . 17 2.2 An ancient dual . 18 2.3 The renaissance of geometry . 22 2.4 Italian school of algebraic geometry . 24 2.5 Foundations reframed . 27 2.5.1 Bourbaki and the struggle for formal refinements . 28 2.5.2 The generic point find its definition . 30 2.6 Beyond foundations . 31 2.6.1 On how generic points looks like . 33 2.7 Conclusion . 35 3 Genericity and arbitrariness 36 3.1 Genericity . 37 3.2 Arbitrariness . 40 3.2.1 Frege’s target . 43 3.3 Structuralist theories of arbitrariness . 48 3.3.1 Arbitrary as function . 49 3.3.2 Aribtrary as generic . 52 4 A logic for genericity 55 4.1 Genericity in mathematics . 55 4.1.1 Generic points . 56 4.1.2 Generic points in (푋, 풫(푋)) ....................... 58 4.1.3 Combinatorial properties . 61 4.2 Generic propositions . 63 4.2.1 Boolean model for generic proposition . 63 4.2.2 S4 logic . 63 4.3 Tying things up . 66 5 Conclusion 68 References 69 A Isabelle Proof of Inf Theorem 71 B Isabelle Proof of Sup Theorem 72 C A PC-Begriffsschrift translator 73 12 Chapter 1 Introduction The limits of my language mean the limits of my world. Tractatus Logico-Philosophicus, 5.6 Since the dawn of human language, genericity has been employed by those who wants to communicate a plethora of things by sign only a single one. In an attempt to differentiate this from mere ambiguity, this linguistical phenomenon has been analyzed by several theories. Because it is useful, we cannot set that apart from our expressivity apparatus. Now, we try to understand how it works. In natural language, genericity often comes with a certain puzzlement. Consider the following generic sentences, taken from [18]: (i) Ravens are black. (ii) Frenchmen eat horsemeat. (iii) Prime numbers are odd. As Nickel observes, if we take generic to be true “just in case at least one member of the kind conforms to it”, then we would take it right in (푖) and (푖푖), but would make a mistake in the case of (푖푖푖), which we would say that is true. On the other hand, if we say that generic is true if, and only if, most members of the kind conform to it, then we would predict rightly in the case of (푖) and (푖푖푖), but wrongly in the case of (푖푖). Both ways entails problems, which is a clue that generic sentences should not be saw as a proposition that hiddes a fixed form. In the Introduction of [16], the authors traces a difference between three types of research on genericity: 13 Linguistic research on generics has developed in three primary directions. Some authors have focused on the generic interpretation of the subject DP. This research has mostly concentrated on the notion of kind and its pos- sible expression in natural languages. Other authors locate the sources of genericity in the VP. And a third group, starting from the assumption that genericity is a feature of the sentence rather than one of its components, have focused on the variety of generic readings of generic sentences.
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