We are wanderers: Abstract geometry reflects spatial navigation.

Abstract

Philosophers throughout history have debated the relations between the abstract geometry of formal mathematics and the physical geometry of the natural world. We provide evidence that abstract geometry reflects the geometry humans and nonhuman animals use for spatial navigation. Across two preregistered experiments, educated adults watched short videos of two points and two line segments forming an open figure on an otherwise blank screen. These simple visuals were described with sparse and minimally different language, creating different spatial contexts. After watching each video, participants were asked to click: anywhere (anywhere condition); to complete the triangle (triangle condition); where the next corner of the object would be (object condition); where the next stop on the agent's path would be (navigation condition); or where the next point on the abstract surface would be (abstract condition). Across spatial contexts, participants produced responses that reflected strikingly different sets of geometric representations; in particular, preserving distance and direction for open paths in the navigation condition but preserving length and angle for closed shapes in the object condition. In the navigation and abstract contexts, however, the elicited geometry was remarkably similar. Human language may thus effectively isolate phylogenetically ancient geometric representations used for navigating the physical world and recognizing the objects in it. Moreover, the cognitive origins of uniquely human abstract geometry may lie in representations used for navigating the physical world. Public Significance StatementSince antiquity, geometry has been the model of human abstract thought. But where do our ideas about geometry come from? Are our geometric concepts rooted in everyday experience? Our work informs debates over the origins of geometry. Using only minimally different language, our experiments were able to get adults to think differently about the same simple geometric forms. We humans are able to think about geometry in many different ways, through our experiences with places or objects. Most surprisingly, however, when we think about abstract geometry, we wander the Euclidean plane like we wander the earth. There is a connection between how we navigate the world and how we think about formal geometry. Perhaps, we suggest, we can harness this connection to inform the development of mathematics education.

How do babies come to know what babies know?

“…In this commentary, I make two main points. First, I describe new evidence from a recent behavioral experiment in my lab that core knowledge about places and forms is indeed still present and active in educated human adults, consistent with Spelke's proposal (Lin & Dillon, 2023). My evidence complements evidence Spelke has put forward insofar as my tasks, unlike the tasks she reviews, relied only on simple and minimally contrastive linguistic descriptionswith no actual navigation or form analysisto elicit core geometry of places and forms.…”

Turtle-like Geometry Learning: How Humans and Machines Differ in Learning Turtle Geometry

“…Recent work in the field of psychology shows that infants possess two different systems in the domain of geometry: a form system which is used for object detection and a place system that is used when navigating (Spelke and Kinzler 2007;Dillon 2023). Furthermore, researchers have shown that humans tend to prioritize the place system over the form system when perceiving abstract geometric shapes (Lin and Dillon 2023).…”

“…Recent work in the field of psychology shows that infants possess two different systems in the domain of geometry: a form system which is used for object detection and a place system that is used when navigating (Spelke and Kinzler 2007;Dillon 2023). Furthermore, researchers have shown that humans tend to prioritize the place system over the form system when perceiving abstract geometric shapes (Lin and Dillon 2023). We believe such findings resonate with the ability of young children (and adults) to learn Turtle geometry, a form of geometry that can be explored through programming, in contrast to Euclidean geometry as traditionally taught in schools.…”