Origins of generic point in algebraic geometry and mathematical practice
Non-deterministic matrices: theory and applications to algebraic semantics
| Grant number: | 16/25891-3 |
| Support Opportunities: | Research Grants - Young Investigators Grants |
| Start date: | January 01, 2018 |
| End date: | December 31, 2021 |
| Field of knowledge: | Humanities - Philosophy - Logic |
| Principal Investigator: | Giorgio Venturi |
| Grantee: | Giorgio Venturi |
| Host Institution: | Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil |
| Associated researchers: | Hugo Luiz Mariano ; Marco Antonio Caron Ruffino ; Mattia Petrolo ; Rodrigo de Alvarenga Freire ; Sourav Tarafder |
| Associated research grant(s): | 19/12527-0 - Algebra-valued models of non-classical set theories, AV.EXT |
| Associated scholarship(s): | 20/08016-7 - Speech acts in Mathematics: hypothetical illocutions,
BP.PD 20/07998-0 - Modal logics and ontologies for possible worlds, BP.IC 20/08214-3 - A study on ontology of numbers, BP.IC + associated scholarships - associated scholarships |
Abstract
With this project, I propose to investigate the notion of arbitrary object using historical, philosophical and logical methodologies. The privileged perspective from which the analysis will be conducted is that of set theory. The project is divided in three complementary axes. First, we propose a general reconstruction of the theorization about arbitrary objects, stressing the role that concepts and their extensions played in shaping axiomatic set theory. Moreover, we plan to reconstruct the origin, and to understand the meaning, of the use of generic objects in the more abstract fields of contemporary Mathematical practice. Later, from a more philosophical perspective, we plan to to confront the concept of arbitrary set with the notions of set based either on Goedel's iterative conception or on Cantor's principle of limitation of size. From a more intensional perspective on the theory of sets, we plan to study whether arbitrary objects or arbitrary reference may clarify the ontological or semantical problems of a process of abstraction à la Frege. To this aim, we propose a philosophy of Mathematical language inspired by Searle's theory of speech acts. Finally, we plan to analyze more formally the notion of arbitrariness by means of that of genericity, on which the technique of Forcing rests. We propose to axiomatize genericity in an abstract setting. Moreover we intent to apply modal logic (RI-logics) in order to capture what is invariant under Forcing (Omega-logic), in the attempt to obtain insights in the solution of the Omega-conjecture. Finally, we propose to use Forcing itself in order to study the relative or absolute character of the notion of genericity. (AU)
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