The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...) which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

Languages around the world use a recurring strategy to discuss abstract concepts: describe them metaphorically, borrowing language from more concrete domains. We “plan ahead” to the future, “count up” to higher numbers, and “warm” to new friends. Past work has found that these ways of talking have implications for how we think, so that shared systems of linguistic metaphors can produce shared conceptualizations. On the other hand, these systematic linguistic metaphors might not just be the cause but also the effect (...) of shared, non‐linguistic ways of thinking. Here, we present a case study of a variety of American English in which a shared, non‐linguistic conceptualization of time has become crystallized as a new system of linguistic metaphors. Speakers of various languages, including English, conceptualize time as a lateral timeline, with the past leftward and the future rightward. Until now, this conceptualization has not been documented in the speech of any language. In two studies, we document how members of the U.S. military, but not U.S. civilians, talk about time using conventionalized lateral metaphors (e.g., “move the meeting right” to mean “move the meeting later”). We argue that, under the right cultural circumstances, implicit mental representations become conventionalized metaphors in language. (shrink)

Speakers of many languages prefer allocentric frames of reference (FoRs) when talking about small‐scale space, using words like “east” or “downhill.” Ethnographic work has suggested that this preference is also reflected in how such speakers gesture. Here, we investigate this possibility with a field experiment in Juchitán, Mexico. In Juchitán, a preferentially allocentric language (Isthmus Zapotec) coexists with a preferentially egocentric one (Spanish). Using a novel task, we elicited spontaneous co‐speech gestures about small‐scale motion events (e.g., toppling blocks) in Zapotec‐dominant (...) speakers and in balanced Zapotec‐Spanish bilinguals. Consistent with prior claims, speakers’ spontaneous gestures reliably reflected either an egocentric or allocentric FoR. The use of the egocentric FoR was predicted—not by speakers’ dominant language or the language they used in the task—but by mastery of words for “right” and “left,” as well as by properties of the event they were describing. Additionally, use of the egocentric FoR in gesture predicted its use in a separate nonlinguistic memory task, suggesting a cohesive cognitive style. Our results show that the use of spatial FoRs in gesture is pervasive, systematic, and shaped by several factors. Spatial gestures, like other forms of spatial conceptualization, are thus best understood within broader ecologies of communication and cognition. (shrink)