Visual thinking and simplicity of proof

Cain, Alan J.

http://hdl.handle.net/10362/75100

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Autores

Cain, Alan J.

Resumo(s)

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 yet more impure than either. This has implications for the supposed simplicity of impure proofs.

Descrição

The author was supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through an Investigador FCT research fellowship (IF/01622/2013/CP1161/CT0001). This work was partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and the projects PTDC/MHC-FIL/2583/2014 and PTDC/MAT-PUR/31174/2017.