Pierre Pica

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

Education Enhances the Acuity of the Nonverbal Approximate Number System

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction.

Flexible intuitions of Euclidean geometry in an Amazonian indigene group

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucus estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics.

Exact Equality and Successor Function: Two Key Concepts on the Path towards Understanding Exact Numbers

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS: the fact that all numbers can be generated by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucú (an Amazonian language), and young Western children (3–4 years old) understand these fundamental properties of numbers.

Atomism and Binding

This book contains several chapters dealing with Binding Theory, that is the theory dealing with the distribution and interpretation of definite descriptions, pronouns and reflexives. The general procedure by which a model is built out of various relatively independent submodules is referred to by the term Atomism.

Non-symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the arrays numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Comparing Biological Motion Perception in Two Distinct Human Societies

Cross cultural studies have played a pivotal role in elucidating the extent to which behavioral and mental characteristics depend on specific environmental influences. Surprisingly, little field research has been carried out on a fundamentally important perceptual ability, namely the perception of biological motion. In this report, we present details of studies carried out with the help of volunteers from the Mundurucu indigene, a group of people native to Amazonian territories in Brazil. We employed standard biological motion perception tasks inspired by over 30 years of laboratory research, in which observers attempt to decipher the walking direction of point-light (PL) humans and animals. Do our effortless skills at perceiving biological activity from PL animations, as revealed in laboratory settings, generalize to people who have never before seen representational depictions of human and animal activity? The results of our studies provide a clear answer to this important, previously unanswered question. Mundurucu observers readily perceived the coherent, global shape depicted in PL walkers, and experienced the classic inversion effects that are typically found when such stimuli are turned upside down. In addition, their performance was in accord with important recent findings in the literature, in the abundant ease with which they extracted direction information from local motion invariants alone. We conclude that the effortless, veridical perception of PL biological motion is a spontaneous and universal perceptual ability, occurring both inside and outside traditional laboratory environments.

The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers

During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them.

Theoretical Implications of the Study of Numbers and Numerals in Mundurucu

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed by the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term ‘low compositionality power’, accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages.

Knowledge of Number and Knowledge of Language: Number as a Test Case for the Role of Language in Cognition

The relationship between language and conceptual thought is an unresolved problem in both philosophy and psychology. It remains unclear whether linguistic structure plays a role in our cognitive processes. This special issue brings together cognitive scientists and philosophers to focus on the role of language in numerical cognition: because of their universality and variability across languages, number words can serve as a fruitful test case to investigate claims of linguistic relativism.