UC San Diego

Mathematics involves thinking and communicating about the absent and abstract. Our primate brains and bodies, by contrast, evolved for the mundane exigencies of the concrete here-and-now. How, then, do we make sense of notions that lie beyond the reach of action and perception? Recent proposals suggest that mathematical cognition recycles neural systems specialized for processing space and action, assembled and coordinated by cultural practices. Every Number in its Place explores this spatialization of arithmetic, the coupling of number and space in calculation, conceptualization, communication, and culture. Inspired by 17th century debates about the reality of imaginary numbers, the five experiments of Chapter 2 demonstrate a novel phenomenon: a sagittal number-line. Thinking about negative and positive integers induces spatial dispositions to move backward and forward, respectively. I argue that these dispositions constitute, in part, our mathematical habitus, dispositions to act and think that reflect and reproduce our conceptual systems. Chapter 3 describes the recruitment of space not just for isolated numbers but for calculation. During mental arithmetic, participants' hand movements revealed systematic spatial biases, as if calculation involves shifts in spatial attention along a mental number-line. This occurred even when the calculation was exact and symbolic, rather than approximate, lending support to proposals that arithmetic co-opts parietal circuits for spatial attention. Mathematics requires not only rote calculation with numbers, but meaningful and reflexive reasoning about numbers. Combining observation and experiment, Chapter 4 analyses spontaneous gestures produced during mathematical reasoning to argue that we conceptualize arithmetic, in part, using a system of complementary spatial metaphors. Chapter 5 investigates the contribution of the communicative body to perpetuating and propagating this spatial understanding. A series of lab- and internet-based experiments demonstrate that co- speech gesture shapes and spreads the mental number-line, a process I call "gestural contagion." Together, Chapters 3 and 4 foreground the body as a nexus for the cultural reproduction of mathematics, both disciplined by and disciplining abstract thought. In sum, this essay is a case study of the production and reproduction of a conceptual system, of the relation between agency and structure, and of the origins of abstraction