Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains.

The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor "state changes are like motions" plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.