The origins of number

 

Invited speakers

Daniel Ansari, University of Westwrn Ontario
Elizabeth Brannon,University of Pennsylvania
Lisa Feigenson, John Hopkins University
Véronique Izard, Paris Descartes University
Mauro Pesenti, Catholic University of Louvain

Conférence d'ouverture par Brian Butterworth :

FOUNDATIONAL CAPACITIES AND ARITHMETICAL DEVELOPMENT
Like many other species, humans, even in infancy, possess a mechanism for extracting numerosity information from the environment, which I have called a foundational capacity. This mechanism is domain-specific, is implemented in a dedicated mechanism and is innate. I argue that the efficient working of this capacity is necessary for typical arithmetical development, and if it works inefficiently, this is sufficient for atypical development – dyscalculia.

par Elizabeth Brannon, University of Pennsylvania

Adult humans quantify, label, and categorize almost every aspect of the world with numbers.  The ability to use numbers is one of the most complex cognitive abilities that humans possess and is often held up as a defining feature of the human mind.  In my talk I will present a body of data that demonstrates that there are strong developmental and evolutionary precursors to adult mathematical cognition that can be uncovered by studying human infants and nonhuman primates. Developmental data and controversies will be discussed in light of comparative research with monkeys and other animals allowing us to see both parallels and discontinuities in the evolutionary and developmental building blocks of adult human cognition. Despite the profound discontinuity between primitive number sense and uniquely human symbolic mathematics I will present evidence that primitive number sense serves as a foundation for symbolic math.

par Mauro Pesenti, Catholic University of Louvain

In this talk, I will present and summarize the many number-finger interactions that my research group has reported in the last years: (i) how the fixed order of fingers on the hand provides human beings with unique facilities to increment numerical changes or represent a cardinal value while solving arithmetic problems; (ii) how pointing actions support object enumeration and how numerical magnitudes interact with pointing-reaching actions; and finally (iii) how number processing has been found to interact with the execution or perception of grasping movements, indicating that the adjustment of the hand grip to match object size shares processes with the computation of number magnitude estimates. From these data, I will argue that the way we express numerical concepts physically, by raising fingers while counting, pointing to objects, or using grip aperture to describe magnitudes, leads to embodied representations of numbers and calculation procedures in the adult brain.

par Daniel Ansari, University of Western Ontario

Humans share with animals the ability to process quantities when they are presented in non-symbolic formats (e.g., collections of objects). Unlike other species, however, over cultural history, humans have developed symbolic representations (such as number words and digits) to represent numerical quantities exactly and abstractly. These symbols and their semantic referents form the foundations for higher-level numerical and mathematical skills. It is commonly assumed that symbols for number acquire their meaning by being mapped onto the pre-existing, phylogenetically ancient system for the approximate representation of non-symbolic number over the course of learning and development. In this talk I will challenge this hypothesis for how numerical symbols acquire their meanings (“the symbol grounding problem”). To do so, I will present a series of behavioral and neuroimaging studies with both children and adults that demonstrate that symbolic and non-symbolic processing of number is dissociated at both the behavioral and brain levels of analysis. I will discuss the implications of these data for theories of the origins of numerical symbol processing.

par Véronique Izard, Paris Descartes University

Human infants already possess representations with numerical content: these representations are sensitive to numerical quantity while abstracting away non-numerical aspects of stimuli, and they can enter into arithmetical operations and inferences in line with the laws and theorems of mathematics. However, while core cognition captures many properties of numbers, children’s early representations are not powerful enough represent our princeps concept of number, the type of numbers children first encounter in language and at school: Integers. In this talk, I will present two series of recent studies where we probed children’s knowledge of fundamental properties of Integers, i.e. properties that serve as foundations for formal descriptions of Integers. First, we studied how children aged 2 ½ to 4 years understand the relation of one-one correspondence between two sets. We found that children do not initially take one-one correspondence to instantiate a relation of numerical equality (a violation of Hume’s principle, at the foundation of set-theoretic descriptions of Integers); instead, they interpret one-one correspondence in terms of set extension. Second, we developed a new task to probe children’s intuitions about the structure of the set of Integer, as described in Dedekind-Peano’s axioms: in particular, we tested whether children understand that Integers form a list structure where all numbers can be generated by a successor function (+1), and whether they understand that this list never loops back on itself. In both sets of studies, we asked both at what age children understand essential properties of Integers, and whether numerical symbols play a role in their acquisition.

par Lisa Feigenson, Johns Hopkins University

The act of quantification (e.g., knowing how many objects are in a scene) requires selecting a relevant entity and storing it in working memory for further processing. Critically, multiple kinds of entities can be selected and stored. In this talk I offer evidence that humans can represent at least three different levels of entities in working memory. They can represent an individual object (e.g., “that bird”). They can represent a collection of items (e.g., “that flock of birds”). And they can represent a set of discrete items (e.g., “the set containing Bird A, Bird B, and Bird C”). each of these types of representations permits a different of quantificational processing. Storing individual objects in working memory permits exact but implicit representation of the number of objects present, up to a maximum of 3 objects. Storing collections of items in working memory permits explicit but inexact representation of the number of items present, with no in principle upper limit. And storing sets of individual items permits exact implicit representation of the number of items present, but is accompanied by a loss of representational precision. Hence, which quantity-relevant computations may be performed in any given situation depends on which level of representation is stored. This framework for thinking about interactions between attention, working memory, and quantification applies throughout development starting in infancy.

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