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.

The foundation of set theory between theory and practice

Venturi G
2014-01-01

Abstract

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.
File in questo prodotto:
File Dimensione Formato  
philosophiascientiae-912.pdf

accesso aperto

Tipologia: Versione finale editoriale
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 685.73 kB
Formato Adobe PDF
685.73 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1163608
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? ND
social impact