‘Chasing’ the Diagram—the Use of Visualizations in Algebraic Reasoning

Abstract

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation.

Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?

“…6 The case of visual proofs or proofs without words discussed here is a rather extreme one. Many recent studies in the philosophy of mathematical practice do not focus on these particular diagrams, but on those that play an important role in (contemporary) mathematical reasoning and that can even be part of a published modern proof (see, for instance, [15], [19] and [16]). See also footnote 8 in this context.…”

Free Rides in Logical Space Diagrams Versus Aristotelian Diagrams

“…Due to the semantic constraints on ADs, we know for sure that -whenever we get from γ to α by means of a dashed C-line in σ 1 , and from α to β by means of a full CD-line in σ 2 -there will be an SA-arrow from γ to β in σ 3 . This is a typical constellation of 'diagram chasing' [2] or 'transitive closure': if you first get from A to B and then from B to C, then you also get from A to C directly. On the target type level, Fig.…”

Diagrams in Intra-Configurational Analysis

“…A common question that many works on mathematical diagrams have tried and still try to answer is whether pieces of mathematical activity that indispensably require diagrams to achieve or justify some relevant result exist [Giaquinto 2007], [Carter 2010], [De Toffoli 2017]. In this context, one of the most studied domains is Euclidean (synthetic) geometry-the first four books of Euclid's Elements in particular.…”