Abstract. This paper explores reasoning about space and time, e.g., in metaphors of time as space; an important method is to find minimal assumptions needed to reach the same conclusions that humans reach. Some mathematical language, including the notion of triad, is introduced for this purpose, formalizing and generalizing the cognitive semantics approaches to conceptual spaces (in the senses of both Fauconnier & Turner and of Gärdenfors), blending, and metaphor; in particular, continuous mathematics is used to model space and time. A new explanation of emergent structure in blend spaces is also discussed, and proposed as a source of creativity. Four main examples illustrate the approach, and an appendix encapsulates the most difficult mathematics. 1

Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego- and temporal-reference-point models. Dynamic models appear to be quasi-universal cross-culturally, as does the generalization that in Ego-reference-point models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; linguistic and gestural data give strong confirmation of this unusual culture-specific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns?