# Arithmetic Notation…now in 3D!

- David Landy,
*University of Richmond*
- Sally Linkenauger,
*University of Virginia*

## 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.

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