Formal notations as diagrams of abstract structure

When we look at drawings, what makes those images realistic is that they engage properties of our perceptual system which usually respond to real physical objects and situations. Treating depictions as things makes it possible to infer facts about images just as one would about their referents. I suggest that people can and do interact with numbers, letters, and formal signs in the same way, treating them as literal physical objects such that learned interactions with those objects guarantee successful formal behavior. One implication is that understanding higher-level thinking requires examining in detail the ways that people perceive, construct, and interact with these physical representations. I present experimental studies which establish that people create and perceive formally irrelevant aspects of the physical layout of simple arithmetic, algebraic, and logical expression. Two formal models attempt to capture these embodied aspects of computation.