Envisioning Transformations – The Practice of Topology

In Brendan Larvor (ed.), Mathematical Cultures. Zurich, Switzerland: Birkhäuser. pp. 25-50 (2016)
Authors
Silvia De Toffoli
Princeton University
Valeria Giardino
Centre National de la Recherche Scientifique
Abstract
The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology – the proof of Alexander’s theorem – is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to the emergence of philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on the notation, and the constant and fruitful back-and-forth from one representation to another in dealing with mathematical content. Finally, some suggestions for further research are given in the conclusions.
Keywords Mathematical practice  Manipulative imagination  Topology
Categories (categorize this paper)
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

An Inquiry Into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-116.
Axioms in Mathematical Practice.D. Schlimm - 2013 - Philosophia Mathematica 21 (1):37-92.
Categorical Foundations and Mathematical Practice.C. McLarty - 2012 - Philosophia Mathematica 20 (1):111-113.
Fuzzy Topology Representation for MV-Algebras.Jialu Zhang & Quanfa Chen - 2009 - Mathematical Logic Quarterly 55 (3):259-270.
The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford University Press.
On the Formal Points of the Formal Topology of the Binary Tree.Silvio Valentini - 2002 - Archive for Mathematical Logic 41 (7):603-618.
Fuzzy Topology Representation for MV‐Algebras.Jialu Zhang & Quanfa Chen - 2009 - Mathematical Logic Quarterly 55 (3):259-270.

Analytics

Added to PP index
2017-08-23

Total downloads
108 ( #60,307 of 2,308,512 )

Recent downloads (6 months)
37 ( #11,408 of 2,308,512 )

How can I increase my downloads?

Monthly downloads

My notes

Sign in to use this feature