Abstract
It has been argued that understanding ‘function’ qua abstract mathematical entity requires (a) that different aspects of this entity be understood as referring to the same mathematical entity (for example that the spatial representation of a function (whether it be a graph or a geometrical figure) and its algebraic form denote the same function, (b) that the abstract algebraic representation be grounded on the more tangible and observable spatial representation (observable in that the spatial representation lays out the relations expressed by the algebraic form in space rendering them available to the senses), which provides an initial concrete meaning to the function, and (c) that functions be not reduced to their spatial representational forms, since that gives rise to various misconceptions. In this paper, we address these ingredients of a proper understanding of the ‘function’ with a view to provide a theoretical framework concerning the relation between numbers and space that will allow the assessment of the different trends in the discussion regarding the interplay of algebraic and spatial representations of functions in understanding the concept ‘function’. In the first section we argue that grounding basic abstract mathematical entities such as natural numbers in spatial configurations is necessary for any adequate understanding of these entities. We adduce two main reasons for this claim. The first, from developmental psychology, concerns the notion of number as it is initially formed. The second concerns the way infants and animals represent numbers as magnitudes, with spatial properties. Our main thesis is that numbers are grounded in space, and we call this phenomenon “the spatial intuition of numbers”.