Centro de Matemática e Aplicações , Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia , 2829-516 Caparica , Portugal.
Philos Trans A Math Phys Eng Sci. 2019 Mar 11;377(2140):20180032. doi: 10.1098/rsta.2018.0032.
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. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.
本文通过分析一组主要涉及费雷尔图（整数划分的可视化表示）的例证证明，以及将其与不使用空间思维的证明进行比较，研究了空间思维如何与简单性在非形式证明中相互作用。分析表明，使用图表和空间思维可以通过（例如）避免技术计算、分类和归纳，以及创建更具可观察性和解释性的证明（这两者都与简单性有关）来有助于简单性。针对希尔伯特第 24 个问题的一部分，本文在一个例子中探讨了两个证明之间的区域，结果表明，在使用空间推理的证明和不使用空间推理的证明之间，存在一个证明，它不如任何一个证明简单，但比任何一个证明都更不纯。这对不纯证明的所谓简单性产生了影响。本文是主题为“简单证明的概念-希尔伯特第 24 个问题”的一部分。
