Arbitrary objects and the concept of infinite

• Grant number: 19/21647-9
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: December 01, 2019
• End date: November 30, 2020
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Giorgio Venturi
• Grantee: Ívore Campos de Mira
• Host Institution: Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 16/25891-3 - Arbitrariness and genericity: or on how to speak of the unspeakable, AP.JP

Abstract

Since antiquity philosophers asked questions about the objects of a mathematical theory: is there any such object? What is the nature of such objects? Is there any difference between mathematical objects and natural objects? Are mathematical objects capable of substantiating the certainty of mathematics? The possible connection between arbitrary and mathematical objects has been noticed since that time. Plato, for instance, proposed that mathematics is about universals. In fact, platonic ideas represent the first interesting historical example of objects with an arbitrary character: objects capable of representing the essence of concrete objects of which they are abstractions. In the same way, the importance of the concept of infinity for mathematics is evident since antiquity: Aristotle distinguished between potential infinity and actual infinity to talk about how many are the numbers, while Eudoxus, another student of Plato, through his exhaustion method, dealt with primitive ideas of what would become the integral calculus. In this research, we plan, in the context of the project Arbitrariness and genericity. Or how to speak of the unspeakable, deepen the understanding of these topics through the analysis of the relation among a theory of arbitrary objects and the classical view of set theory as a theory of the infinite. (AU)