Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and 25 participants from the Amazon (age (...) 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The spatial content of the (...) formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

Des branches entières des mathématiques sont fondées sur des liens posés entre les nombres et l’espace : mesure de longueurs, définition de repères et de coordonnées, projection des nombres complexes sur le plan… Si les nombres complexes, comme l’utilisation de repères, sont apparus relativement récemment (vers le XVIIe siècle), la mesure des longueurs est en revanche un procédé très ancien, qui remonte au moins au 3e ou 4e millénaire av. J-C. Loin d’être fortuits, ces liens entre les nombres et l’espace (...) reflèteraient une intuition fondamentale, universelle, façonnée au cours des millénaires par la sélection naturelle, et qui aurait servi de guide et d’inspiration aux mathématiciens au fil des siècles. (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that (...) are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 (...) or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic (...) mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics education. (...) By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

In this article, I criticize Véronique Zanetti on the topic of moral compromise. As I understand Zanetti, a compromise could only be called a “moral compromise” if (i) it does not originate under coercive conditions, (ii) it involves conflict whose subject matter is moral, and (iii) “the parties support the solution found for what they take to be moral reasons rather than strategic interests.” I offer three criticisms of Zanetti. First, Zanetti ignores how some parties may not have reason to (...) seek social peace at all. Zanetti’s claim that there is consensus on the aim of social peace can involve idealising away from disagreement in a manner that Zanetti accuses Rawls of. Second, even if there is consensus on the aim of seeking social peace, this leaves open the possibility of disagreement about which society different people should belong to. This idealises away from real world conflict concerning borders. Indeed, Zanetti does not mention that her ‘central example’ of moral disagreement, the German Abortion compromise, was enacted in the wake of German reunification. Third, there are at least two things that can be called the ‘German Abortion compromise.’ The compromise that Zanetti speaks of was imposed by the German Federal Constitutional Court. The court declared unconstitutional a law passed in 1992 that had been negotiated in parliament. Zanetti does not dwell on this lack of democratic credentials. Even the substance of the court-imposed solution is itself a dubious example of a moral compromise between parties based on what is acceptable to their reason. -/- (penultimate version - If you would like the .pdf of the final version for personal scholarly use, please contact me through the email address on my profile or my CV.). (shrink)

A response to a declaration in 'Le Monde', 'Luttons efficacement contre les théories du complot' by Gérald Bronner, Véronique Campion-Vincent, Sylvain Delouvée, Sebastian Dieguez, Karen Douglas, Nicolas Gauvrit, Anthony Lantian, and Pascal Wagner-Egger, published on June the 6th, 2016.

This volume contains new and original papers on Martin Heidegger's complex relation to Friedrich Nietzsche's philosophy. The authors not only critically discuss the many aspects of Heidegger's reading of Nietzsche, they also interpret Heidegger's thought from a Nietzschean perspective. Here is presented for the first time an overview of not only Heidegger's and Nietzsche's philosophy but also an overview of what is alive - and dead - in their thinking. Many authors through a reading of Heidegger and Nietzsche deal with (...) current issues such as technology, ecology, and politics. This volume is of interest for everyone interested in Heidegger's and Nietzsche's thought. Contributors include: Babette Babich, Charles Bambach, Robert Bernasconi, Virgilio Cesarone, Stuart Elden, Michael Eldred, Markus Enders, Charles Feitosa, Véronique Fóti, Luanne T. Frank, Jeffery Kinlaw, Theodore Kisiel, William D. Melaney, Eric Sean Nelson, Abraham Olivier, Friederike Rese, Karlheinz Ruhstorfer, Harald Seubert, Robert Sinnerbrink, Robert Switzer, Jorge Uscatescu Barrón, Nancy A. Weston, Dale Wilkerson, Angel Xolocotzi, Jens Zimmermann. (shrink)

A reply to Gérald Bronner, Véronique Campion-Vincent, Sylvain Delouvée, Sebastian Dieguez, Nicolas Gauvrit, Anthony Lantian, and Pascal Wagner-Egger's piece, '“They” Respond: Comments on Basham et al.’s “Social Science’s Conspiracy-Theory Panic: Now They Want to Cure Everyone”.