文章基本信息

标题：Hilbert, Completeness and Geometry
其他标题：Hilbert, Completeness and Geometry
本地全文：下载
作者：Giorgio Venturi
期刊名称：Rivista Italiana di Filosofia Analitica Junior
电子版ISSN：2037-4445
出版年度：2018
卷号：9
期号：2
页码：152-180
DOI：10.13130/2037-4445/11092
语种：English
出版社：Rivista Italiana di Filosofia Analitica Junior
摘要：This paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert’s conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert’s foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert’s position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert’s for the completeness of set theory.