# Conférence « Sources of numerical knowledge » – 9 février 2018

La conférence se déroulera de 12h à 13h, en salle 128, à la MSHS.

**Résumé :**

Professional mathematicians often evoke mental images, situated in space and time, when describing their thought process. In line with these reports, cognitive science has provided evidence that non-verbal representations are critically involved in mathematical thinking, in mathematicians as well as in the general population (Amalric & Dehaene, 2016; Dehaene, 2011; Dehaene, Bossini, & Giraux, 1993; Monti, Parsons, & Osherson, 2012; Varley et al., 2005).

Such non-verbal representations provide foundations to mathematics: they support learning in children, and they perhaps also entered into the genesis of mathematical concepts. To search for the origins of mathematical knowledge, this field has been studying various populations: non-human animals, as well as humans of various cultural

backgrounds and ages, focusing particularly on people who have not (yet) received education in mathematics. In these populations, representations with mathematical content have been tested in the following two senses:

– Representations encoding properties that have been formalized in mathematics (numerousness, shape…); and encoding these properties in an abstract way, i.e. independently from other properties that are not relevant in mathematics (size of objects, material…).

– Representations supporting inferences in line with the laws and theorems of mathematics. In this conference, Véronique Izard (Laboratoire Psychologie de la Perception – UMR 8242) will mostly focus about our knowledge of numbers. She will first review findings providing evidence that numerical content is present in our Core Cognition – a set of cognitive systems that constitute infants’ first cognitive abilities, and then remain active throughout lifetime. Second, while core intuitions capture some properties of numbers, other aspects go beyond their representational power: in particular, children are not initially endowed with resources to discriminate between exact large numbers. In the second part of the talk, she will present two studies where we investigated the development of Integer concepts, by probing children’s understanding of fundamental properties that have served to ground mathematical formalization of Integers. The first study focuses on the relation of numerical equality, and how it is instantiated by one-one correspondence (Hume’s principle, grounding settheoretical formalizations of number). The

second study probes children’s understanding of structural properties highlighted by Peano-Dedekind axioms: the fact that the list of all Integers can be generated by iterating a successor function, and the fact that the Integer list does not loop back on itself. Time permitting, she may also talk about the cognitive foundations for geometry.