Article Authors Metrics Comments Media Coverage Abstract Author Summary Introduction Results Discussion Supporting information Acknowledgments Author Contributions References Reader Comments (0) Media Coverage (0) Figures Abstract During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to (...) predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them. (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.

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)

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event (...) depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (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)

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)

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)

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)

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)

This paper clarifies the usage of Albert Hirschman’s categories of market behaviour as of exit and voice in debates about workplace democracy by taking seriously his critique of the neoclassical analysis of competition. Pro-market liberals are generally hostile to the idea of workplace democracy and tend to favour top-down hierarchies as a way of organising labour. This hostility is generally inspired by the neoclassical analysis of exploitation and efficiency, which leads them to defend distributions achieved through exit-based competitive equilibria. Following (...) Hirschman, I propose to consider a hypothetical alternative: a democratic equilibrium, reached through the use of voice. I show that it would present the same appealing characteristics than its competitive counterpart while also accounting for the non-ideal conditions in which markets operate. Support for free markets should entail support for workplace democracy minimally understood as a strengthening of voice. (shrink)

The most influential justification of individual property rights is the Propertarian Argument. It is the idea that the institution of private property renders everyone better off, and crucially, even the worst-off members of society. A recent critique of the Argument is that it relies on an anthropologically false hypothesis – the idea, following Thomas Hobbes, that life in the state of nature is one of widespread scarcity and violence to which property rights are a solution. The present article seeks to (...) reformulate this Anthropological Objection in a way that more directly addresses Hobbes’ original argument. It then shows that private property is justified to the extent that it allows anyone to reclaim their free time. (shrink)

There are two opposing views concerning intuitive cases of wage exploitation. The first denies that they are cases of exploitation at all. It is based on the nonworseness claim: there is nothing wrong with a discretionary mutually beneficial employment relationship. The second is the reasonable view: some employment relationships can be exploitative even if employers have no duty towards their employees. This article argues that the reasonable view does not completely defeat defences of wage exploitation, because these do not rely (...) solely on the nonworseness claim. They also rely on the idea, popularised by Alan Wertheimer, that exploitation is a form of disequilibrium price occurring in defective markets. The article then proceeds to criticise Wertheimer’s account through neoclassical, new institutional, and Austrian economics. It concludes that considerations for economic efficiency are irrelevant to assessing intuitions regarding exploitation. (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)

The notion of antifragility, an attribute of systems that makes them thrive under variable conditions, has recently been proposed by Nassim Taleb in a business context. This idea requires the ability of such systems to ‘tinker’, i.e., to creatively respond to changes in their environment. A fairly obvious example of this is natural selection-driven evolution. In this ubiquitous process, an original entity, challenged by an ever-changing environment, creates variants that evolve into novel entities. Analyzing functions that are essential during stationary-state (...) life yield examples of entities that may be antifragile. One such example is proteins with flexible regions that can undergo functional alteration of their side residues or backbone and thus implement the tinkering that leads to antifragility. This in-built property of the cell chassis must be taken into account when considering construction of cell factories driven by engineering principles. (shrink)

What Stanislas Debaene dubs "the number sense" is a natural ability humans share with other animals, enabling us to "count" to four virtually instantaneously. This so-called "accumulator" provides "a direct intuition of what numbers mean". Beyond four, our ability to perceive numbers becomes approximate, though concepts enable us to move beyond approximation. Because humans typically learn number concepts in early childhood, we easily forget that our brains retain the number sense throughout life. This book examines the biological basis for (...) this intuitive ability, with nine chapters organized into three readily graspable groups of three. Aside from its frustrating lack of a clear referencing system, the book is a pleasure to read. (shrink)

In this chapter, we discuss a selection of current views of the neural correlates of consciousness (NCC). We focus on the different predictions they make, in particular with respect to the role of prefrontal cortex (PFC) during visual experiences, which is an area of critical interest and some source of contention. Our discussion of these views focuses on the level of functional anatomy, rather than at the neuronal circuitry level. We take this approach because we currently understand more about experimental (...) evidence at this coarse level and because these results are appropriate for arbitrating between current theoretical frameworks. We discuss the Two-Visual-Systems Hypothesis (Milner & Goodale 1995; 2006), Local Recurrency (Lamme 2010; Lamme 2006), Higher Order (Lau 2008; Lau & Rosenthal 2011) and Global Workspace theories (Baars 1997; Baars 2005; Dehaene & Naccache 2001; Dehaene 2014). Despite the apparent stark differences between conscious and unconscious perceptual processing, available evidence suggests that their neural substrates must be largely shared. This indicates that the difference in neural activity between conscious and unconscious perceptual processing is likely to be subtle and highly specialized. We argue that current experimental evidence about the involvement of specific activity in prefrontal cortex supports the higher order neural theory of consciousness. In consequence, imaging techniques that focus only on marked differences between conscious and unconscious level of activity are likely to be insensitive to the relevant neural activity patterns that underlie conscious experiences. Finally, it follows from the evidence we discuss that the functional advantages of conscious over unconscious perceptual processing may be more limited than commonly thought. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review (...) literature from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

The problem of animal consciousness has profound implications on our concept of nature and of our place in the natural world. In philosophy of mind and cognitive neuroscience the problem of animal consciousness raises two main questions (Velmans, 2007): the distribution question (“are there conscious animals beside humans?”) and the phenomenological question (“what is it like to be a non-human animal?”). In order to answer these questions, many approaches take into account similarities and dissimilarities in animal and human behavior, e.g. (...) the use of language or tools and mirror self-recognition (Allen and Bekoff, 2007), however behavioral arguments don’t seem to be conclusive (Baars, 2005). Cognitive neuroscience is providing comparative data on structural and functional similarities, respectively called “homologies” and “analogies”. Many experimental results suggest that the thalamocortical system is essential for consciousness (Edelman and Tononi, 2000; Tononi, 2008). The argument from homology states that the general structure of thalamocortical system remained the same in the last 100-200 million years, for it is neuroanatomically similar in all the present and past mammals and it didn’t change much during phylogeny (Allen and Bekoff, 2007). The argument from analogy states that the key functional processes correlated with consciousness in humans are still present in all other mammals and many other animals (Baars, 2005). These processes are information integration through effective cortical connectivity (Massimini et al., 2005; Rosanova et al., 2012) and elaboration of information at a global level (Dehaene and Changeux, 2011). On this basis, the Cambridge Declaration on Consciousness states that all mammals, birds, and many other animals (such as octopuses) possess the neurological substrates of consciousness (Low et al., 2012). Conscious experience is private (Chalmers, 1995; Nagel, 1974) therefore the answer to the phenomenological question may be impossible. Nevertheless, cognitive neuroscience may provide an answer to the distribution question, showing that conscious experience is not limited to humans since it is a major biological adaptation going back millions of years. (shrink)

A new strategy for moving forward in the characterization of the Global Neuronal Workspace (GNW) is proposed. According to Dehaene, Changeux and colleagues, broadcasting is the main function of the GNW. However, the dynamic network properties described by recent graph-theoretic GNW models are consistent with many large-scale communication processes that are different from broadcasting. We propose to apply a different graph-theoretic approach, originally developed for optimizing information dissemination in communication networks, which can be used to identify the pattern of (...) frequency and phase-specific directed functional connections that the GNW would exhibit only if it were a broadcasting network. (shrink)

Attention has a long history in philosophy, despite its near absence in the twentieth century. This chapter provides an overview of philosophical research on attention. It begins by explaining the concept of "selection from limitation," contrasting it with the more recent "selection for action." It reviews historical texts that discuss attention, focusing on those in the Western canon whose understanding of "attention" aligns with contemporary usage. It then describes the differential treatment of attention in phenomenology and behaviorism in the last (...) century. Finally, it discusses contemporary research by topic: attention, perception, and knowledge; attention and consciousness; attention and action; and attention and the self. It includes work by Allport,Aristotle, Astell, Augustine, Bergson, Berkeley, Block, Bradley, Buddhaghosa, Campbell, Cherry, De Brigard, Dehaene and Naccache, Descartes, Dickie, Du Bois, Ganeri, Hegel, Heidegger, James, Kant, Koch and Tsuchiya, LaBerge, Locke, Mack and Rock, Malebranche, Merleau-Ponty, Mole, Montemayor and Haladjian, Neumann, Pashler, Posner, Prinz, Reid, Rensink, Reynolds and Heeger, Treisman, Watzl, Wittgenstein, Wollstonecraft, and Wu. (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

Consciousness is known to be limited in processing capacity and often described in terms of a unique processing stream across a single dimension: time. In this paper, we discuss a purely temporal pattern code, functionally decoupled from spatial signals, for conscious state generation in the brain. Arguments in favour of such a code include Dehaene et al.'s long-distance reverberation postulate, Ramachandran's remapping hypothesis, evidence for a temporal coherence index and coincidence detectors, and Grossberg's Adaptive Resonance Theory. A time-bin resonance (...) model is developed, where temporal signatures of conscious states are generated on the basis of signal reverberation across large distances in highly plastic neural circuits. The temporal signatures are delivered by neural activity patterns which, beyond a certain statistical threshold, activate, maintain, and terminate a conscious brain state like a bar code would activate, maintain, or inactivate the electronic locks of a safe. Such temporal resonance would reflect a higher level of neural processing, independent from sensorial or perceptual brain mechanisms. (shrink)

Consciousness is known to be limited in processing capacity and often described in terms of a unique processing stream across a single dimension: time. In this paper, we discuss a purely temporal pattern code, functionally decoupled from spatial signals, for conscious state generation in the brain. Arguments in favour of such a code include Dehaene et al.’s long-distance reverberation postulate, Ramachandran’s remapping hypothesis, evidence for a temporal coherence index and coincidence detectors, and Grossberg’s Adaptive Resonance Theory. A time-bin resonance (...) model is developed, where temporal signatures of conscious states are generated on the basis of signal reverberation across large distances in highly plastic neural circuits. The temporal signatures are delivered by neural activity patterns which, beyond a certain statistical threshold, activate, maintain, and terminate a conscious brain state like a bar code would activate, maintain, or inactivate the electronic locks of a safe. Such temporal resonance would reflect a higher level of neural processing, one that is independent from sensorial or perceptual brain mechanisms. (shrink)

Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...) At least three factors are involved in Einstein's and Wigner's miracles: the physical world, mathematics, and human cognition. One way to relate these factors is to ask how the universe could possibly be structured in such a way that mathematics would be applicable to it, and we would be able to understand that application. This is roughly Wigner's question. Alternatively, the way of the mathematical naturalist is to argue that we abstract certain properties from the world, perhaps using our bodies and physical tools, thereby articulating basic mathematical concepts, which we continue building into the complex formal structures of mathematics. John Stuart Mill, Penelope Maddy, and Rafael Nuñez teach this strategy of cognitive abstraction, in very different manners. But what if the very concepts and basic principles of mathematics were built into our cognitive structure itself? Given such a cognitive a priori mathematical endowment, would the miracles of the link between world and cognition (Einstein) and mathematics and world (Wigner) not vanish, or at least significantly diminish? This is the stance of Stanislas Deheane and Elizabeth Brannon's 2011 anthology, following a venerable rationalist tradition including Plato and Immanuel Kant. (shrink)