Mathematical intuition and the cognitive roots of mathematical concepts

Giuseppe Longo, Arnaud Viarouge. Mathematical intuition and the cognitive roots of mathematical concepts. Invited paper, Topoi, Special issue on Mathematical knowledge: Intuition, visualization, and understanding (Horsten L., Starikova I., eds), Vol. 29, n. 1, pp. 15-27, 2010. (TopoiIntuitCogni.pdf)International audienceThe 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 by this too 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 the philosophical frame as well as to the some recent advances in Cognition that support our claim, the cognitive origin and the constitutive role of mathematical intuition

Evidence for the role of inhibition in numerical comparison: A negative priming study in 7- to 8-year-olds and adults

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Event-Related Potentials Reveal the Impact of Conflict Strength in a Numerical Stroop Paradigm

Numerical cognition provides an opportunity to study the underlying processes of selective attention to numerical information in the face of conflicting, non-numerical, information of different magnitudes. For instance, in the numerical Stroop paradigm, participants are asked to judge pairs of Arabic digits whose physical size can either be congruent (e.g., 3 vs. 5) or incongruent (e.g., 3 vs. 5) with numerical value. Congruency effects when deciding which of the two digits is numerically larger are thought to reflect the inhibition of the irrelevant physical size. However, few studies have investigated the impact of the salience of the irrelevant non-numerical information on these congruency effects and their neural substrates. EEG was recorded in 32 adults during a numerical Stroop task with two levels of salience (low, high) of the irrelevant size dimension. At the behavioral level, we observed larger congruency effects in the high salience condition (i.e., when the difference in size between the two digits is larger). At the neural level, at centro-parietal electrodes, we replicated previous studies showing a main effect of congruency on event-related potential (ERP) amplitudes between 280 and 370 ms post-stimulus, as well as a main effect of salience around 200 ms post-stimulus. Crucially, congruency and salience interacted both between 230 and 250 ms (P2), and between 290 and 340 ms (P3). These results provide support for separate processes underlying the increase in congruency effect, which can be attributed to higher demands in both the inhibition of the irrelevant dimension, and the attention to the relevant numerical information

Attention to number requires magnitude-specific inhibition

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The progressive 6-year-old conserver: Numerical saliency and sensitivity as core mechanisms of numerical abstraction in a Piaget-like estimation task

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Les bases cognitives de l'intuition mathématique

La cognition numérique rend possible une approche expérimentale de l'intuition mathématique, pouvant être mise en parallèle des conceptions de philosophes et de mathématiciens, voire apporter des réponses à des questions épistémologiques. Deux aspects de l intuition mathématique sont révélés par ces travaux. Le premier est l'existence de contraintes de la structure de notre système cognitif sur nos intuitions des nombres. Le second est la présence d'une composante spatiale de ces intuitions. Ces aspects sont repris dans deux études comportementales. La première indique une compression de l'échelle de représentation des nombres, la seconde précise les cadres de référence spatiaux impliqués dans les interactions spatio-numériques. Ces données vont dans le sens d'une intuition mathématique non figée, constituée à partir d'une pluralité d'expériences. Nous proposons que cette dynamique de l intuition soit à l'origine d intuitions complexes intervenant dans la pratique des mathématiques.Numerical cognition allow an experimental approach to mathematical intuition, which can be conceived as paralleling some conceptions developed by philosophers and mathematicians, and even as answering some epistemological issues. Two aspects of mathematical intuition are revealed by these studies. The first is the existence of constraints on the structure of our cognitive system on our intuitions of numbers. The second is the presence of a strong spatial component in these intuitions. These two aspects are considered in two behavioral studies we conducted. The first study shows a compressed representational scale of numbers, the second study specifies the spatial reference frames implicated in the spatio-numerical interactions. These data also support the idea of a non-static intuition, constituted from a plurality of experiences. We propose that this dynamic of intuition is at the origin of more complex intuitions, which play a crucial role in the practice of mathematics.PARIS3-BU (751052102) / SudocSudocFranceF

The organization of spatial reference frames involved in the SNARC effect

The SNARC effect refers to faster reaction times for larger numbers with right-sided responses and for smaller numbers with left-sided responses, even when numerical magnitude is irrelevant. Although the SNARC is generally thought to reflect a mapping between numbers and space, the question of which spatial reference frame(s) are critical for the effect has not been systematically explored. We propose a dynamic hierarchical organization of the reference frames (from a global left–right frame to body- and object-related frames), where the influence of each frame can be modulated by experimental context. We conducted two experiments based on predictions derived from this organizational system. Experiment 1 compared instructions that differed only in focusing participants ’ attention on either the response buttons or the hands. Instructions focusing on a hand-based reference frame eliminated the SNARC. Experiment 2 provided the opportunity for an object-centred reference frame to manifest itself in the SNARC. Although we did not observe an effect of an object-centred reference frame, we observed the influence of other reference frames in a context where an object-centred reference frame was emphasized. Altogether, these results support the proposed organization of the reference frames

The cognitive mechanisms of the SNARC effect: an individual differences approach.

Access to mental representations of smaller vs. larger number symbols is associated with leftward vs. rightward spatial locations, as represented on a number line. The well-replicated SNARC effect (Spatial-Numerical Association of Response Codes) reveals that simple decisions about small numbers are facilitated when stimuli are presented on the left, and large numbers facilitated when on the right. We present novel evidence that the size of the SNARC effect is relatively stable within individuals over time. This enables us to take an individual differences approach to investigate how the SNARC effect is modulated by spatial and numerical cognition. Are number-space associations linked to spatial operations, such that those who have greater facility in spatial computations show the stronger SNARC effects, or are they linked to number semantics, such that those showing stronger influence of magnitude associations on number symbol decisions show stronger SNARC effects? Our results indicate a significant correlation between the SNARC effect and a 2D mental rotation task, suggesting that spatial operations are at play in the expression of this effect. We also uncover a significant correlation between the SNARC effect and the distance effect, suggesting that the SNARC is also related to access to number semantics. A multiple regression analysis reveals that the relative contributions of spatial cognition and distance effects represent significant, yet distinct, contributions in explaining variation in the size of the SNARC effect from one individual to the next. Overall, these results shed new light on how the spatial-numerical associations of response codes are influenced by both number semantics and spatial operations