Visualisation and proof: a brief survey of philosophical perspectives

Abstract

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.

Mathematically Gifted High School Students’ Approaches to Developing Visual Proofs (VP) and Preliminary Ideas about VP

“…A study of Borwein and Jörgenson (2001) is the only study, which roughly mentions an investigation system/criteria that could be useful while examining VPs. Here, the researchers discuss there are three requirements to accept a visual proof: 1-reliability, 2-consistency, and 3-repeatability (Borwein & Jörgenson, 2001 On the other hand, Hanna and Sidoli (2007) claim that these criteria cannot be completely applicable either to visual proof or to proof in general. It is particularly due to the "reliability" criteria, which is not expressed clear enough to show or decide whether a visual proof is reliable or not (Hanna & Sidoli, 2007;Hanna, 2000).…”

On the Persuasiveness of Visual Arguments in Mathematics

“…whether proofs must be, at least in principle, completely independent of any picture). A brief review of these different philosophical positions on the status of pictures in proofs was presented by Hanna and Sidoli (2007). Rather, we focus on a particular aspect of this discussion: the formation of belief from visual arguments.…”

“…In this paper we focus upon one further factor which may influence the persuasiveness of visual mathematical arguments: the presence or absence of descriptive text. Hanna and Sidoli (2007) pointed out that in the philosophical debate regarding the role of visual arguments in mathematics some authors give particular importance to the verbal/symbolic text that accompanies the picture, which may be seen as the result of "extracting" the information implicitly presented in the visual representation. Here we present two experiments which were designed to interrogate the effect of the presence or absence of a passage of descriptive text on mathematics undergraduates' and research-active mathematicians' evaluations of a visual argument.…”

Formas del discurso y razonamiento configural de estudiantes para maestros en la resolución de problemas de geometría

“…Un aspecto del conocimiento de la geometría por parte del maestro está relacionado con el desarrollo de la visualización (Battista, 2007(Battista, , 2008Brown y Weatley, 1997;Presmeg, 2006) y los procesos de exploración e indagación vinculados a esta, que pueden favorecer el que los estudiantes establezcan relaciones entre las definiciones y las propiedades geométricas (Clemente y Llinares, 2013;Hanna y Sidoli, 2007). Por lo que la relación entre lo visual y el sistema lógico-deductivo es un primer paso para que los estudiantes puedan establecer relaciones entre las definiciones y propiedades geométricas (Hanna, 1998;Hershkowitz, 1990).…”