Foundation of Mathematics between Theory and Practice

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to...

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Bibliographic Details
Main Author: Giorgio Venturi
Format: Article
Language:deu
Published: Éditions Kimé 2014-03-01
Series:Philosophia Scientiæ
Online Access:http://journals.openedition.org/philosophiascientiae/912
Description
Summary:In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics.
ISSN:1281-2463
1775-4283