Abstract
When people reason formally, they often make use of special notations—algebra and arithmetic are familiar examples. These notations are often treated as mere shorthand—a concise way of referring to meaningful mathematical concepts. Other authors have argued that people treat notations as pictures—literal diagrams of an imagined set of objects (Dörfler, 2003; Landy Goldstone, 2009). If notations depict objects that exist in space, then it makes sense to wonder how they are arranged not just in the two visible dimensions, but in depth. In four experiments, we find a consistent pattern: properties that increase mathematical precedence also tend to make objects appear closer in space. This alignment of formal pressures and informal pressures suggests that perceived depth may play a role in supporting computational reasoning processes. Although our primary focus is documenting the existence of depth illusions in notations, we also evaluate several sources of information that might guide depth judgments: availability of an object for computational actions, formal syntactic structure, relative symbol salience and voluntary attention shifts. We consider relationships between these nonexclusive possible sources of information in guiding how people judge depth in mathematics.
