Debates over how geometry is understood and learned date back at least to the days of Plato, with more recent scholars concluding that only humans possess the foundations of this understanding. However, a new analysis by New York University psychology professor Moira Dillon concludes that geometry's foundations are shared by humans and a variety of other animals—from rats to chickens to fish.
"Our ability to think geometrically may not come from a built-in, uniquely human 'math module' in the brain, but rather from the same cognitive systems that help humans, as well as animals, find their way home," explains Dillon, whose work appears in the journal Trends in Cognitive Sciences. "Put another way, our understanding of geometry may very well come from wandering rather than from worksheets."
While Plato and, later, Descartes and Kant all debated the origins of geometry and the role of cognition in its beginnings, only in the latter half of the 20th century did scientists start testing how it is learned.
The more prominent assessments link the cognitive origins of geometry to a version of the "language-of-thought" hypothesis. Here, multiple formal systems in the mind—or mental "languages"—help us humans flourish in topics that are unique to us, like mathematics and music.
More broadly, these assessments view such languages as a singularly human endeavor, allowing us to build on our understanding of simple concepts to comprehend more complex ones. The proposed mental language for geometry centers specifically on Euclidean geometry—the properties of lines, shapes, and points on flat surfaces—and posits that simple concepts like "parallelism" and "perpendicularity" are already built into the human mind from birth.
However, Dillon draws upon decades of research in showing that the foundations of our geometric understanding are instead rooted in navigation-like mental processes shared by humans and animals that merely approximate Euclidean geometry, not capturing it exactly—what she calls the Wanderers Hypothesis for Geometry.
"Animals that have never studied angles or triangles can navigate efficiently and simulate that navigation in their minds to plan their routes—and even babies seem to understand something about distance, direction, and shape," adds Dillon, whose 2023 study found that infants could outperform AI in certain cognitive tasks. "These abilities rely on geometry that captures some, but not all, of the properties of Euclidean geometry."
So what sets us humans apart from other animals?
Dillon suggests that the answer is not in a geometry-specific language, but our regular, natural language.
"Language allows humans to call upon the geometry used for navigation and use it in new ways, for example, to solve geometry problems in our minds without actually going anywhere," she explains. "Language allows us to mentally wander unlike any other animal can."
The Trends in Cognitive Sciences article considers studies on how both adults and children across cultures learn geometry and perceive shapes, experimental results in which both humans and animals perceive and navigate spaces, and the design of AI systems, such as Google DeepMind's AlphaGeometry.
Together, Dillon's review of these studies reveals that our human ability to learn geometry may originate from evolutionarily inherited, navigation-like mental processes that get transformed for us only because we humans are the only animals who can talk.
