Divisive language.

What language devises, it might divide. By exploring the relations among the core geometries of the physical world, the abstract geometry of Euclid, and language, I give new insight into both the persistence of core knowledge into adulthood and our access to it through language. My extension of Spelke's language argument has implications for pedagogy, philosophy, and artificial intelligence.

Infants' sensitivity to shape changes in 2D visual forms.

Research in developmental cognitive science reveals that human infants perceive shape changes in 2D visual forms that are repeatedly presented over long durations. Nevertheless, infants' sensitivity to shape under the brief conditions of natural viewing has been little studied. Three experiments tested for this sensitivity by presenting 128 seven-month-old infants with shapes for the briefer durations under which they might see them in dynamic scenes. The experiments probed infants' sensitivity to two fundamental geometric properties of scale- and orientation-invariant shape: relative length and angle. Infants detected shape changes in closed figures, which presented changes in both geometric properties. Infants also detected shape changes in open figures differing in angle when figures were presented at limited orientations. In contrast, when open figures were presented at unlimited orientations, infants detected changes in relative length but not in angle. The present research therefore suggests that, as infants look around at the cluttered and changing visual world, relative length is the primary geometric property by which they perceive scale- and orientation-invariant shape.

Places in the Brain: Bridging Layout and Object Geometry in Scene-Selective Cortex.

Diverse animal species primarily rely on sense (left-right) and egocentric distance (proximal-distal) when navigating the environment. Recent neuroimaging studies with human adults show that this information is represented in 2 scene-selective cortical regions-the occipital place area (OPA) and retrosplenial complex (RSC)-but not in a third scene-selective region-the parahippocampal place area (PPA). What geometric properties, then, does the PPA represent, and what is its role in scene processing? Here we hypothesize that the PPA represents relative length and angle, the geometric properties classically associated with object recognition, but only in the context of large extended surfaces that compose the layout of a scene. Using functional magnetic resonance imaging adaptation, we found that the PPA is indeed sensitive to relative length and angle changes in pictures of scenes, but not pictures of objects that reliably elicited responses to the same geometric changes in object-selective cortical regions. Moreover, we found that the OPA is also sensitive to such changes, while the RSC is tolerant to such changes. Thus, the geometric information typically associated with object recognition is also used during some aspects of scene processing. These findings provide evidence that scene-selective cortex differentially represents the geometric properties guiding navigation versus scene categorization.

Young Children's Use of Surface and Object Information in Drawings of Everyday Scenes.

Pictorial symbols such as photographs, drawings, and maps are ubiquitous in modern cultures. Nevertheless, it remains unclear how children relate these symbols to the scenes that they represent. The present work investigates 4-year-old children's (N = 144) sensitivity to extended surface layouts and objects when using drawings of a room to find locations in that room. Children used either extended surfaces or objects when interpreting drawings, but they did not combine these two types of information to disambiguate target locations. Moreover, children's evaluations of drawings depicting surfaces or objects did not align with their use of such information in those drawings. These findings suggest that pictures of all kinds serve as media in which children deploy symbolic spatial skills flexibly and automatically.

Core foundations of abstract geometry.

Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.

Core geometry in perspective.

Research on animals, infants, children, and adults provides evidence that distinct cognitive systems underlie navigation and object recognition. Here we examine whether and how these systems interact when children interpret 2D edge-based perspectival line drawings of scenes and objects. Such drawings serve as symbols early in development, and they preserve scene and object geometry from canonical points of view. Young children show limits when using geometry both in non-symbolic tasks and in symbolic map tasks that present 3D contexts from unusual, unfamiliar points of view. When presented with the familiar viewpoints in perspectival line drawings, however, do children engage more integrated geometric representations? In three experiments, children successfully interpreted line drawings with respect to their depicted scene or object. Nevertheless, children recruited distinct processes when navigating based on the information in these drawings, and these processes depended on the context in which the drawings were presented. These results suggest that children are flexible but limited in using geometric information to form integrated representations of scenes and objects, even when interpreting spatial symbols that are highly familiar and faithful renditions of the visual world.

We are wanderers: Abstract geometry reflects spatial navigation.

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. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Seeing the Forest but Naming the Trees: An Object-Over-Place Bias in Learning Noun Labels.

Objects and places are foundational spatial domains represented in human symbolic expressions, like drawings, which show a prioritization of depicting small-scale object-shape information over the large-scale navigable place information in which objects are situated. Is there a similar object-over-place bias in language? Across six experiments, adults and 3- to 4-year-old children were asked either to extend a novel noun in a labeling phrase, to extend a novel noun in a prepositional phrase, or to simply match pictures. To dissociate specific object and place information from more general figure and ground information, participants either saw scenes with both place information (a room) and object information (a block in the room), or scenes with two kinds of object information that matched the figure-ground relations of the room and block by presenting an open container with a smaller block inside. While adults showed a specific object-over-place bias in both extending novel noun labels and matching, they did not show this bias in extending novel nouns following prepositions. Young children showed this bias in extending novel noun labels only. Spatial domains may thus confer specific and foundational biases for word learning that may change through development in a way that is similar to that of other word-learning biases about objects, like the shape bias. These results expand the symbolic scope of prior studies on object biases in drawing to object biases in language, and they expand the spatial domains of prior studies characterizing the language of objects and places.

Tactile localization on digits and hand: structure and development.

Localization of tactile stimuli to the hand and digits is fundamental to somatosensory perception. However, little is known about the development or genetic bases of this ability in humans. We examined tactile localization in normally developing children, adolescents, and adults and in people with Williams syndrome (WS), a genetic disorder resulting in a wide range of severe visual-spatial deficits. Normally developing 4-year-olds made large stimulus-localization errors, sometimes across digits, but nevertheless their errors revealed a structured internal representation of the hand. In normally developing individuals, errors became exponentially smaller over age, reaching the adult level by adolescence. In contrast, people with WS showed large localization errors regardless of age and a significant proportion of cross-digit errors, a profile similar to that of normally developing 4-year-olds. Thus, tactile localization reflects internal organization of the hand even early in normal development, undergoes substantial development in normal children, and is susceptible to developmental, but not organizational, impairment under genetic deficit.

Mind-bending geometry: Children's and adults' intuitions about linearity on spheres.

Humans appear to intuitively grasp definitions foundational to formal geometry, like definitions that describe points as infinitely small and lines as infinitely long. Nevertheless, previous studies exploring human's intuitive natural geometry have consistently focused on geometric principles in planar Euclidean contexts and thus may not comprehensively characterize humans' capacity for geometric reasoning. The present study explores whether children and adults can reason about linearity in spherical contexts. We showed 48 children (age range: 6-8 years) and 48 adults from the U.S. Northeast two different paths between the same two points on pictures of spheres and asked them to judge which path was the most efficient for an actor to get from a starting point to a goal object. In one kind of trial, both paths looked curved in the pictures, and in another kind of trial, the correct curved-looking path was paired with an incorrect straight-looking path. Adults were successful on both kinds of trials, and although children often chose the incorrect straight-looking path, they were surprisingly successful at identifying the efficient path when comparing two that were curved. Children thus may build on a natural geometry that gives us humans intuitions that are not limited to the formal axioms of Euclidean geometry or even to the Euclidean plane. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Commonsense psychology in human infants and machines.

Human infants are fascinated by other people. They bring to this fascination a constellation of rich and flexible expectations about the intentions motivating people's actions. Here we test 11-month-old infants and state-of-the-art learning-driven neural-network models on the "Baby Intuitions Benchmark (BIB)," a suite of tasks challenging both infants and machines to make high-level predictions about the underlying causes of agents' actions. Infants expected agents' actions to be directed towards objects, not locations, and infants demonstrated default expectations about agents' rationally efficient actions towards goals. The neural-network models failed to capture infants' knowledge. Our work provides a comprehensive framework in which to characterize infants' commonsense psychology and takes the first step in testing whether human knowledge and human-like artificial intelligence can be built from the foundations cognitive and developmental theories postulate.

Euclid's Random Walk: Developmental Changes in the Use of Simulation for Geometric Reasoning.

Euclidean geometry has formed the foundation of architecture, science, and technology for millennia, yet the development of human's intuitive reasoning about Euclidean geometry is not well understood. The present study explores the cognitive processes and representations that support the development of humans' intuitive reasoning about Euclidean geometry. One-hundred-twenty-five 7- to 12-year-old children and 30 adults completed a localization task in which they visually extrapolated missing parts of fragmented planar triangles and a reasoning task in which they answered verbal questions about the general properties of planar triangles. While basic Euclidean principles guided even young children's visual extrapolations, only older children and adults reasoned about triangles in ways that were consistent with Euclidean geometry. Moreover, a relation beteen visual extrapolation and reasoning appeared only in older children and adults. Reasoning consistent with Euclidean geometry may thus emerge when children abandon incorrect, axiomatic-based reasoning strategies and come to reason using mental simulations of visual extrapolations.

Rooms without walls: Young children draw objects but not layouts.

Drawing is the epitome of uniquely human expression, with few known limits beyond those of our perceptual and motor systems and the cultures in and for which we draw. The present study evaluates whether the drawings of young children nevertheless reveal an early emerging bias in the depiction of 2 different foundational spatial categories: layouts and objects. Across 2 experiments following preregistered designs and analysis plans, 4-year-old children either sat in a colorful "fort" or looked at a small "toy" version of the fort and were asked to draw exactly what they saw. Children's drawings often omitted the walls composing the fort's layout but included the corresponding object parts for the toy. Symbolic representations of space in young children's drawings thus privilege small-scale objects over large and fixed layout geometry. A distinction between the intuitive geometries of layouts and objects leads to their differential treatment in both humans and other animals during everyday navigation. This distinction may also underlie the differential treatment of layouts and objects in children's drawings. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Bringing Home Baby Euclid: Testing Infants' Basic Shape Discrimination Online.

Online developmental psychology studies are still in their infancy, but their role is newly urgent in the light of the COVID-19 pandemic and the suspension of in-person research. Are online studies with infants a suitable stand-in for laboratory-based studies? Across two unmonitored online experiments using a change-detection looking-time paradigm with 96 7-month-old infants, we found that infants did not exhibit measurable sensitivities to the basic shape information that distinguishes between 2D geometric forms, as had been observed in previous laboratory experiments. Moreover, while infants were distracted in our online experiments, such distraction was nevertheless not a reliable predictor of their ability to discriminate shape information. Our findings suggest that the change-detection paradigm may not elicit infants' shape discrimination abilities when stimuli are presented on small, personal computer screens because infants may not perceive two discrete events with only one event displaying uniquely changing information that draws their attention. Some developmental paradigms used with infants, even those that seem well-suited to the constraints and goals of online data collection, may thus not yield results consistent with the laboratory results that rely on highly controlled settings and specialized equipment, such as large screens. As developmental researchers continue to adapt laboratory-based methods to online contexts, testing those methods online is a necessary first step in creating robust tools and expanding the space of inquiry for developmental science conducted online.

Geometric categories in cognition.

At the scale in which we live, space is continuous. Nevertheless, our perception and cognition parse the world into categories, whether physical, like scene or object, or abstract, like infinitesimal point or 7. The present study focuses on 2 categories of special angles in planar geometry, parallels and perpendiculars, and we evaluate how these categories might be reflected in adults' basic angle discrimination. In the first experiment, participants were most precise when detecting 2 parallel or perpendicular lines among other pairs of lines at different relative orientations. Detection was also enhanced for 2 connected lines whose angle approached 90°, with precision peaking at 90°. These patterns emerged despite large variations in the scales and orientations of the angle exemplars. In the second experiment, the enhanced detection of perpendiculars persisted when stimuli were rotated in depth, indicating a capacity to discriminate shapes based on perpendicularity in 3 dimensions despite large variation in angles' 2-dimensional projections. The results suggest that 2 categorical concepts which lie at the foundation of Euclidean geometry, parallelism and perpendicularity, are reflected in our discrimination of simple visual forms, and they pave the way for future studies exploring the developmental and evolutionary origins of these cognitive categories. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

From map reading to geometric intuitions.

The origins and development of our geometric intuitions have been debated for millennia. The present study links children's developing intuitions about the properties of planar triangles to their developing abilities to read purely geometric maps. Six-year-old children are limited when navigating by maps that depict only the sides of a triangle in an environment composed of only the triangle's corners and vice versa. Six-year-old children also incorrectly judge how the angle size of the third corner of a triangle varies with changes to the other two corners. These limitations in map reading and in judgments about triangles are attenuated, respectively, by 10 and 12 years of age. Moreover, as children get older, their map reading predicts their geometric judgments on the triangle task. Map reading thus undergoes developmental changes that parallel an emerging capacity to reason explicitly about the distance and angle relations essential to euclidean geometry. (PsycINFO Database Record

The statistical shape of geometric reasoning.

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task - planar triangle completion. An analysis of the distribution of participants' errors in localizing a fragmented triangle's missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants' estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our findings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

Cognitive science in the field: A preschool intervention durably enhances intuitive but not formal mathematics.

Many poor children are underprepared for demanding primary school curricula. Research in cognitive science suggests that school achievement could be improved by preschool pedagogy in which numerate adults engage children's spontaneous, nonsymbolic mathematical concepts. To test this suggestion, we designed and evaluated a game-based preschool curriculum intended to exercise children's emerging skills in number and geometry. In a randomized field experiment with 1540 children (average age 4.9 years) in 214 Indian preschools, 4 months of math game play yielded marked and enduring improvement on the exercised intuitive abilities, relative to no-treatment and active control conditions. Math-trained children also showed immediate gains on symbolic mathematical skills but displayed no advantage in subsequent learning of the language and concepts of school mathematics.

Children's expectations about training the approximate number system.

Humans possess a developmentally precocious and evolutionarily ancient approximate number system (ANS) whose sensitivity correlates with uniquely human symbolic arithmetic skills. Recent studies suggest that ANS training improves symbolic arithmetic, but such studies may engender performance expectations in their participants that in turn produce the improvement. Here, we assessed 6- to 8-year-old children's expectations about the effects of numerical and non-numerical magnitude training, as well as states of satiety and restfulness, in the context of a study linking children's ANS practice to their improved symbolic arithmetic. We found that children did not expect gains in symbolic arithmetic after exercising the ANS, although they did expect gains in ANS acuity after training on any magnitude task. Moreover, children expected gains in symbolic arithmetic after a good night's sleep and their favourite breakfast. Thus, children's improved symbolic arithmetic after ANS training cannot be explained by their expectations about that training.