Core knowledge of number and space as building bricks of advanced mathematics

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The case of numbers and arithmetic

Numbers have first been studied by cognitive sciences. Over the past decades, the existence of a “number sense”, shared by many animal species including human babies and adults, has been suggested. Indeed, number appears to be one of the fundamental parameters through which human perceive the external world. Some recent studies have shown that the human brain automatically extracts sets numerosity, i.e. the numerical perceptual parameter contained in object sets. A visual illusion on sets of dots, created by Burr and Ross, suggested that there exists a brain system that extracts and can adapt to numerical information (Burr and Ross, 2008). Number actually seems to be more easily perceived than other visual dimensions such as surface, size or spacing (Park et al., 2016). In this last study, authors have shown that 3 orthogonal dimensions – numerosity, size and spacing – could represent the entire parameter space describing the perception of dot arrays. Authors then probed how variations in these three properties quantitatively changed participants’ event-related-potentials (ERPs), and suggested that very precocious ERP activity was more sensitive to numerosity than to other visual dimension.
The brain system perceiving numerosity seems to be present in all human adults and babies. Indeed, 5-years-old infants proved to be able to manipulate abstract amodal representations of numbers to compare quantities or perform simple nonsymbolic additions (Barth et al., 2005). Moreover, Jordan and Brannon (2006) have shown that 7-month-old babies are able to match the number of faces they saw with the number of voices they heard. A similar result was also found in 6-month-old babies (Feigenson, 2011). In a seminal study, 5-month-old also proved to be able to perform basic arithmetical operations with small numbers (Wynn, 1992a). Even 2-days-old human babies are already able to represent numerical information (Izard et al., 2009). In this last experiment, babies were first exposed to auditory series containing a fixed number of syllables (“tu-tu-tu-tu”; “ra-ra-ra-ra”). They were then presented with visual stimuli containing a certain number of objects. Babies looked longer when visual and auditory numbers were matched than when they sufficiently differed (e.g. 4 vs 12). All these findings tend to show that infants are able to detect numerosity, in an abstract manner independent of modality.
Moreover, ethology has revealed that many animal species, from insects, amphibians, birds and fishes, to mammals, including horses, felines and non-human primates, also possess numerical abilities. For example, studies of lions and hyenas in the wild have shown that they can adapt their behavior according to their estimations of the relative number of their intruders. This results in greater vigilance when they have lower numerical advantage, or more risky attitude and even attack when they evaluate pertaining to the largest group (Benson-Amram et al., 2011). Agrillo et al. (2010) have also shown that newborn fishes can discriminate small quantities up to 3, and can identify larger quantities after 40 days of life when they are reared in groups. Rugani et al. (2009) have used imprinting to familiarize newborn chicks to certain numbers, meaning that chicks were reared from birth with a certain number (here 5) of identical objects that were spontaneously considered as “social companions”. After a few days of life, chicks were placed in front of two opaque screens behind which experimenter made imprinted objects disappear either one by one or all at a glance, so that one screen hid 3 objects and the other hid 2 objects. In both conditions, chicks preferentially headed to the screen hiding the biggest amount of imprinted objects, thus suggesting that they were able to discriminate sets of 3 versus 2 objects, without any training, even when sets were not directly visible but memorized. Finally, a lot of research has now been conducted with monkeys. For example, Cantlon et al. (2016) have shown that rhesus monkeys can approximately perform additions and subtractions. Moreover, Matsuzawa and colleagues have shown that chimpanzees can be trained to recognize and associate Arabic numerals from 0 to 19 to corresponding sets of objects (Biro and Matsuzawa, 2001). They can also understand ordinal aspects of numbers and perform very well in tasks requiring sorting numerals in ascending order (Inoue and Matsuzawa, 2007).
Interestingly, numerical perception in all animal species shares the same characteristic: a distance effect measured on a logarithmic scale (Feigenson et al., 2004). In other words, the larger and closer numbers are, the more difficult is the task (figure 0.1). Moyer and Landauer (1967) were the first to verify that Weber’s law applies to number discrimination in adults, whose reaction time was systematically influenced by both the distance and the absolute magnitude of the values represented by two Arabic numerals they were asked to compare. These effects have then been observed in different numerical notations (Buckley and Gillman, 1974; Dehaene et al., 1990; Hinrichs et al., 1981).
Adapted from (Dehaene, 2007) reanalyzing data from (Cantlon and Brannon, 2006). Both error rates and reaction times show distance effects in both humans and monkeys. Indeed, error rate and reaction time increase whenever the log ratio between compared numerosities gets closer to 0 (i.e. compared numerosities are close).
These effects have also been observed in monkeys. In particular, untrained monkeys have proved to be able to discriminate numerosities with accuracy depending on their ratio (Hauser et al., 2003) Interestingly, Cantlon and Brannon (2006) have directly compared humans’ and rhesus macaques’ performance and have shown that macaques were able to choose the smaller of two sets of dots, regardless of covariate parameters such as density, surface or perimeter, with a distance effect similar to humans performing the same comparison task. Indeed, both groups exhibited decrease in accuracy and increase in response time as the ratio between sets approached 1 (figure 0.1). Furthermore, classical effects of human arithmetic have been found in rhesus macaques trained to perform non-symbolic additions and subtractions (Cantlon et al., 2016) and untrained monkeys who spontaneously compute additions of large numbers (Flombaum et al., 2005). Adapting Wynn’s seminal paradigm (Wynn, 1992a) originally introduced to study additions and subtractions in human infants, Flombaum et al. (2005) have shown that rhesus macaques looked longer at impossible compared to possible results of additions of two sets of lemons, only when values were large and differed by a ratio of 1:2 but not when they differed by a ratio of 2:3. According to (Cantlon et al., 2016), monkeys exhibited a ratio effect for addition and subtraction; they exhibited a residual size effect (i.e. systematic decline of accuracy as the magnitude of operands increase) after the ratio effect was regressed out; and they also performed better when the two operands in additions were identical, revealing a classical tie effect. Altogether, these results tend to show that the human brain system for numerosity is inherited from evolutionarily ancient system.
This system has therefore been considered as one of the fundamental “core systems” that all humans possess. According to Spelke’s “core knowledge theory”, there are five such innate domain-specific and encapsulated systems for objects, actions, social partners, numbers and space (or geometry).

The case of space and geometry

The idea that some geometric intuitions are available in human minds from birth can already been found in Antique Greece. In the Meno, Plato leads a young uneducated slave to discover by himself how to double the surface of square, suggesting that some geometric properties are spontaneously accessible. In the past decades, cognitive studies conducted in animals, babies and uneducated adults have revealed that all humans are endowed with evolutionarily ancient basic geometrical intuitions about spatial relations, shapes and their properties.
Three main paradigms have been used in these studies: reorientation tasks, map tasks and intruder tasks. In reorientation tasks, the subject explores a room that has a specific geometric shape, in which a target is hidden (it can be a toy for kids, food for animals). The subject is then disoriented and reintroduced into the room where he is asked to grab the target. In map tasks, subjects are presented with a minimal abstract map constituted of geometrical shapes showing the location of an object they will have to search for or place in a room whose configuration is depicted by the map. Finally, in intruder tasks, subjects are showed a slide with 6 different visual objects that all share a specific geometric property but one. This different one, the intruder, has to be picked by the subject.
These different types of tasks have revealed the existence of two separate core knowledge systems for geometry: one navigational system that extracts information about distances and directions, and one system responsible for the detection of shapes and their properties such as length, angles, symmetries or topology.
(A) adapted from (Lee and Spelke, 2008): exemplar of reorientation task comparing different environment layouts over a rectangular shape. (B) adapted from (Dillon et al., 2013): exemplar of map task. In this experiment, children used six different maps to navigate within triangular arrays. (C), (D) adapted from (Dehaene et al., 2006): intruder task. (C) Examples of slides in which 5 images share a geometric property that is absent from the last image. (D) Strong correlation between performances of Munduruku and American children and adults in this intruder task.
In reorientation tasks, subjects proved to make primary use of geometrical cues such as distance and orientation to navigate the environment, before using visual landmarks such as colors or distinctive signs. Interestingly, these geometric information failed to be extracted from 2D layouts, and  information about length and angles led subjects to systematic and typical errors (Lee and 20 Spelke, 2008; Spelke and Lee, 2012, figure 0.2). For example, 30-month-old children will look for a sticker in every four corners of a square room delimited by walls of different lengths, but they will look for the sticker only in two diagonal pertinent corners in a fragmented rectangle made of walls of the same length but at different distances (Lee et al., 2012). Similar findings underlying spontaneous sensitivity to geometrical cues have been exhibited in several animal species including monkeys (Deipolyi et al., 2001), rats (Cheng, 1986), chicks (Chiandetti and Vallortigara, 2007), fishes (Sovrano et al., 2002) and even ants (Wystrach and Beugnon, 2009). Adaptation of such navigation tasks to computer testing has revealed that even 5-month-old children devoid of experience with independent locomotion, were sensitive to geometrical cues present in an enclosed triangular layout (Lourenco and Huttenlocher, 2008).
Intruder tasks have nevertheless suggested that humans possess spontaneous intuitions of 2D shapes and their properties. In particular, a seminal study have revealed that, although Amazonian Munduruku people are largely deprived of formal schooling and possess an impoverished lexicon for numerical and geometrical concepts, they can spontaneously identify a wide range of geometrical concepts such as shapes (circle, square, right-angled triangle, etc.), Euclidean properties (parallelism, alignment, etc.), topological properties (closure, connectedness, etc.), metric properties (distance, proportion, etc.) and symmetries (Dehaene et al., 2006). Notably, arguing for a certain universality of the patterns of difficulty of the tested geometrical concepts, strong correlations were found between Mundurukus children and adults’ performance, and American children and adults’ performance (Dehaene et al., 2006, figure 0.2). This original intruder task was then adapted to test specifically for the human ability to extract information about angles, length and direction (or sense) from 2D displays (Izard et al., 2011a). In trials where the intruder varied only in size, angle or sense, as well as in trials where another dimension interfered so that various deviants could be picked, all age groups (from 3 to 30) proved to detect better angle and size intruders than sense deviants. In particular young 3/4-years-old children used only angle and size, thus confirming the existence of two separate geometrical systems, one that recollects direction and orientation from 3D navigational environment, and another that extracts length and angle from 2D shapes (Spelke et al., 2010). Note that these results find support in previous habituation tasks showing the sensitivity to angle and length of young and even a few-hours-old infants (Newcombe et al., 1999; Slater et al., 1991) Finally, map tasks have revealed that children and uneducated adults are able to read and use geometrical information contained in abstract maps, even though it is the first time they are presented with such tool, to locate a target object (Dehaene et al., 2006; Izard et al., 2014). Interestingly, 4-years-old children provided with a geometrical map show flexible use of the two core knowledge systems to navigate in a triangular room delimited either by three distant walls or three distinct corners (figure 0.2). Although there is no evidence for any transfer from one system to another, they were able to extract in both situations pertinent information respectively about distance and angle (Dillon et al., 2013).

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Other intuitive mathematical components: probabilities and inferences

Recent studies have suggested that babies are able, very early in their development, to internalize and update probabilities of external events, to evaluate the plausibility of simultaneous hypotheses and to use probabilities to generate predictions and compare them to incoming external data. In particular, babies exhibit sensitivity to statistical regularities and are able to make bidirectional probabilistic inferences.
First, an important study conducted by Saffran et al. (1996) has revealed that 8-month-old infants are able to learn temporal statistical regularities. Infants were presented with a succession of syllables constituted of four different 3-syllabic “words” randomly chained, such that within a given “word” the transition probability between syllables was equal to 1, but transition probabilities between the last syllable of a “word” and the first syllable of another “word” was equal to 1/3. Authors showed that infants’ looking time was greater for new or rare isolated words than for words of the initial sequence. Their result therefore suggested that young infants spontaneously and quickly built an internal representation of statistical information available in the sequence and used it to detect novel words that did not follow initial probabilities. Moreover, Marcus et al. (1999) have suggested that this capacity for statistical learning does not only apply to specific items but can also underlie the acquisition of more abstract “algebraic patterns”, i.e. the abstract rule, underlying a set of specific sequences. By 7 months of age, infants can already understand that the set {aab, ccd, eeg} includes sequences systematically composed with a repetition of any two items followed by a third one. Such capacity for regularity learning does not seem to be grounded in a specific sensory modality but is rather abstract. Indeed, a few hours-old newborns were already able to identify regularities in a visual sequence of geometrical shapes forming pairs presented in random order (Bulf et al., 2011).
Second, using situations in which subjects were confronted to random sampling of collections of objects with different properties, some studies have now suggested that young children possess a certain sense of probabilistic inference (Denison et al., 2013; Denison and Xu, 2010; Kushnir et al., 2010; Teglas et al., 2011; Téglás et al., 2007; Xu and Denison, 2009; Xu and Garcia, 2008). In 2007, Téglás and Bonatti have shown that 12-month-old babies can anticipate the probability of a forthcoming event. In their experiment, babies are presented with objects of different colors colliding in a box, e.g. 3 blue objects and a yellow one. The box then becomes opaque and one object gets out of the box. When the least probable object gets out of the box, e.g. the only yellow one; babies look longer, therefore indicating their surprise (figure 0.3). Authors showed that neither physical characteristics of the final screen nor frequency played a role in infants’ behavior, which was instead driven by a feeling of improbability (Téglás et al., 2007).
(A) Adapted from (Téglás et al., 2007): mean looking time of infants (c) when presented with probable (b) or improbable (d) outcome of a lottery (a). (B) Adapted from (Xu and Garcia, 2008): infants look longer when the population of red and white balls mismatched the sample.
In 2008, similar results have been found in 8-month-old babies. In their experiment, Xu and Garcia had first familiarized babies with boxes containing a majority of either red or white balls. At the beginning of each trial, the content of boxes was hidden. The experimenter then designated one box, closed his eyes, and picked 5 balls from the box, 4 in one color and 1 in the other color. After revealing the content of the box, babies looked longer when it did not match the sample (figure 0.3). Conversely, babies could also use information about the whole population to predict what samples were most probable. Indeed, when the content of the box was visible from start, babies looked longer when the picked sample was improbable (Xu and Garcia, 2008).
These findings have suggested that young babies are able to perform bidirectional probabilistic inferences. In 2014, Fontanari and collaborators have also shown that Mayan subjects, devoid of any mathematical training, were able to indicate better than chance from which set a red object was more likely to be picked, even in situation where set size was incongruent with the proportion of red objects relative to black objects. In a second experiment, authors also showed that when subjects were asked to indicate whether two objects picked out of one specific set would have the same color or not, the subjective probability of their answer was tuned to the objective probability computed by enumerating the relative number of colored objects in the set (Fontanari et al., 2014).
Human capacity to perform probabilistic inference appears to be grounded in two other mathematical abilities. First, it could be linked to a specific intuition for proportions, i.e. ratios between set numerosities (Denison and Xu, 2014). Recently, Hubbard, Matthews and colleagues have shown that humans have perceptual access to non-symbolic ratio magnitudes and can estimate and compare ratios, in symbolic and non-symbolic contexts, with typical distance effect (Lewis et al., 2016; Matthews and Chesney, 2015).
The ability of young infants to perform probabilistic inference also leans on intuitions of logical reasoning. Cognitive sciences have started to address this issue in infants. Gopnik and colleagues have notably shown that 2.5-years-old children are able to perform quick and sophisticated inferences. In their experiments, children were presented with a “blicket” detector and various objects that were to be labeled or not as “blicket”. In the first condition, one object alone activated the “blicket” detector, another object alone did not, and both objects together activated the detector. In the second condition, one objet systematically activated the detector and another object activated the detector only 66% of the time. In the first condition, toddlers correctly labeled only the first objet as “blicket” and both objects in the second condition (Gopnik et al., 2001).
Some work led on great apes has revealed that humans share such intuitions of logical inference with other animal species. In particular, Call placed two cups in front of orangutans, chimpanzees, gorillas or bonobos, presented them with food, hid the food in one of the two cups and closed them, before revealing the content of either both or only one cup. Results notably suggested that, when presented with the empty cup, great apes were able to reason by exclusion to look for food in the other cup (Call, 2004). Certain forms of deduction, such as transitive inference (“A < B < C D”), have also been observed in rats (Davis, 1992), birds (Bond et al., 2003), and fishes (Grosenick et al., 2007).

Neural correlates of numerical processing

Early neuropsychological and fMRI works have led to the hypothesis that parts of the intraparietal sulcus (IPS) play a central role in the representation of numbers (figure 0.4).

Table of contents :

Chapter 1.Origins of the brain networks for advanced mathematics in expert mathematicians
1. Introduction to the article
2. Abstract
3. Introduction
4. Methods
4.1. Participants
4.2. Visual runs
4.3. Auditory runs
4.4. Localizer scan
4.5. Post-MRI questionnaire
4.6. fMRI data acquisition and analysis
5. Results
5.1. Behavioral results
5.1.1. Behavioral results in auditory runs
5.1.2. Behavioral results in visual runs
5.1.3. Subjective variables reported during the post-MRI questionnaire
5.2. fMRI activations associated with mathematical reflection
5.3. Variation in brain activation across mathematical problems
5.4. fMRI activations associated with meaningful mathematical reflection
5.5. Controls for task difficulty
5.6. Dissociation with the areas activated during non-mathematical reflection
5.7. Activation profile in language areas
5.8. Relationships between mathematics, calculation, and number detection
5.9. Activations during the sentence-listening period
5.10. Differences between mathematicians and controls in ventral visual cortex
6. Discussion
7. Supplementary tables
Chapter 2. Dissociated cortical networks for mathematical and non-mathematical knowledge
1. Introduction to the article
2. Abstract
3. Introduction
4. Experiment 1: Simple mathematical facts
4.1. Introduction
4.2. Methods
4.3. Results
4.3.1. Behavior
4.3.2. Dissociation between brain activations to math and nonmath reflection
4.3.3. Effect of difficulty
4.3.4. Differences between math types
4.3.5. Activation profile in language areas
4.3.6. Activation profile in math-responsive areas
4.4. Conclusion
5. Experiment 2: effect of minimal combinatorial operations such as quantifiers and negation. 99
5.1. Introduction
5.2. Method
5.3. Results
5.3.1. Behavior
5.3.2. Math versus nonmath dissociation
5.3.3. Activation profile in auditory and language areas
5.3.4. Effect of quantifiers
5.3.5. Effect of negation
5.4. Conclusion
6. Discussion
7. Methods
7.1. Ethics statement
7.2. Stimuli
7.3. Procedure
7.4. Syntax localizer
7.5. fMRI data acquisition and analysis
8. Supplementary materials
Chapter 3. On the role of visual experience in mathematical development: Evidence from blind mathematicians 
1. Introduction to the article 
2. Abstract
3. Introduction 
4. Methods 
4.1. Participants
4.2. Description of the blind participants
4.3. Experiment 1
4.4. Experiment 2
4.5. Procedure
4.6. fMRI data acquisition and analysis
5. Results
5.1. Experiment 1: advanced mathematical statements
5.2. Experiment 2. Simpler mathematical facts
6. Discussion 
7. Supplementary information
Chapter 4. The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers 
1. Introduction to the article
2. Abstract 
3. Introduction 
3.1. Language
3.2. Stimulus sequences
4. Experiment 
4.1. Methods
4.1.1. Ethics statement
4.1.2. Participants
4.1.3. Procedure
4.1.4. Stimuli
4.1.5. Statistical analysis
4.2. Results
4.3. Discussion
5. Experiment 
5.1. Methods
5.1.1. Participants
5.1.2. Procedure
5.1.3. Stimuli
5.2. Results
5.3. Discussion
6. Experiment 
6.1. Methods
6.1.1. Participants
6.1.2. Stimuli and procedure
6.2. Results
6.3. Discussion
7. Experiment
7.1. Methods
7.1.1. Participants
7.1.2. Stimuli and procedure
7.2. Results
7.3. Discussion
8. Detailed fitting of the “language of geometry” model 
8.1. Model description
8.2. Fits to adults’ data
8.3. Fits to children’s data
8.4. Fits to Munduruku data
9. Discussion
9.1. Geometrical primitives
9.2. Embedded expressions
9.3. Minimal description length as a predictor of spatial memory
10. Supporting information

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