Mental Number Line

Humans show a remarkable tendency to describe and think of numbers as being placed on a mental number line (MNL), with smaller numbers located on the left and larger ones on the right. Faster responses to small numbers are indeed performed on the left side of space, while responses to large numbers are facilitated on the right side of space (spatial-numerical association of response codes, SNARC effect). This phenomenon is considered the experimental demonstration of the MNL and has been extensively replicated throughout a variety of paradigms. Nevertheless, the majority of previous literature has mainly investigated this effect by means of response times and accuracy, whereas studies considering more subtle and automatic measures such as kinematic parameters are rare (e.g., in a reaching-to-grasp movement, the grip aperture is enlarged in responding to larger numbers than in responding to small numbers). In this brief review we suggest that numerical magnitude can also affect the what and how of action execution (i.e., temporal and spatial components of movement). This evidence could have large implications in the strongly debated issue concerning the effect of experience and culture on the orientation of MNL. In the 19th century Galton first noted that humans visualize and think of numbers as represented on a mental number line (MNL), usually oriented from left-to-right. Along the MNL smaller numbers are located on the left side and larger ones on the right side (Galton, 1880; Dehaene, 2011). The first, and often replicated, experimental demonstration of the MNL is the fact that adult humans are faster at processing small numbers when responses are executed on the left side of space, and at processing large numbers when responses are executed on the right side of space (spatial-numerical association of response codes, SNARC effect; Dehaene et al., 1993). Since this first evidence, large body of studies have investigated the cognitive representation of numbers using chronometric methods, but the impact of number processing on motor control has been scarcely explored. According to a recent influential hypothesis, cognitive representations of perceptual and semantic information cannot be fully understood without considering their impact on actions (Gallese and Lakoff, 2005). In this vein, here we present a critical review to highlight how the existing knowledge on MNL could be fostered by studies that critically analyze the link between motor actions and numbers.

We reviewed literature to understand when a spatial map for time is available in the brain. We carefully defined the concepts of metrical map of time and of conceptual representation of time as the Mental time line (MTL) in order to formulate our position. It is that both metrical map and conceptual representation of time are spatial in nature. The former should be innate, related to motor/implicit timing, it should represent all magnitudes with an analogic and bi-dimensional structure. The latter (MTL) should be learned, available at about 8-10 years-old and related to cognitive/explicit time. It should have uni-dimensional, linear and directional structure (left-to-right in Western culture). We bear the centrality of the development of number cognition, of time semantic concepts and of reading/writing habits for the development of ordinality and linearity of the mental time line.

It is widely accepted that different number-related tasks, including solving simple addition and
subtraction, may induce attentional shifts on the so-called mental number line, which represents
larger numbers on the right and smaller numbers on the left. Recently, it has been shown that
different number-related tasks also employ spatial attention shifts along with general cognitive
processes. Here we investigated for the first time whether number line estimation and complex
mental arithmetic recruit a common mechanism in healthy adults. Participants’ performance in
two-digit mental additions and subtractions using visual stimuli was compared with their
performance in a mental bisection task using auditory numerical intervals. Results showed
significant correlations between participants’ performance in number line bisection and that in twodigit
mental arithmetic operations, especially in additions, providing a first proof of a shared
cognitive mechanism (or multiple shared cognitive mechanisms) between auditory number
bisection and complex mental calculation.

Decision times during processing of positive number symbols (1, 2, 3 etc.) inform our understanding of mental representations of integers (Holyoak, 1978; Dehaene et al., 1993; Fischer and Shaki, 2014). Effects of number magnitude on cognition include distance effects (faster discrimination for larger numerical differences in a number pair), size effects (faster processing of smaller numbers), Spatial-Numerical Association of Response Codes (SNARC; faster left/right responses to small/large numbers), linguistic markedness (MARC; faster left/right responses to odd/even numbers) and semantic congruity effects (faster smaller/larger decisions over smaller/larger number pairs). Results converge on the notion of a spatially oriented mental number line (MNL) where numerically smaller number concepts exist to the left of larger number concepts. How do these performance signatures help us to understand the cognitive representation of negative number symbols (−1, −2, −3 etc.)? Unlike natural number symbols, negative number symbols lack corresponding real entities that support sensory-motor learning. We discuss a recent proposal by Varma and Schwartz (2011) with implications for developmental research.

Many metaphor theorists argue that our mental experience of time is grounded in our understanding
of space, including motion through space. Results from recent experiments – in which people think
about motion, which in turn influences their thinking about time – support this position. Still, many
questions remain about the nature of the metaphorical connection between time and space. Can the
mere suggestion of motion influence how people reason about time, and if so, when and how? Three
experiments investigated how thinking about “abstract” motion through sequences of numbers or letters
would influence reasoning about time. Our results extend earlier psychological work on the link
between time and space by showing that even motion in non-physical domains can influence temporal
reasoning. The results provide further evidence that metaphorical understanding is grounded in
our everyday physical and conceptual experience.

Space, time, and number are fundamental to how we act within and reason about the world. These three experiential domains are systematically intertwined in behavior, language, and the brain. Two main theories have attempted to account for cross-domain interactions. A Theory of Magnitude (ATOM) posits a domain-general magnitude system. Conceptual Metaphor Theory (CMT) maintains that cross-domain interactions are manifestations of asymmetric mappings that use representations of space to structure the domains of number and time. These theories are often viewed as competing accounts. We propose instead that ATOM and CMT are complementary, each illuminating different aspects of cross-domain interactions. We argue that simple representations of magnitude cannot, on their own, account for the rich, complex interactions between space, time and number described by CMT. On the other hand, ATOM is better at accounting for low-level and language-independent associations that arise early in ontogeny. We conclude by discussing how magnitudes and metaphors are both needed to understand our neural and cognitive web of space, time and number.

We reviewed literature to understand when a spatial map for time is available in the brain. We carefully defined the concepts of metrical map of time and of conceptual representation of time as the mental time line (MTL) in order to formulate our position. It is that both metrical map and conceptual representation of time are spatial in nature. The former should be innate, related to motor/implicit timing, it should represent all magnitudes with an analogic and bi-dimensional structure. The latter MTL should be learned, available at about 8-10 years-old and related to cognitive/explicit time. It should have uni-dimensional, linear and directional structure (left-to-right in Western culture). We bear the centrality of the development of number cognition, of time semantic concepts and of reading/writing habits for the development of ordinality and linearity of the MTL.

Mangalam and Karve raise concerns on whether our results demonstrate a mental number line, suggesting auxiliary experiments. Further data analyses show that their methodological concerns are not founded. Harshaw suggests that a side bias could have affected our results. We show that this concern is also unfounded.