# Foundations of Mathematics: Philosophical Inquiries on Real Numbers

**Reading Group - A.A. 2016-2017 - I-II Semester**

**VII.**

**Title: **Foundations of Mathematics: Philosophical Inquiries on Real Numbers

**Convener: **F. Baracco

**Contact**: fla.baracco@gmail.com

**Committed: **C. Mariani

**Attendees: **L. Conti, D. Quadrellaro, G. Martino, L. Zanetti.

The aim of this reading group is to investigate the notion of real number. At the turn of the XX century several proposals were suggested to develop an account of real numbers. It was usually thought that geometric (intuitive) reasoning was misleading and a rigorous arithmetical definition of real numbers was necessary. The arithmetization of analysis marked a turning point in the history of mathematics: geometry was not regarded as a solid basis for mathematical reasoning anymore whereas it was analysis itself to be considered suitable to grasp geometric intuitions. Focusing on this historical period seems to be a promising starting points for our philosophical inquiry on real numbers. We aim to investigate some of these accounts shedding light on philosophical issues concerning the development of mathematical account of real numbers.

Each section will take several hours, since the morning to the early afternoon. In this way we wish to promote an effective dialectic on these topics. Below is shown the main reference that each attendee is supposed to read before joining the reading group. Depending on which issues will be raised during each section, we aim to select some references that will be the basis for the discussion in the following section (some of them are listed below).

**Main Reference (english version will be provided if necessary)**

Mangione C., Bozzi S. (2012), *Storia della Logica*, cap. III-IV, CUEM

**Further References (english version will be provided if necessary)**

- Dauben J. W. (1979),
*George Cantor*, Princeton University Press - Feferman S. (1964),
*The Number Systems: Foundation of Algebra and Analysis*, - Frege G. (1903),
*Basic Law of Arithmetic*, vol. II, Oxford University Press - Frege G. (1988),
*Ricerche Logiche*, Guerini - Hale B. (2000),
*Real by Abstraction*, Philosophia Mathematica, 8, (2), pp. 100-123 - Husserl E. (2015),
*Ricerche Logiche*, il Saggiatore - Kline M. (1972),
*Mathematical Thought from Ancient to Modern Times*, Oxford University Press - Weyl H. (1918),
*The Continuum*, Dover