Diagrams in the Mind ?

Clearly we can solve problems by thinking about them. Sometimes we have the impression that in doing so we use words, at other times diagrams or images. Often we use both. What is going on when we use mental diagrams or images? This question is addressed in relation to the more general multi-pronged question: what are representations , what are they for, how many different types are they, in how many different ways can they be used, and what difference does it make whether they are in the mind or on paper? The question is related to deep problems about how vision and spatial manipulation work. It is suggested that we are far from understanding what’s going on. In particular we need to explain how people understand spatial structure and motion, and I’ll try to suggest that this is a problem with hidden depths, since our grasp of spatial structure is inherently a grasp of a complex range of possibilities and their implications. Two classes of examples discussed at length illustrate requirements for human visualisation capabilities. One is the problem of removing undergarments without removing outer garments. The other is thinking about infinite discrete mathematical structures. We can think with diagrams Consider the trick performed by Mr Bean (actually the actor Rowan Atkinson): removing his (stretchable) underpants without removing his trousers. 1 Is that really possible? Think about it if you haven’t previously done so.2 Is it possible to remove the underpants without removing the trousers (i.e. the waistband of the trousers remains The first draft of this paper located Mr Bean in a launderette. Toby Smith corrected me, pointing out that the shy Mr Bean was on the beach, and wished to remove his underpants then put on his swimming trunks, both without removing his trouse rs. On 29th July 1995 I posted Mr Bean’s problem as a followup to a discussion of achievements of AI in several internet new s groups (comp.ai, comp.ai.philosophy, sci.logic, sci.cog nitive) and received a number of interesting and entertaining comments . Chris Malcolm pointed out the similarity with the bra and swe ater problem, i.e. removing a bra without removing the sweater wo rn above it. Readers are invited to reinvent the jokes that were then posted, about which problem was easier for whom under which conditions. In particular, someone pointed out the distinc tio between difficulty due to unfamiliarityvsdifficulty due to being distracted. I have previously given audiences the task of finding out how many possible numbers of intersection (or tangent) points t here can be between a triangle and a circle in the same plane. It’s easier than Mr Bean’s problem, but many people miss out some cases unless prompted. constantly around the person’s waist), allowing only continuous changes of shape of the body and the underpants and trousers, e.g. stretching, bending, twisting, but no separation of anything into disconnected parts, no creation of new holes, etc.? Does it matter whether the waistband of the trousers is tight or not? Many people can answer this question by thinking about it and visualising the processes required, even if they have not seen Rowan Atkinson’s performance.