A two-level model of embodied mathematical thinking. Body schema, body image and language

Where does mathematics come from? This question has been addressed, among others, by Lakoff and Núñez (2000), who proposed an embodied account of our mathematical cognition. In their perspective, mathematics is not an arbitrary discipline since it is grounded in universal conceptual metaphors. However, their account has been criticized. The main objections concern how the creation of these metaphors from random connections that are stabilised through repetition can lead to something so universal and globally shared. In other words, metaphors introduce objects in a target domain that, however, already exists and is well defined as a domain. For example, the authors contend that metaphors are useful for understanding and performing tasks that are more difficult than the simple act of counting (they are used to introduce the concept of number). Nevertheless, counting requires the understanding of the concept of number (cardinal and ordinal alike). The same goes for the grounding metaphor “numbers as object collections”, where addition and other actual tasks in the source domain require arithmetic abilities (e.g., the representation of a basis requires “the creation of subgroups”. But these can be created only starting from a mathematical criterion). The same goes even for the grounding metaphor “arithmetic as object collection” and “classes as containers”; in both cases, the authors do not specify how the target domains of “integers” and “classes” are established (Graziano, 2018). According to several authors (e.g., Sperber and Wilson, 2006; Wilson and Carston, 2006), the main limitation of conceptual metaphors is the so-called “conceptual reductionism” put forward by Lakoff, consisting in ignoring the role of language in the conceptualisation processes related to the processing and structuring of the “process of thought”, while focussing only on the issues related to mental metaphors. We will propose an alternative, but still embodied, account of mathematical cognition which is based on the idea that we can identify two levels of embodiment (Cuccio, 2018) which are the bases for two levels of mathematical abilities. The first level is grounded in our body schema. The notion of body schema refers to the set of sensorimotor processes that enable us to interact with the environment. The hypothesis that is presented here is that body schemas are the basic and ontogenetically primary source of metonymic correlation via a mapping from sensorimotor processes to perception (and, from there, tocognition). Invisible embodied metonymies give life to cognition by founding our perceptual experiences (the target domain) on our sensorimotor processes (the source domain). A first, very basic and universal level of mathematical cognition is grounded on body schemas. The second level of embodiment is grounded in body images. Gallagher (2005, 25) defined the body image as “a complete set of intentional states and dispositions – perceptions, beliefs and attitudes – in which the intentional object is one’s own body”. This is the level of metaphorical cognition. Conceptual metaphors have a body image as a source domain since, in this case, the source of the metaphor is our body as an object of representation and knowledge: the source of the metaphorical mapping is a representation of the body, seen as a body percept, body affect or body concept. As such, the metaphorical mapping takes place at the conceptual level, from one representation to another. For this reason, in the case of metaphors with a body image as their source domain, the contribution of the body is mediated by our cultural, environmentally situated and linguistically structured representations of the body itself. In the account here proposed, a second level of more sophisticated mathematical abilities is grounded in conceptual metaphors. In this view, language thus plays an important role in the conceptualisation of mathematics.