Depictive expressions of thought predate written language by thousands of years. They have evolved in communities through a kind of informal user testing that has refined them. Analyzing common visual communications reveals consistencies that illuminate how people think as well as guide design; the process can be brought into the laboratory and accelerated. Like language, visual communications abstract and schematize; unlike language, they use properties of the page (e.g., proximity and place: center, horizontal/up–down, vertical/left–right) and the marks on it (e.g., (...) dots, lines, arrows, boxes, blobs, likenesses, symbols) to convey meanings. The visual expressions of these meanings (e.g., individual, category, order, relation, correspondence, continuum, hierarchy) have analogs in language, gesture, and especially in the patterns that are created when people design the world around them, arranging things into piles and rows and hierarchies and arrays, spatial-abstraction-action interconnections termed spractions. The designed world is a diagram. (shrink)

In the present study, we explore how reading habits (e.g., reading from left to right in French or reading from right to left in Arabic) influence thescanningand theconstructionof mental models in spatial reasoning. For instance, when participants are given a problem like A is to the left of B; B is to the left of C, what is the relation between A and C? They are assumed to construct the model: A B C. If reading habits influence the scanning process, (...) then readers of French should inspect models from left to right, whereas readers of Arabic should inspect them from right to left. The prediction following this analysis is that readers of French should be more inclined to produce “left” conclusions (i.e., A is to the left of C), whereas readers of Arabic should be more inclined to produce “right” conclusions (i.e., C is to the right of A). Furthermore, one may expect that readers of French show a greater ease in constructing models following a left-to-right direction than models following a right-to-left direction, whereas an opposite pattern might be expected for readers of Arabic. We tested these predictions in two experiments involving French and Yemeni participants. Experiment 1 investigated the formulation of conclusions from spatial premises, and Experiment 2, which was based on non-linguistic stimuli, examined the time required to construct mental models from left to right and from right to left. Our results show clear differences between the two groups. As expected, the French sample showed a strong left-to-right bias, but the Yemeni sample did not show the reverse bias. Results are discussed in terms of cultural influences and universal mechanisms. (shrink)

Three-to-five-year-old French children were asked to add or remove objects to or from linear displays. The hypothesis of a universal tendency to represent increasing number magnitudes from left to right led to predict a majority of manipulations at the right end of the rows, whatever children's hand laterality. Conversely, if numbers are not inherently associated with space, children were expected to favour laterality-consistent manipulations. The results showed a strong tendency to operate on the right end of the rows in right-handers, (...) but no preference in left-handers. These findings suggest that the task elicited a left-to-right oriented representation of magnitudes that counteracted laterality-related responses in left-handed children. The young age of children and the lack of a developmental trend towards right preference weaken the hypothesis of a cultural origin of this oriented representation. The possibility that our results are due to weaker brain lateralisation in left-handers compared to right-handers is addressed in Discussion section. (shrink)

Across cultures people construct spatial representations of time. However, the particular spatial layouts created to represent time may differ across cultures. This paper examines whether people automatically access and use culturally specific spatial representations when reasoning about time. In Experiment 1, we asked Hebrew and English speakers to arrange pictures depicting temporal sequences of natural events, and to point to the hypothesized location of events relative to a reference point. In both tasks, English speakers (who read left to right) arranged (...) temporal sequences to progress from left to right, whereas Hebrew speakers (who read right to left) arranged them from right to left, replicating previous work. In Experiments 2 and 3, we asked the participants to make rapid temporal order judgments about pairs of pictures presented one after the other (i.e., to decide whether the second picture showed a conceptually earlier or later time-point of an event than the first picture). Participants made responses using two adjacent keyboard keys. English speakers were faster to make “earlier” judgments when the “earlier” response needed to be made with the left response key than with the right response key. Hebrew speakers showed exactly the reverse pattern. Asking participants to use a space-time mapping inconsistent with the one suggested by writing direction in their language created interference, suggesting that participants were automatically creating writing-direction consistent spatial representations in the course of their normal temporal reasoning. It appears that people automatically access culturally specific spatial representations when making temporal judgments even in nonlinguistic tasks. (shrink)

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...) mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic. (shrink)

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that (...) embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it. (shrink)

Núñez and Fias raised concerns on whether our results demonstrate a linear number-space mapping. Patro and Nuerk urge caution on the use of animal models to understand the origin of the orientation of spatial–numerical association. Here, we discuss why both objections are unfounded.

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...) with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)

The present study examined participants’ performance on a temporal judgment task while holding language constant but varying their lifetime and immediate reading experience of horizontal and vertical texts. Chinese participants from Taiwan and China were randomly assigned to a reading task involving horizontally or vertically arranged texts. A temporal judgment task followed the reading task, asking the participants to judge if the event depicted in a second picture occurred earlier or later than that in a first picture. Responses were faster (...) when the left keys represented the ‘earlier’ responses than when the right keys did. Half of the participants responded with horizontally oriented keys while the rest with vertically oriented keys. For the Taiwan participants, the overall starc effect was greater when the response keys were vertical than horizontal, but no difference was observed for the China participants. A questionnaire indicates that the two groups of participants had similar lifetime experiences of reading horizontal texts, but the Taiwan participants read vertical texts in their life far more frequently than the China participants. Immediate reading experiences interacted with lifetime experiences in modulating the vertical bias. For the Taiwan participants, the vertical bias was strong following the vertical prime, but disappeared following the horizontal prime. For the China participants, the horizontal prime led to no vertical bias whereas the vertical prime brought about a horizontal bias. We conclude that the directionality of orthography and speakers’ immediate and lifetime reading experiences can better explain the vertical bias in the Chinese speakers’ temporal thinking. The findings, however, may be interpreted as constituting a different manifestation of linguistic relativity and recast under a broader framework of the extended-mind hypothesis of human cognition. (shrink)

The directionality of space-number association (SNA) is shaped by cultural experiences. It usually follows the culturally dominant reading direction. Smaller numbers are generally associated with the starting side for reading (left side in Western cultures), while larger numbers are associated with the right endpoint side. However, SNAs consistent with cultural reading directions are present before children can actually read and write. Therefore, these SNAs cannot only be shaped by the direction of children’s own reading/writing behavior. We propose six distinct processes (...) – one biological and five cultural/educational – underlying directional SNAs before formal reading acquisition: (i) Brain lateralization, (ii) Monitoring adult reading behavior, (iii) Pretend reading and writing, and rudimentary reading and writing skills, (iv) Dominant attentional directional preferences in a society, not directly related to reading direction, (v) Direct spatial-numerical learning, (vi) Other spatial-directional processes independent of reading direction. In this mini-review, we will differentiate between these processes, elaborate when in development they might emerge, discuss how they may create the SNAs observed in preliterate children and propose how they can be studied in the future. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent logicism is compatible with intuitionism.

The present contribution argues that transitive reasoning, as exemplified in paradigms of linear order construction in mental space, is associated with spatial effects. Starting from robust findings from the early 70s, research so far has widely discussed the symbolic distance effect. This effect shows that after studying pairs of relations, e.g., “A > B,” “B > C,” and “D > E,” participants are more correct, and faster in correct responding, the wider the “distance” between two elements within the chain A (...) > B > C > D > E. The SDE has often been given spatial interpretations, but alternatively, non-spatial models of the effect are also viable on the empirical basis so far, which means the question about spatial contributions to the construction of analog representations of rank orders is still open. We suggest here that laterality effects can add the necessary additional information to support the idea of spatial processes. We introduce anchoring effects in terms of showing response advantages for congruent versus incongruent pairings of presentation location on a screen on the one hand, and the hypothetical spatial arrangement of the order in mental space, on the other hand. We report pertinent findings and discuss anchoring paradigms with respect to their internal validity as well as their being rooted in basic mechanisms of trained reading/writing direction. (shrink)

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a (...) formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures. (shrink)

In 2011, van Dijck and Fias described a positional SNARC effect: the SPoARC. To-be-remembered items presented centrally on a screen seemed to acquire a left-to-right spatial dimension. If confirmed, this spatialization could be crucial for immediate memory theories. However, given the intricate links between visual and spatial dimensions, this effect could be due to the visual presentation, which could have probed the left-to-right direction of reading/writing. To allow a generalization of this effect, we adapted van Dijck and Fias's task using (...) an auditory version of Sternberg's paradigm. Lists of five consonants were auditorily presented at a rate of 3 s/item. A SPoARC effect was observed. The consequences are discussed first from an immediate memory perspective, putting forward the view that order could be coded through spatialization, and then in terms of similarities between SPoARC and SNARC. (shrink)