Metaphor and the Philosophical Implications of Embodied Mathematics

Abstract

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.

Embodiment of infinity in mathematics

“…Lakoff and Núñez (2000) argue that BASIC METAPHOR OF INFNITY helps us ground our understanding of infinity. In this metaphor, the source domain is a completed iterative process and the target domain is an iterative process that goes on endlessly (Winter and Yoshimi, 2020). Each completed stage of this repeating process has an end point.…”

Matematyka w metaforach? O wyjaśnianiu pojęć matematycznych za pomocą metafor kognitywnych

“…Społeczne użycie języka skutkować może również formułowaniem metafor, które na zasadzie sprzężenia zwrotnego mogą prowadzić do formułowania kolejnych, jeszcze bardziej abstrakcyjnych myśli, w tym także w dziedzinie matematyki (zob. Winter, Yoshimi, 2020)…”

Iconic Mathematics: Math Designed to Suit the Mind

“…Ordinary symbolic mathematics is highly abstract, but mounting evidence suggests that mental number representations and mathematical operations are embodied —that is, grounded in Lakoff and Núñez (2000) , or at least shaped or affected by Winter and Yoshimi (2020) , the sensory experiences our bodies provide to us (for reviews, see Fischer and Brugger, 2011 ; Fischer and Shaki, 2018 ; Soylu et al, 2018 ; Barrocas et al, 2020 ; see also especially Fischer et al, 2021 ; Glenberg, 2021 ) and other studies not covered by the reviews: Hilton, 2019 ; Proverbio and Carminati, 2019 ; van den Berg et al, 2021 . For example, whereas Germans are accustomed to counting to ten using two hands, the Chinese manage the same with just one, and as if forced to mentally switch hands, Germans take longer than the Chinese to identify the smaller (or larger) of such numbers as 4 and 6 but not 2 and 4.…”

“…Relatedly, addition can be pictured as rightward, and subtraction as leftward, movement along this line ( Knops et al, 2009 ; Marghetis et al, 2014 ). Alternative kinds of the embodiment have been observed in mathematical cognition too ( Winter and Yoshimi, 2020 ).…”