Compositionality in language and arithmetic

The lack of conceptual analysis within cognitive science results in multiple models of the same phenomena. However, these models incorporate assumptions that contradict basic structural features of the domain they are describing. This is particularly true about the domain of mathematical cognition. In this paper we argue that foundational theoretic aspects of psychological models for language and arithmetic should be clarified before postulating such models. We propose a means to clarify these foundational concepts by analyzing the distinctions between metric and linguistic compositionality, which we use to assess current models of mathematical cognition. Our proposal is consistent with the scientific methodology that determines that careful conceptual analysis should precede theoretical descriptions of data. Scientific theories must not only be true, but also coherent and systematic. Part of their being coherent and systematic rests on the conceptual clarity of the most fundamental terms that these theories explain. This means that scientific theories are not only determined by experimentation and quantification, but also by conceptual analysis. Actually, it is conceptual clarity that distinguishes a set of random facts from a systematic scientific explanation. Cognitive psychology has produced an impressive corpus of data. However, few attempts are made to clarify the meaning of the concepts that are being used in interpreting these data, whether the use of concepts in different theories is consistent and how these concepts relate to one another. One of the central theoretical issues in psychology is the nature of measurement in psychological experiments and its relation to cognitive capacities. Some of the most influential psychologists have addressed this issue directly by defining mapping relations between mental content and units of measurement. For instance, Stevens (1951) proposed different scales (ex: nominal, ordinal, interval, ratio) and distinguished them in terms of their formal properties. A fascinating aspect of the \\server05\productn\T\THE\27-1\THE107.txt unknown Seq: 2 4-JAN-08 8:01 54 Journal of Theoretical and Philosophical Psy. Vol. 27, No. 1, 2007 topic of measurement in psychology is that certain cognitive computations appear to preserve metric and scale structure, while others remain qualitative and are not metrically structured. Recently, mathematical, temporal and spatial cognition gained interest in different areas of psychology, and it is certainly a fundamental concern in cognitive science. However, the models that have been proposed to account for mathematical cognition and the representation of quantities reveal serious conceptual confusion, a situation that could be alleviated by revisiting the fundamental assumptions of the theory of measurement. For instance, these models differ with respect to their approach to the acquisition of numerical concepts and the representational structure underlying discrete quantities. According to one model, the content of discrete quantities is determined by linguistic representations in the course of psycholinguistic development (Carey, 2004). However, other models favor a visual-attentional mechanism that automatically picks out the discrete quantities provided that there is a limit to the size of the perceived set, independently of linguistic representations (Xu & Spelke, 2000). These two models assume that the representation of small and large quantities has different formats and are processed differently in the mind. This assumption has specific predictions concerning the psychophysical properties of discrete quantity representations and it is another dimension in which these models of mathematical cognition differ. Another model, which differs from the visual-attentional and psycholinguistic accounts, assumes a common metric underlying the numerical representations of all sizes. This model assumes that all numbers are represented in the same format (on the same mental continuum) and differs from the previous models with respect to the development of mathematical cognition. For instance, according to the ‘common metric’ account, the innate structure of magnitude based representations (Gallistel and Gelman, 1992) allows cognitive agents to represent the numerical attributes of the physical world with no functional dependence on their linguistic or visual-perceptual capacities. As a consequence of the linguistic independence postulated by this account, the ‘common metric’ model generalizes to mathematical cognition processes in human and non-human animals. But even within the models that propose a common metric, there are divergences concerning the type of mapping between the physical and mental quantities. Some theorists (Dehaene, 2001) assume a logarithmic mapping with constant variability, while others (Gallistel & Gelman, 1992) assume a linear mapping with scalar variability. These views also have different predictions about the psychophysical properties of the discrete quantities. In this paper, we evaluated the assumptions of these models by critically assessing their implications with respect to the theory of measure\\server05\productn\T\THE\27-1\THE107.txt unknown Seq: 3 4-JAN-08 8:01