Rafael Núñez

Where mathematics comes from: How the embodied mind brings mathematics into being.

This book is about mathematical ideas, about what mathematics means-and why. Abstract ideas, for the most part, arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world. Where Mathematics Comes From argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms.

With the Future Behind Them: Convergent Evidence From Aymara Language and Gesture in the Crosslinguistic Comparison of Spatial Construals of Time

Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego- and temporal-reference-point models. Dynamic models appear to be quasi-universal cross-culturally, as does the generalization that in Ego-reference-point models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; linguistic and gestural data give strong confirmation of this unusual culture-specific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns?

The tangle of space and time in human cognition

Everyday concepts of duration, of sequence, and of past, present, and future are fundamental to how humans make sense of experience. In culture after culture, converging evidence from language, co-speech gesture, and behavioral tasks suggests that humans handle these elusive yet indispensable notions by construing them spatially. Where do these spatial construals come from and why do they take the particular, sometimes peculiar, spatial forms that they do? As researchers across the cognitive sciences pursue these questions on different levels--cultural, developmental--in diverse populations and with new methodologies, clear answers will depend upon a shared and nuanced set of theoretical distinctions. Time is not a monolith, but rather a mosaic of construals with distinct properties and origins.

Time after time: The psychological reality of the Ego- and Time-Reference-Point distinction in metaphorical construals of time.

Research in cognitive linguistics and in processing of temporal metaphors has traditionally distinguished between Moving-Ego and Moving-Time mappings: Either the Ego is construed as moving regarding fixed temporal landmarks or Time is construed as moving regarding the Ego. Both of these metaphors involve time events in reference to an Ego, which specifies the present time Now.We build on recent theoretical suggestions for a more fundamental classification of temporal metaphors: Ego- and Time-Reference-Point metaphors (Ego-RP and Time-RP). The distinction focuses on the role of reference points in ascribing orientation, rather than on the identity of a moving entity (Ego or Time). Using visual priming experiments we provide evidence of the psychological reality of the Time-RP metaphor, a temporal metaphor with no reference to an Ego.

Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics

Robotics, artificial intelligence and, in general, any activity involving computer simulation and engineering relies, in a fundamental way, on mathematics. These fields constitute excellent examples of how mathematics can be applied to some area of investigation with enormous success. This, of course, includes embodied oriented approaches in these fields, such as Embodied Artificial Intelligence and Cognitive Robotics. In this chapter, while fully endorsing an embodied oriented approach to cognition, I will address the question of the nature of mathematics itself, that is, mathematics not as an application to some area of investigation, but as a human conceptual system with a precise inferential organization that can be investigated in detail in cognitive science. The main goal of this piece is to show, using techniques in cognitive science such as cognitive semantics and gestures studies, that concepts and human abstraction in general (as it is exemplified in a sublime form by mathematics) is ultimately embodied in nature.

No Innate Number Line in the Human Brain

Many authors in the field of numerical cognition have adopted a rather nativist view that all humans share the intuition that numbers map onto space and, more specifically, that an oriented left-to-right mental number line (MNL) is localized bilaterally in the intraparietal sulcus of the human brain. We review results from archaeological and historical (diachronic) studies as well as cross-cultural (synchronic) ones and contends that these claims are not well founded. The data actually suggest that the MNL is not innate. We argue that the MNL—and number-to-space mappings in general—emerges outside of natural selection proper requiring top-down dynamics that are culturally and historically mediated through high-order cognitive mechanisms such as fictive motion, conceptual mappings, and external representational media. These mechanisms, which are not intrinsically numerical and usually are acquired through education, are not genetically determined and are biologically realized through the systematic consolidation of specific brain phenotypes that support number-to-space mappings.

Doing arithmetic by hand: Hand movements during exact arithmetic reveal systematic, dynamic spatial processing

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.

Gesture and Metaphor Comprehension: Electrophysiological Evidence of Cross-Modal Coordination by Audiovisual Stimulation.

In recent years, studies have suggested that gestures influence comprehension of linguistic expressions, for example, eliciting an N400 component in response to a speech/gesture mismatch. In this paper, we investigate the role of gestural information in the understanding of metaphors. Event related potentials (ERPs) were recorded while participants viewed video clips of an actor uttering metaphorical expressions and producing bodily gestures that were congruent or incongruent with the metaphorical meaning of such expressions. This modality of stimuli presentation allows a more ecological approach to meaning integration. When ERPs were calculated using gesture stroke as time-lock event, gesture incongruity with metaphorical expression modulated the amplitude of the N400 and of the late positive complex (LPC). This suggests that gestural and speech information are combined online to make sense of the interlocutors linguistic production in an early stage of metaphor comprehension. Our data favor the idea that meaning construction is globally integrative and highly context-sensitive.

Number concepts without number lines in an indigenous group of Papua New Guinea.

Background The generic concept of number line, which maps numbers to unidimensional space, is a fundamental concept in mathematics, but its cognitive origins are uncertain. Two defining criteria of the number line are that (i) there is a mapping of each individual number (or numerosity) under consideration onto a specific location on the line, and (ii) that the mapping defines a unidimensional space representing numbers with a metric — a distance function. It has been proposed that the number line is based on a spontaneous universal human intuition, rooted directly in brain evolution, that maps number magnitude to linear space with a metric. To date, no culture lacking this intuition has been documented. Methodology/Principal Findings By means of a number line task, we investigated the universality proposal with the Yupno of Papua New Guinea. Unschooled adults did exhibit a number-to-space mapping (criterion i) but, strikingly, despite having precise cardinal number concepts, they located numbers only on the endpoints, thus failing to use the extent of the line. The produced mapping was bi-categorical and metric-free, in violation of criterion ii. In contrast, Yupnos with scholastic experience used the extent of the segment according to known standards, but they did so not as evenly as western controls, exhibiting a bias towards the endpoints. Conclusions/Significance Results suggest that cardinal number concepts can exist independently from number line representations. They also suggest that the number line mapping, although ubiquitous in the modern world, is not universally spontaneous, but rather seems to be learned through — and continually reinforced by — specific cultural practices.

Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?

Numbers are fundamental entities in mathematics, but their cognitive bases are unclear. Abundant research points to linear space as a natural grounding for number representation. But, is number representation fundamentally spatial? We disentangle number representation from standard number-to-line reporting methods, and compare numerical estimations in educated participants using line-reporting with three nonspatial reporting conditions (squeezing, bell-striking, and vocalizing). All three cases of nonspatial-reporting consistently reproduced well-established results obtained with number-line methods. Furthermore, unlike line-reporting-and congruent with the psychophysical Weber-Fechner law-nonspatial reporting systematically produced logarithmic mappings for all nonsymbolic stimuli. Strikingly, linear mappings were obtained exclusively in conditions with culturally mediated elements (e.g., words). These results suggest that number representation is not fundamentally spatial, but builds on a deeper magnitude sense that manifests spatially and nonspatially mediated by magnitude, stimulus modality, and reporting condition. Number-to-space mappings-although ubiquitous in the modern world-do not seem to be rooted directly in brain evolution but have been culturally privileged and enhanced.