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 is going on. In particular we need to explain how people understand spatial structure and motion, and how we can think about objects in terms of a basic topological structure with more or less additional metrical information. I shall try to explain why 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, such as infinite ordinals. More questions are asked than answered.