Cognitive scientists Spelke and Kintzler (2007) and Carey (2009) propose objects, actions, space and numbers as ‘core domains of knowledge’ that underpin the framework of concepts people use to describe and communicate about the world. Gärdenfors (2019, 2020) argues that humans make sense of domains by appealing to various types of invariances in sensory signals. In this talk, I present work by Quinon and Gärdenfors (manuscript) in which the aim is to extend the analysis in terms of invariances to the domain of numbers. We focus on several perspectives relating invariances: cognitive modeling, formal mathematical and experimental.

As theoretical background, we assume that numbers are properties of collections (Simons 1982, 2007, 2011; Johansson 2015; Angere 2014). We observe that the domain of number is determined by two types of invariances. First, the concept of collection itself depends on being invariant under the location of its objects. Second, the determinant invariance of the domain of number is the fungibility of objects: If an object in a collection is exchanged for another object, the collection will still contain the same number of objects. Fungibility will be shown to be closely related to one-to-one correspondences.

We first introduce the concept of a collection and show how it differs from the concept of a set. Then we present the invariance of location of objects that applies to collections and we introduce fungibility as a second type of invariance. We illustrate our theoretical analysis by empirical material from experiments of developmental psychologists.

This is joint work with Peter Gärdenfors (Lund).