Pierced by the number line: Integers are associated with back-to-front sagittal space

Pierced by the number line: Integers are associated with back-to-front sagittal space Tyler Marghetis* Kendall Youngstrom tmarghet@ucsd.edu Department of Cognitive Science University of California, San Diego kyoungst@ucsd.edu Department of Cognitive Science University of California, San Diego Abstract enculturation into spatial practices; indeed, its orientation switches in cultures where numbers and words are read right-to-left (Shaki, Fischer, & Petrusic, 2009). The horizontal number line is but one in a system of spatial-numerical associations, including a bottom-to-top association with vertical space (e.g. Hartmann, Grabherr, & Mass, 2012). Spatial dispositions may even play a role in more complex tasks: Mental arithmetic, for instance, induces systematic dispositions to respond spatially, with addition biasing responses rightward and subtraction biasing responses leftward (Knops et al, 2009), even during exact, symbolic calculation (Marghetis et al, 2014). Mathematical cognition, paragon of abstraction, is marked by systematic associations between number and space, often described as “mental number lines.” In three experiments, we demonstrate a novel effect: a sagittal (back-to-front) number line for negative and positive integers. In a paradigm requiring full-body movements, participants judged numerical magnitude (Exp. 1-2) and parity (Exp. 3). Across all three experiments, participants associated numerical magnitude with locations in front of and behind the body. Responses to negative integers were faster when moving backward than forward; responses to positive integers, faster forward than backward. This sagittal number line appears to require the involvement of negative numbers (Exp. 1-2) and is most pronounced when judging magnitude (Exp. 3). In sum, reasoning about integers induces systematic dispositions to act along the sagittal axis. Such dispositions may reflect our mathematical habitus, habits of action and thought that reflect and enact our conceptual systems. * Keywords: numerical cognition; space; embodiment; integers; sagittal number line A back-to-front sagittal number line? These spatial dispositions seem to reflect asymmetries in spatial experience. The vertical axis is oriented in virtue of the asymmetry of our bodies, as well as by experiential correlations between more and up (Lakoff & Nunez, 2000). And while our bodies are bilaterally symmetric, the horizontal axis is oriented by cultural practices such as reading and counting (e.g., Shaki, Fischer, & Petrusic, 2009). It is surprising, therefore, that little attention has been given to dispositions along an especially prominent axis: the sagittal axis, running from behind the body to the front. The sagittal axis, after all, is striking in its asymmetry. Things in front can be seen, heard, touched; things behind are far more difficult to access. The sagittal axis, moreover, is associated with another abstract domain: Time (see Nunez & Cooperrider, 2013, for review). In English, language typically reserved for sagittal space is used to talk about time (e.g. look forward to the future); co-speech gestures also reflect this pattern; and temporal reasoning induces dispositions to move forward or backward (ibid). Language, gesture, and thought all enact a sagittal timeline. What about number? In English, decreasing or increasing counting is most often described spatially as counting down or up, but also sometimes as counting backward or forward. But there is little evidence that we actually conceptualize number using the sagittal axis, using a back-to-front sagittal number line (SNL). Some suggestive evidence comes from the so-called “vertical” SNARC, perhaps more accurately described as a radial effect: Responses to smaller numbers are faster in near space, and to bigger numbers, in far space (Ito & Hatta, 2004). However, this radial SNARC effect has always been tested with response buttons placed directly in front of the body, thus confounding location along the sagittal axis with distance from the body; larger numbers may just prime responses that are farther away. Hartmann et al (2012) tried to find evidence for an unambiguously back- to-front representation of number. Their Swiss participants SNARC; Introduction “[Space] provides a location for all things that come into being. […] [E]verything that exists must of necessity be somewhere, in some place and occupying some space, and that which doesn’t exist somewhere, whether on earth or in heaven, doesn’t exist at all.” – Plato, Timaeus Mathematics is paradigmatically abstract, and its objects are beyond the reach of perception and interaction. And yet we still manage to engage in mathematical reasoning—for some, with more than a modicum of success and confidence. This may depend, in part, on weaving mathematics into concrete practices, habits, and concepts (Kitcher, 1984; Lakoff & Nunez, 2000). Indeed, numbers are tightly linked to space in both action and thought (Hubbard et al, 2005). A now-classic finding is the “SNARC” effect: among literate individuals from cultures that read left-to-right, smaller numbers induce dispositions to act in left space, and larger numbers, in right space (Dehaene et al, 1993). Negative integers, too, induce spatial dispositions, although task- demands influence whether they are located to the “left” of zero, in line with their relative numerical magnitude, or mixed in with positive integers on the basis of their absolute value (e.g. Fischer, 2003; Ganor-Stern & Tzelgov, 2008). This horizontal “mental number line” reflects a lifetime of Corresponding author.