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Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world-in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for instance, could involve shifts in spatial attention along a mental "number-line", the product of cultural artefacts and practices that systematically spatialize number and arithmetic. Here, we investigate this hypothesized spatial processing during exact, symbolic arithmetic (e.g., 4 + 3 = 7). Participants added and subtracted single-digit numbers and selected the exact solution from responses in the top corners of a computer monitor. While they made their selections using a computer mouse, we recorded the movement of their hand as indexed by the streaming x, y coordinates of the computer mouse cursor. As predicted, hand movements during addition and subtraction were systematically deflected toward the right and the left, respectively, as if calculation involved simultaneously simulating motion along a left-to-right mental number-line. This spatial-arithmetical bias, moreover, was distinct from-but correlated with-individuals' spatial-numerical biases (i.e., spatial-numerical association of response codes, SNARC, effect). These results are the first evidence that exact, symbolic arithmetic prompts systematic spatial processing associated with mental calculation. We discuss the possibility that mathematical calculation relies, in part, on an integrated system of spatial processes.

Deictic time words like "yesterday" and "tomorrow" pose a challenge to children not only because they are abstract, and label periods in time, but also because their denotations vary according to the time at which they are uttered: Monday's "tomorrow" is different than Thursday's. Although children produce these words as early as age 2 or 3, they do not use them in adult-like ways for several subsequent years. Here, we explored whether children have partial but systematic meanings for these words during the long delay before adult-like usage. We asked 3- to 8-year-olds to represent these words on a bidirectional, left-to-right timeline that extended from the past (infancy) to the future (adulthood). This method allowed us to independently probe knowledge of these words' deictic status (e.g., "yesterday" is in the past), relative ordering (e.g., "last week" was before "yesterday"), and remoteness from the present (e.g., "last week" was about 7 times longer ago than "yesterday"). We found that adult-like knowledge of deictic status and order emerge in synchrony, between ages 4 and 6, but that knowledge of remoteness emerges later, after age 7. Our findings suggest that children's early use of deictic time words is not random, but instead reflects the gradual construction of a structured lexical domain.

The canonical history of mathematics suggests that the late 19th-century "arithmetization" of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitivehistorical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice.

Metaphorical expressions are pervasive in natural language and pose a substantial challenge for computational semantics. The inherent compositionality of metaphor makes it an important test case for compositional distributional semantic models (CDSMs). This paper is the first to investigate whether metaphorical composition warrants a distinct treatment in the CDSM framework. We propose a method to learn metaphors as linear transformations in a vector space and find that, across a variety of semantic domains, explicitly modeling metaphor improves the resulting semantic representations. We then use these representations in a metaphor identification task, achieving a high performance of 0.82 in terms of F-score.