Visual thinking and simplicity in proof

TL;DR

This paper explores how spatial thinking and visual representations, like Ferrers diagrams, can enhance the simplicity and explanatory power of proofs, contrasting visual and non-visual approaches.

Contribution

It analyzes the role of spatial reasoning in proof simplicity, highlighting how diagrams can reduce complexity and improve surveyability in mathematical proofs.

Findings

Diagrams can avoid technical calculations and case divisions.

Spatial thinking can create more explanatory proofs.

An example shows intermediate proofs can be less simple than both visual and non-visual proofs.

Abstract

This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers, and comparing them to proofs that do not use spatial thinking. The analysis shows that using diagrams and spatial thinking can contribute to simplicity by (for example) avoiding technical calculations, division into cases, and induction, and creating a more surveyable and explanatory proof (both of which are connected to simplicity). In response to one part of Hilbert's 24th Problem, the area between two proofs is explored in one example, showing that between a proof that uses spatial reasoning and one that does not, there is a proof that is less simple than either.