An Inquiry into the Practice of Proving in Low-Dimensional Topology

In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-116 (2015)
Authors
Silvia De Toffoli
Princeton University
Valeria Giardino
Centre National de la Recherche Scientifique
Abstract
The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology.
Keywords Visualization  Topological pictures  Manipulative imagination
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

Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures. Zurich, Switzerland: Birkhäuser. pp. 25-50.
Fuzzy Topology Representation for MV‐Algebras.Jialu Zhang & Quanfa Chen - 2009 - Mathematical Logic Quarterly 55 (3):259-270.
On Some Uses and Abuses of Topology in the Social Analysis of Technology.N. Marres - 2012 - Theory, Culture and Society 29 (4-5):288-310.
Diagrams and Proofs in Analysis.Jessica Carter - 2010 - International Studies in the Philosophy of Science 24 (1):1 – 14.
The nature of the topological intuition.L. B. Sultanova - 2016 - Liberal Arts in Russia 5 (1):14-21.
Laver’s Results and Low-Dimensional Topology.Patrick Dehornoy - 2016 - Archive for Mathematical Logic 55 (1-2):49-83.
Natural Topology.Frank Waaldijk - 2012 - Brouwer Society.
The Topology of Justification.Sergei Artemov & Elena Nogina - 2008 - Logic and Logical Philosophy 17 (1-2):59-71.
Similarity, Topology, and Physical Significance in Relativity Theory.Samuel C. Fletcher - 2016 - British Journal for the Philosophy of Science 67 (2):365-389.

Analytics

Added to PP index
2017-08-23

Total downloads
85 ( #78,956 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