Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures

Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Processing symbolic magnitude information conveyed by number words and by scalar adjectives

Humans not only process and compare magnitude information such as size, duration, and number perceptually, but they also communicate about these properties using language. In this respect, a relevant class of lexical items are so-called scalar adjectives like “big,” “long,” “loud,” and so on which refer to magnitude information. It has been proposed that humans use an amodal and abstract representation format shared by different dimensions, called the generalised magnitude system (GMS). In this paper, we test the hypothesis that scalar adjectives are symbolic references to GMS representations, and, therefore, GMS gets involved in processing their meaning. Previously, a parallel hypothesis on the relation between number symbols and GMS representations has been tested with the size congruity paradigm. The results of these experiments showed interference between the processing of number symbols and the processing of physical (font-) size. In the first three experiments of the present study (total N = 150), we used the size congruity paradigm and the same/different task to look at the potential interaction between physical size magnitude and numerical magnitude expressed by number words. In the subsequent three experiments (total N = 149), we looked at a parallel potential interaction between physical size magnitude and scalar adjective meaning. In the size congruity paradigm, we observed interference between the processing of the numerical value of number words and the meaning of scalar adjectives, on the one hand, and physical (font-) size, on the other hand, when participants had to judge the number words or the adjectives (while ignoring physical size). No interference was obtained for the reverse situation, i.e., when participants judged the physical font size (while ignoring numerical value or meaning). The results of the same/different task for both number words and scalar adjectives strongly suggested that the interference that was observed in the size congruity paradigm was likely due to a response conflict at the decision stage of processing rather than due to the recruitment of GMS representations. Taken together, it can be concluded that the size congruity paradigm does not provide evidence in support the hypothesis that GMS representations are used in the processing of number words or scalar adjectives. Nonetheless, the hypothesis we put forward about scalar adjectives is still is a promising potential line of research. We make a number of suggestions for how this hypothesis can be explored in future studies.

Impact of sharing full versus averaged social information on social influence and estimation accuracy

The recent developments of social networks and recommender systems have dramatically increased the amount of social information shared in human communities, challenging the human ability to process it. As a result, sharing aggregated forms of social information is becoming increasingly popular. However, it is unknown whether sharing aggregated information improves people's judgments more than sharing the full available information. Here, we compare the performance of groups in estimation tasks when social information is fully shared versus when it is first averaged and then shared. We find that improvements in estimation accuracy are comparable in both cases. However, our results reveal important differences in subjects' behaviour: (i) subjects follow the social information more when receiving an average than when receiving all estimates, and this effect increases with the number of estimates underlying the average; (ii) subjects follow the social information more when it is higher than their personal estimate than when it is lower. This effect is stronger when receiving all estimates than when receiving an average. We introduce a model that sheds light on these effects, and confirms their importance for explaining improvements in estimation accuracy in all treatments.

A unifying Bayesian framework accounting for spatiotemporal interferences with a deceleration tendency

Spatial and temporal levels of information processing interfere with each other. The Kappa effect is a well-known spatiotemporal interference in which the estimated time between two lights increases as the distance between them increases, showing a deceleration tendency. A classical model attributes this interference to constant speeds and predicts a linear relation, whereas a slowness model attributes the interference to slow speeds and proposes that the tendency is due to the uncertainty of stimuli locations. This study integrated a unifying Bayesian framework with the classical model and argued that this tendency is the result of the Weber-Fechner law. This hypothesis was tested via two time discrimination tasks that manipulated the uncertainty of stimuli locations and the distance between stimuli. Experiment 1 showed that the estimated time was not modulated by the uncertainty of the stimuli locations. Experiment 2 revealed that the behavioral predictions made by the Bayesian model on logarithmic scales were more accurate than those made by the linear model. Our results suggest that the deceleration tendency in the Kappa effect is the result of the Weber-Fechner law.

Accuracy-Risk Trade-Off Due to Social Learning in Crowd-Sourced Financial Predictions

A critical question relevant to the increasing importance of crowd-sourced-based finance is how to optimize collective information processing and decision-making. Here, we investigate an often under-studied aspect of the performance of online traders: beyond focusing on just accuracy, what gives rise to the trade-off between risk and accuracy at the collective level? Answers to this question will lead to designing and deploying more effective crowd-sourced financial platforms and to minimizing issues stemming from risk such as implied volatility. To investigate this trade-off, we conducted a large online Wisdom of the Crowd study where 2037 participants predicted the prices of real financial assets (S&P 500, WTI Oil and Gold prices). Using the data collected, we modeled the belief update process of participants using models inspired by Bayesian models of cognition. We show that subsets of predictions chosen based on their belief update strategies lie on a Pareto frontier between accuracy and risk, mediated by social learning. We also observe that social learning led to superior accuracy during one of our rounds that occurred during the high market uncertainty of the Brexit vote.

Biological scaling analyses are more than statistical line fitting

The magnitude of many biological traits relates strongly and regularly to body size. Consequently, a major goal of comparative biology is to understand and apply these 'size-scaling' relationships, traditionally quantified by using linear regression analyses based on log-transformed data. However, recently some investigators have questioned this traditional method, arguing that linear or non-linear regression based on untransformed arithmetic data may provide better statistical fits than log-linear analyses. Furthermore, they advocate the replacement of the traditional method by alternative specific methods on a case-by-case basis, based simply on best-fit criteria. Here, I argue that the use of logarithms in scaling analyses presents multiple valuable advantages, both statistical and conceptual. Most importantly, log-transformation allows biologically meaningful, properly scaled (scale-independent) comparisons of organisms of different size, whereas non-scaled (scale-dependent) analyses based on untransformed arithmetic data do not. Additionally, log-based analyses can readily reveal biologically and theoretically relevant discontinuities in scale invariance during developmental or evolutionary increases in body size that are not shown by linear or non-linear arithmetic analyses. In this way, log-transformation advances our understanding of biological scaling conceptually, not just statistically. I hope that my Commentary helps students, non-specialists and other interested readers to understand the general benefits of using log-transformed data in size-scaling analyses, and stimulates advocates of arithmetic analyses to show how they may improve our understanding of scaling conceptually, not just statistically.

Cultural Evolution of Genetic Heritability

Behavioral genetics and cultural evolution have both revolutionized our understanding of human behavior-largely independent of each other. Here we reconcile these two fields under a dual inheritance framework, offering a more nuanced understanding of the interaction between genes and culture. Going beyond typical analyses of gene-environment interactions, we describe the cultural dynamics that shape these interactions by shaping the environment and population structure. A cultural evolutionary approach can explain, for example, how factors such as rates of innovation and diffusion, density of cultural sub-groups, and tolerance for behavioral diversity impact heritability estimates, thus yielding predictions for different social contexts. Moreover, when cumulative culture functionally overlaps with genes, genetic effects become masked, unmasked, or even reversed, and the causal effects of an identified gene become confounded with features of the cultural environment. The manner of confounding is specific to a particular society at a particular time, but a WEIRD (Western, educated, industrialized, rich, democratic) sampling problem obscures this boundedness. Cultural evolutionary dynamics are typically missing from models of gene-to-phenotype causality, hindering generalizability of genetic effects across societies and across time. We lay out a reconciled framework and use it to predict the ways in which heritability should differ between societies, between socioeconomic levels and other groupings within some societies but not others, and over the life course. An integrated cultural evolutionary behavioral genetic approach cuts through the nature-nurture debate and helps resolve controversies in topics such as IQ.

The history of number words in the world's languages-what have we learnt so far?

For over 100 years, researchers from various disciplines have been enthralled and occupied by the study of number words. This article discusses implications for the study of deep history and human evolution that arise from this body of work. Phylogenetic modelling shows that low-limit number words are preserved across thousands of years, a pattern consistently observed in several language families. Cross-linguistic frequencies of use and experimental studies also point to widespread homogeneity in the use of number words. Yet linguistic typology and field documentation reports caution against positing a privileged linguistic category for number words, showing a wealth of variation in how number words are encoded across the world. In contrast with low-limit numbers, the higher numbers are characterized by a rapid and morphologically consistent pattern of expansion, and behave like grammatical phrasal units, following language-internal rules. Taken together, the evidence suggests that numbers are at the cross-roads of language history. For languages that do have productive and consistent number systems, numerals one to five are among the most reliable available linguistic fossils of deep history, defying change yet still bearing the marks of the past, while higher numbers emerge as innovative tools looking to the future, derived using language-internal patterns and created to meet the needs of modern speakers. This article is part of the theme issue 'Reconstructing prehistoric languages'.

Range and distribution effects on number line placement

People's placement of numbers on number lines sometimes shows linear and sometimes compressive scaling. We investigated whether people's placement of numbers was affected by their range and distribution, as indicated by Parducci's (Psychological Review, 72, 407-418, 1965) range-frequency theory. Experiment 1 found large compressive effects when the endpoints were 1 and 10¹⁶. Experiment 2 showed compression when 14 logarithmically distributed numbers were placed on a line marked 1-1,000 and close to linear scaling when the numbers were linearly distributed. Thus, we found both range and frequency effects on compression. Where compression arose, it was not as pronounced as that predicted by logarithmic scaling, but analyses of the results from Experiments 1 and 2 indicate this was not explained by participants switching between linear and logarithmic scaling.

Location- and object-based attention enhance number estimation

Humans and non-humans can extract an estimate of the number of items in a collection very rapidly, raising the question of whether attention is necessary for this process. Visual attention operates in various modes, showing selectivity both to spatial location and to objects. Here, we tested whether each form of attention can enhance number estimation, by measuring whether presenting a visual cue to increase attentional engagement will lead to a more accurate and precise representation of number, both when attention is directed to location and when it is directed to objects. Results revealed that enumeration of a collection of dots in the location previously cued led to faster, more precise, and more accurate judgments than enumeration in un-cued locations, and a similar benefit was seen when the cue and collection appeared on the same object. This work shows that like many other perceptual tasks, numerical estimation may be enhanced by the spread of active attention inside a pre-cued object.

How to better communicate the exponential growth of infectious diseases

Exponential growth bias is the phenomenon whereby humans underestimate exponential growth. In the context of infectious diseases, this bias may lead to a failure to understand the magnitude of the benefit of non-pharmaceutical interventions. Communicating the same scenario in different ways (framing) has been found to have a large impact on people's evaluations and behavior in the contexts of social behavior, risk taking and health care. We find that framing matters for people's assessment of the benefits of non-pharmaceutical interventions. In two commonly used frames, most subjects in our experiment drastically underestimate the number of cases avoided by adopting non-pharmaceutical interventions. Framing growth in terms of doubling times rather than growth rates reduces the bias. When the scenario is framed in terms of time gained rather than cases avoided, the median subject assesses the benefit of non-pharmaceutical interventions correctly. These findings suggest changes that could be adopted to better communicate the exponential spread of infectious diseases.

Boys' advantage on the fractions number line is mediated by visuospatial attention: Evidence for a parietal-spatial contribution to number line learning

The study tested the hypotheses that boys will have an advantage learning the fractions number line and this advantage will be mediated by spatial abilities. Fractions number line and, as a contrast, fractions arithmetic performance were assessed for 342 adolescents, as was their intelligence, working memory, and various spatial abilities. Boys showed smaller placement errors on the fractions number line (d = -0.22) and correctly solved more fractions arithmetic problems (d = 0.23) than girls. Working memory and intelligence predicted performance on both fractions measures, and a measure of visuospatial attention uniquely predicted number line performance and fully mediated the sex difference. Visuospatial working memory uniquely predicted fractions arithmetic performance and fully mediated the sex difference. The results help to clarify the nuanced relations between spatial abilities and formal mathematics learning and the sex differences that often emerge in mathematical domains that have a visuospatial component.

Numerical averaging in mice

Rodents can be trained to associate different durations with different stimuli (e.g., light/sound). When the associated stimuli are presented together, maximal responding is observed around the average of individual durations (akin to averaging). The current study investigated whether mice can also average independently trained numerosities. Mice were initially trained to make 10 or 20 lever presses on a single (run) lever to obtain a reward and each fixed-ratio schedule was signaled either with an auditory or visual stimulus. Then, mice were trained to press another lever to obtain the reward after they responded on the run lever for the minimum number of presses [Fixed Consecutive Number (FCN)-10 or -20 trials] signaled by the corresponding discriminative stimulus. Following this training, FCN trials with the compound stimulus were introduced to test the counting behavior of mice when they encountered conflicting information regarding the number of responses required to obtain the reward. Our results showed that the numbers of responses on these compound test trials were around the average of the number of responses in FCN-10 and FCN-20 trials particularly when the auditory stimulus was associated with a fewer number of required responses. The counting strategy explained the behavior of the majority of the mice in the FCN-Compound test trials (as opposed to the timing strategy). The number of responses in FCN-Compound trials was accounted for equally well by the arithmetic, geometric, and Bayesian averages of the number of responses observed in FCN-10 and FCN-20 trials.

Logarithmic encoding of ensemble time intervals

Although time perception is based on the internal representation of time, whether the subjective timeline is scaled linearly or logarithmically remains an open issue. Evidence from previous research is mixed: while the classical internal-clock model assumes a linear scale with scalar variability, there is evidence that logarithmic timing provides a better fit to behavioral data. A major challenge for investigating the nature of the internal scale is that the retrieval process required for time judgments may involve a remapping of the subjective time back to the objective scale, complicating any direct interpretation of behavioral findings. Here, we used a novel approach, requiring rapid intuitive ‘ensemble’ averaging of a whole set of time intervals, to probe the subjective timeline. Specifically, observers’ task was to average a series of successively presented, auditory or visual, intervals in the time range 300–1300 ms. Importantly, the intervals were taken from three sets of durations, which were distributed such that the arithmetic mean (from the linear scale) and the geometric mean (from the logarithmic scale) were clearly distinguishable. Consistently across the three sets and the two presentation modalities, our results revealed subjective averaging to be close to the geometric mean, indicative of a logarithmic timeline underlying time perception.

Differences and ratios in a nonsymbolic 'Artificial algebra': Effects of extended training

Grace et al. (2018) showed that humans could estimate ratios and differences of stimulus magnitudes by feedback and without explicit instruction in a nonsymbolic 'artificial algebra' task, but that responding depended on both operations even though only one was trained. Here we asked whether control by the trained operation would increase over several sessions, that is, if perceptual learning would occur. Observers (n = 16) completed four sessions in which feedback was based on either ratios or differences for stimulus pairs that varied in brightness (Experiment 1) or line length (Experiment 2). Results showed that control by the trained and untrained operations increased and decreased, respectively, over the sessions, indicating perceptual learning. For about two thirds of individual sessions, regressions indicated significant control by both differences and ratios, suggesting that the perceptual system automatically computes two operations. The similarity of results across experiments with both intensive (brightness) and extensive (line length) stimulus dimensions suggests that differences and ratios are computed centrally, perhaps as part of a general system for processing magnitudes (cf. Walsh, 2003).

The impact of incorrect social information on collective wisdom in human groups

A major problem resulting from the massive use of social media is the potential spread of incorrect information. Yet, very few studies have investigated the impact of incorrect information on individual and collective decisions. We performed experiments in which participants had to estimate a series of quantities, before and after receiving social information. Unbeknownst to them, we controlled the degree of inaccuracy of the social information through 'virtual influencers', who provided some incorrect information. We find that a large proportion of individuals only partially follow the social information, thus resisting incorrect information. Moreover, incorrect information can help improve group performance more than correct information, when going against a human underestimation bias. We then design a computational model whose predictions are in good agreement with the empirical data, and sheds light on the mechanisms underlying our results. Besides these main findings, we demonstrate that the dispersion of estimates varies a lot between quantities, and must thus be considered when normalizing and aggregating estimates of quantities that are very different in nature. Overall, our results suggest that incorrect information does not necessarily impair the collective wisdom of groups, and can even be used to dampen the negative effects of known cognitive biases.

Knowledge before Belief

Research on the capacity to understand others' minds has tended to focus on representations of beliefs, which are widely taken to be among the most central and basic theory of mind representations. Representations of knowledge, by contrast, have received comparatively little attention and have often been understood as depending on prior representations of belief. After all, how could one represent someone as knowing something if one doesn't even represent them as believing it? Drawing on a wide range of methods across cognitive science, we ask whether belief or knowledge is the more basic kind of representation. The evidence indicates that non-human primates attribute knowledge but not belief, that knowledge representations arise earlier in human development than belief representations, that the capacity to represent knowledge may remain intact in patient populations even when belief representation is disrupted, that knowledge (but not belief) attributions are likely automatic, and that explicit knowledge attributions are made more quickly than equivalent belief attributions. Critically, the theory of mind representations uncovered by these various methods exhibit a set of signature features clearly indicative of knowledge: they are not modality-specific, they are factive, they are not just true belief, and they allow for representations of egocentric ignorance. We argue that these signature features elucidate the primary function of knowledge representation: facilitating learning from others about the external world. This suggests a new way of understanding theory of mind-one that is focused on understanding others' minds in relation to the actual world, rather than independent from it.

A Unified Theory of Psychophysical Laws in Auditory Intensity Perception

Psychophysical laws quantitatively relate perceptual magnitude to stimulus intensity. While most people have accepted Stevens's power function as the psychophysical law, few believe in Fechner's original idea using just-noticeable-differences (jnd) as a constant perceptual unit to educe psychophysical laws. Here I present a unified theory in hearing, starting with a general form of Zwislocki's loudness function (1965) to derive a general form of Brentano's law. I will arrive at a general form of the loudness-jnd relationship that unifies previous loudness-jnd theories. Specifically, the "slope," "proportional-jnd," and "equal-loudness, equal-jnd" theories, are three additive terms in the new unified theory. I will also show that the unified theory is consistent with empirical data in both acoustic and electric hearing. Without any free parameters, the unified theory uses loudness balance functions to successfully predict the jnd function in a wide range of hearing situations. The situations include loudness recruitment and its jnd functions in sensorineural hearing loss and simultaneous masking, loudness enhancement and the midlevel hump in forward and backward masking, abnormal loudness and jnd functions in cochlear implant subjects. Predictions of these loudness-jnd functions were thought to be questionable at best in simultaneous masking or not possible at all in forward masking. The unified theory and its successful applications suggest that although the specific form of Fechner's law needs to be revised, his original idea is valid in the wide range of hearing situations discussed here.

Toward a Refined Mindfulness Model Related to Consciousness and Based on Event-Related Potentials

Neuroimaging, behavioral, and self-report evidence suggests that there are four main cognitive mechanisms that support mindfulness: (a) self-regulation of attention, (b) improved body awareness, (c) improved emotion regulation, and (d) change in perspective on the self. In this article, we discuss these mechanisms on the basis of the event-related potential (ERP). We reviewed the ERP literature related to mindfulness and examined a data set of 29 articles. Our findings show that the neural features of mindfulness are consistently associated with the self-regulation of attention and, in most cases, reduced reactivity to emotional stimuli and improved cognitive control. On the other hand, there appear to be no studies of body awareness. We link these electrophysiological findings to models of consciousness and introduce a unified, mechanistic mindfulness model. The main idea in this refined model is that mindfulness decreases the threshold of conscious access. We end with several working hypotheses that could direct future mindfulness research and clarify our results.

Sequential Presentation Protects Working Memory From Catastrophic Interference

Neural network models of memory are notorious for catastrophic interference: Old items are forgotten as new items are memorized (French, 1999; McCloskey & Cohen, 1989). While working memory (WM) in human adults shows severe capacity limitations, these capacity limitations do not reflect neural network style catastrophic interference. However, our ability to quickly apprehend the numerosity of small sets of objects (i.e., subitizing) does show catastrophic capacity limitations, and this subitizing capacity and WM might reflect a common capacity. Accordingly, computational investigations (Knops, Piazza, Sengupta, Eger & Melcher, 2014; Sengupta, Surampudi & Melcher, 2014) suggest that mutual inhibition among neurons can explain both kinds of capacity limitations as well as why our ability to estimate the numerosity of larger sets is limited according to a Weber ratio signature. Based on simulations with a saliency map-like network and mathematical proofs, we provide three results. First, mutual inhibition among neurons leads to catastrophic interference when items are presented simultaneously. The network can remember a limited number of items, but when more items are presented, the network forgets all of them. Second, if memory items are presented sequentially rather than simultaneously, the network remembers the most recent items rather than forgetting all of them. Hence, the tendency in WM tasks to sequentially attend even to simultaneously presented items might not only reflect attentional limitations, but also an adaptive strategy to avoid catastrophic interference. Third, the mean activation level in the network can be used to estimate the number of items in small sets, but it does not accurately reflect the number of items in larger sets. Rather, we suggest that the Weber ratio signature of large number discrimination emerges naturally from the interaction between the limited precision of a numeric estimation system and a multiplicative gain control mechanism.

Gender differences in the development of semantic and spatial processing of numbers

This study recruited kindergarteners and first graders to investigate gender and grade differences in semantic and spatial processing of number magnitude. Results based on the Bayesian statistics showed that (1) there was extreme evidence in favour of grade differences in both semantic processing and spatial processing; (2) there were no gender differences in semantic processing; and (3) boys developed earlier than girls in spatial processing of numbers, especially for the more difficult task. These results are discussed in terms of gender differences in cognitive mechanisms underlying semantic and spatial processing of number magnitude.

Large as being on top of the world and small as hitting the roof: a common magnitude representation for the comparison of emotions and numbers

Previous work on the direct speed-intensity association (SIA) on comparative judgement tasks involved spatially distributed responses over spatially distributed stimuli with high motivational significance like facial expressions of emotions. This raises the possibility that the inferred stimulus-driven regulation of lateralized motor reactivity described by SIA, which was against the one expected on the basis of a valence-specific lateral bias, was entirely due to attentional capture from motivational significance (beyond numerical cognition). In order to establish the relevance of numerical cognition on the regulation of attentional capture we ran two complementary experiments. These involved the same direct comparison task on stimulus pairs that were fully comparable in terms of their analog representation of intensity but with different representational domain and motivational significance: symbolic magnitudes with low motivational significance in experiment 1 vs. emotions with rather high motivational significance in experiment 2. The results reveal a general SIA and point to a general mechanism regulating comparative judgements. This is based on the way spatial attention is captured toward locations that contain the stimulus which is closest in term of relative intensity to the extremal values of the series, regardless from its representational domain being it symbolic or emotional.

Domain-general cognitive functions fully explained growth in nonsymbolic magnitude representation but not in symbolic representation in elementary school children

In this study, we aimed to compare developmental changes in nonsymbolic and symbolic magnitude representations across the elementary school years. For this aim, we used a four-wave longitudinal study with a one-year interval in schoolchildren in grades 1-4 in Russia and Kyrgyzstan (N = 490, mean age was 7.65 years at grade 1). The results of mixed-effects growth models revealed that growth in the precision of symbolic representation was larger than in the nonsymbolic representation. Moreover, growth in nonsymbolic representation was fully explained by growth in fluid intelligence (FI), visuospatial working memory (VSWM) and processing speed (PS). The analysis demonstrated that growth in nonsymbolic magnitude representation was significant only for pupils with a high level of FI and PS, whereas growth in precision of symbolic representation did not significantly vary across pupils with different levels of FI or VSWM.

Searching for the Critical p of Macphail's Null Hypothesis: The Contribution of Numerical Abilities of Fish

In 1985, Macphail argued that there are no differences among the intellects of non-human vertebrates and that humans display unique cognitive skills because of language. Mathematical abilities represent one of the most sophisticated cognitive skills. While it is unquestionable that humans exhibit impressive mathematical skills associated with language, a large body of experimental evidence suggests that Macphail hypothesis must be refined in this field. In particular, the evidence that also small-brained organisms, such as fish, are capable of processing numerical information challenges the idea that humans display unique cognitive skills. Like humans, fish may take advantage of using continuous quantities (such as the area occupied by the objects) as proxy of number to select the larger/smaller group. Fish and humans also showed interesting similarities in the strategy adopted to learn a numerical rule. Collective intelligence in numerical estimation has been also observed in humans and guppies. However, numerical acuity in humans is considerably higher than that reported in any fish species investigated, suggesting that quantitative but not qualitative differences do exist between humans and fish. Lastly, while it is clear that contextual factors play an important role in the performance of numerical tasks, inter-species variability can be found also when different fish species were tested in comparable conditions, a fact that does not align with the null hypothesis of vertebrate intelligence. Taken together, we believe that the recent evidence of numerical abilities in fish call for a deeper reflection of Macphail's hypothesis.

Identification of Differentially Expressed Genes to Establish New Biomarker for Cancer Prediction

The goal of the human genome project is to integrate genetic information into different clinical therapies. To achieve this goal, different computational algorithms are devised for identifying the biomarker genes, cause of complex diseases. However, most of the methods developed so far using DNA microarray data lack in interpreting biological findings and are less accurate in disease prediction. In the paper, we propose two parameters risk_factor and confusion_factor to identify the biologically significant genes for cancer development. First, we evaluate risk_factor of each gene and the genes with nonzero risk_factor result misclassification of data, therefore removed. Next, we calculate confusion_factor of the remaining genes which determines confusion of a gene in prediction due to closeness of the samples in the cancer and normal classes. We apply nondominated sorting genetic algorithm (NSGA-II) to select the maximally uncorrelated differentially expressed genes in the cancer class with minimum confusion_factor. The proposed Gene Selection Explore (GSE) algorithm is compared to well established feature selection algorithms using 10 microarray data with respect to sensitivity, specificity, and accuracy. The identified genes appear in KEGG pathway and have several biological importance.

Number Representations Drive Number-Line Estimates

The number-line task has been extensively used to study the mental representation of numbers in children. However, studies suggest that proportional reasoning provides a better account of children's performance. Ninety 4- to 6-year-olds were given a number-line task with symbolic numbers, with clustered dot arrays that resembled a perceptual scaling task, or with spread-out dot arrays that involved numerical estimation. Children performed well with clustered dot arrays, but poorly with symbolic numbers and spread-out dot arrays. Performances with symbolic numbers and spread-out dot arrays were highly correlated and were related to counting skill; neither was true for clustered dot arrays. Overall, results provide evidence for the role of mental representation of numbers in the symbolic number-line task.

Counteracting estimation bias and social influence to improve the wisdom of crowds

Aggregating multiple non-expert opinions into a collective estimate can improve accuracy across many contexts. However, two sources of error can diminish collective wisdom: individual estimation biases and information sharing between individuals. Here, we measure individual biases and social influence rules in multiple experiments involving hundreds of individuals performing a classic numerosity estimation task. We first investigate how existing aggregation methods, such as calculating the arithmetic mean or the median, are influenced by these sources of error. We show that the mean tends to overestimate, and the median underestimate, the true value for a wide range of numerosities. Quantifying estimation bias, and mapping individual bias to collective bias, allows us to develop and validate three new aggregation measures that effectively counter sources of collective estimation error. In addition, we present results from a further experiment that quantifies the social influence rules that individuals employ when incorporating personal estimates with social information. We show that the corrected mean is remarkably robust to social influence, retaining high accuracy in the presence or absence of social influence, across numerosities and across different methods for averaging social information. Using knowledge of estimation biases and social influence rules may therefore be an inexpensive and general strategy to improve the wisdom of crowds.

The Enculturated Move From Proto-Arithmetic to Arithmetic

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation.

Effect of Presentation Format on Judgment of Long-Range Time Intervals

Investigations in the temporal estimation domain are quite vast in the range of milliseconds, seconds, and minutes. This study aimed to determine the psychophysical function that best describes long-range time interval estimation and evaluate the effect of numerals in duration presentation on the form of this function. Participants indicated on a line the magnitude of time intervals presented either as a number + time-unit (e.g., “9 months”; Group I), unitless numerals (e.g., “9”; Group II), or tagged future personal events (e.g., “Wedding”; Group III). The horizontal line was labeled rightward (“Very short” = >“Very long”) or leftward (“Very long” = >“Very short”) for Group I and II, but only rightward for Group III. None of the linear, power, logistic or logarithmic functions provided the best fit to the individual participant data in more than 50% of participants for any group. Individual power exponents were different only between the tagged personal events (Group III) and the other two groups. When the same analysis was repeated for the aggregated data, power functions provided a better fit than other tested functions in all groups with a difference in the power function parameters again between the tagged personal events and the other groups. A non-linear mixed effects analysis indicated a difference in the power function exponent between Group III and the other groups, but not between Group I and II. No effect of scale directionality was found in neither of the experiments in which scale direction was included as independent variable. These results suggest that the judgment of intervals in a number + time-unit presentation invoke, at least in part, processing mechanisms other than those used for time-domain. Consequently, we propose the use of event-tagged assessment for characterizing long-range interval representation. We also recommend that analyses in this field should not be restricted to aggregated data given the qualitative variation between participants.

Sex Differences in Number Magnitude Processing Strategies Are Mediated by Spatial Navigation Strategies: Evidence From the Unit-Decade Compatibility Effect

The hybrid model of number magnitude processing suggests that multi-digit numbers are simultaneously processed holistically (whole number magnitudes) and in a decomposed manner (digit magnitudes). Thus, individual tendencies and situational factors may affect which type of processing becomes dominant in a certain individual in a given situation. The unit-decade compatibility effect has been described as indicative of stronger decomposed number processing. This effect occurs during the comparison of two-digit numbers. Compatible items in which the larger number contains the larger unit digit are easier to solve than incompatible items in which the larger number contains the smaller unit digit. We have previously described women show a larger compatibility effect than men. Furthermore, the compatibility effect is modulated by situational factors like the vertical spacing of the presented numbers. However, it has not been addressed whether situational factors and sex affect the unit-decade compatibility effect interactively. We have also demonstrated that the unit-decade compatibility effects relates to global-local processing, which in turn also affects spatial processing strategies. However, a link between spatial processing strategies and the unit-decade compatibility effect has not yet been established. In the present study we investigate, whether sex differences in the unit-decade compatibility effect (i) depend on the vertical spacing between numbers, (ii) are mediated via sex hormone levels of participants, and (iii) relate to sex differences in spatial processing strategies. 42 men and 41 women completed a two-digit number comparison task as well as a spatial navigation task. The number comparison task modulates compatibility and vertical spacing in a 2 × 2 design. The results confirm a larger compatibility effect in women compared to men and with dense compared to sparse spacing. However, no interactive effect was observed, suggesting that these factors modulate number magnitude processing independently. The progesterone/testosterone ratio was related to the compatibility effect, but did not mediate the sex difference in the compatibility effect. Furthermore, spatial processing strategies were related to the compatibility effect and did mediate the sex difference in the compatibility effect. Participants with a stronger focus on landmarks in the spatial navigation task showed a larger compatibility effect.

Precise relative-quantity judgement in the striped field mouse Apodemus agrarius Pallas

Applying the classical experimental scheme of training animals with food rewards to discriminate between quantities of visual stimuli, we demonstrated that not only can striped field mice Apodemus agrarius discriminate between clearly distinctive quantities such as 5 and 10, but some of these mice also exhibit high accuracy in discriminating between quantities that differ only by one. The latter include both small (such as 2 versus 3) and relatively large (such as 5 versus 6, and 8 versus 9) quantities of elements. This is the first evidence of precise relative-quantity judgement in wild rodents. We found striking individual variation in cognitive performance among striped field mice, which possibly reflects individual cognitive variation in natural populations. We speculate that high accuracy in differentiating large quantities is based on the adaptive ability of wild rodents to capture subtle changes in their environment. We suggest that the striped field mouse may be a powerful model species to develop advanced cognitive tests for comparative studies of numerical competence in animals and for understanding evolutionary roots of quantity processing.

Number, time, and space are not singularly represented: Evidence against a common magnitude system beyond early childhood

Our ability to represent temporal, spatial, and numerical information is critical for understanding the world around us. Given the prominence of quantitative representations in the natural world, numerous cognitive, neurobiological, and developmental models have been proposed as a means of describing how we track quantity. One prominent theory posits that time, space, and number are represented by a common magnitude system, or a common neural locus (i.e., Bonn & Cantlon in Cognitive Neuropsychology, 29(1/2), 149-173, 2012; Cantlon, Platt, & Brannon in Trends in Cognitive Sciences, 13(2), 83-91, 2009; Meck & Church in Animal Behavior Processes, 9(3), 320, 1983; Walsh in Trends in Cognitive Sciences, 7(11), 483-488, 2003). Despite numerous similarities in representations of time, space, and number, an increasing body of literature reveals striking dissociations in how each quantity is processed, particularly later in development. These findings have led many researchers to consider the possibility that separate systems may be responsible for processing each quantity. This review will analyze evidence in favor of a common magnitude system, particularly in infancy, which will be tempered by counter evidence, the majority of which comes from experiments with children and adult participants. After reviewing the current data, we argue that although the common magnitude system may account for quantity representations in infancy, the data do not provide support for this system throughout the life span. We also identify future directions for the field and discuss the likelihood of the developmental divergence model of quantity representation, like that of Newcombe (Ecological Psychology, 2, 147-157, 2014), as a more plausible account of quantity development.

Regular Distribution Inhibits Generic Numerosity Processing

This study investigated the role of pattern regularity in approximate numerical processing. Experiment 1 demonstrated that the change in stimulus size has a distinct effect on the adaptation aftereffect for random and regular patterns. For regular patterns, adapting to large patterns and being tested with small patterns caused stronger aftereffects than the reverse treatment, in which the participants adapted to small patterns and were tested with large patterns. For random patterns, this effect was absent. Experiment 2 revealed a distinct connectedness effect on the numerosity processing of random and regular patterns. For random patterns, reference stimuli were perceived to contain fewer items when the dots were connected by lines than when they were not connected, and the number of items in the connected reference was further underestimated when the participants adapted to unconnected patterns with the same number of dots. For regular patterns, this effect was absent. Distinct mechanisms were thus suggested for the numerosity coding of random and regular patterns. For random patterns, the change in primary texture features would be abstracted from numerosity processing, while connectedness could affect this coding by affecting the processing of numerical unit individuation. For regular patterns, generic numerosity processing is inhibited, and numerical judgments appear to be inferred from the visual processing results of texture features such as dot size or the distance between adjacent dots.

Dynamics and development in number-to-space mapping

Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults' estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults' numerosity estimation.

A Mathematical Model of How People Solve Most Variants of the Number-Line Task

Current understanding of the development of quantity representations is based primarily on performance in the number-line task. We posit that the data from number-line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number-line task: the production and estimation variations of the bounded and unbounded number-line tasks. Comparison of performance in the bounded and unbounded number-line tasks provides a unique and direct way to assess the role of strategy in number-line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number-line task produce equivalent biases except for one: the estimation variation of the bounded number-line task. We discuss the important implications of these findings.

Probing the architecture of visual number sense with parietal tRNS

Theoretical accounts of the visual number sense (VNS), i.e., an ability to discriminate approximate numerosities, remain controversial. A proposal that the VNS represents a process of numerosity extraction, leading to an abstract number representation in the brain, has been challenged by the view that the VNS is non-numerical in its essence and amounts to a weighted integration of continuous magnitude features that typically change with numerosity. In the present study, using two-alternative forced-choice paradigm, we aimed to distinguish between these proposals by probing brain areas implicated in the VNS with transcranial random noise stimulation (tRNS). We generated predictions for the stimulation-related changes in behavioural performance which would be compatible with alternative mechanisms proposed for the VNS. First, we investigated whether the superior parietal (SP) area hosts a numerosity code or whether its function is to modulate weighting of continuous stimulus features. We predicted that stimulation may affect the VNS precision if the SP role is representational, and that it may affect decision threshold if its role is modulatory. Second, we investigated whether the intraparietal (IP) area hosts a numerosity code independently of codes for continuous stimulus features, or whether their representations overlap. If the numerosity code is independent, we predicted that IP stimulation may improve the VNS but not continuous magnitude judgements. Our results were consistent with the hypotheses of a modulatory role of the SP and of the independence of the numerosity code in the IP, whereby suggesting that VNS is an emergent abstract property based on continuous magnitude statistics.

Variability in the Alignment of Number and Space Across Languages and Tasks

While the domains of space and number appear to be linked in human brains and minds, their conceptualization still differs across languages and cultures. For instance, frames of reference for spatial descriptions vary according to task, context, and cultural background, and the features of the mental number line depend on formal education and writing direction. To shed more light on the influence of culture/language and task on such conceptualizations, we conducted a large-scale survey with speakers of five languages that differ in writing systems, preferences for spatial and temporal representations, and/or composition of number words. Here, we report data obtained from tasks on ordered arrangements, including numbers, letters, and written text. Comparing these data across tasks, domains, and languages indicates that, even within a single domain, representations may differ depending on task characteristics, and that the degree of cross-domain alignment varies with domains and culture.

Comparing Symbolic and Nonsymbolic Number Lines: Consistent Effects of Notation Across Output Measures

The mental number line (MNL) is a popular metaphor for magnitude representation in numerical cognition. Its shape has frequently been reported as being nonlinear, based on nonlinear response functions in magnitude estimation. We investigated whether this shape reflects a phenomenon of the mapping from stimulus to internal magnitude representation or of the mapping from internal representation to response. In five experiments, participants (total N = 66) viewed stimuli that represented numerical magnitude either in a symbolic notation (i.e., Arabic digits) or in a nonsymbolic notation (i.e., clouds of dots). Participants estimated these magnitudes by either adjusting the position of a mark on a ruler-like response bar (nonsymbolic response) or by typing the corresponding number on a keyboard (symbolic response). Responses to symbolic stimuli were markedly different from responses to nonsymbolic stimuli, in that they were mostly powershaped. We investigated whether the nonlinearity could be explained by effects of previous trials, but such effects were (a) not strong enough to explain the nonlinear responses and (b) existed only between trials of the same input notation, suggesting that the nonlinearity is due to input mappings. Introducing veridical feedback improved the accuracy of responses, thereby showing a calibration based on the feedback. However, this calibration persisted only temporarily, and responses to nonsymbolic stimuli remained nonlinear. Overall, we conclude that the nonlinearity is a phenomenon of the mapping from nonsymbolic input format to internal magnitude representation and that the phenomenon is surprisingly robust to calibration.

Comparing Numerical Comparison Tasks: A Meta-Analysis of the Variability of the Weber Fraction Relative to the Generation Algorithm

Since more than 15 years, researchers have been expressing their interest in evaluating the Approximate Number System (ANS) and its potential influence on cognitive skills involving number processing, such as arithmetic. Although many studies reported significant and predictive relations between ANS and arithmetic abilities, there has recently been an increasing amount of published data that failed to replicate such relationship. Inconsistencies lead many researchers to question the validity of the assessment of the ANS itself. In the current meta-analysis of over 68 experimental studies published between 2004 and 2017, we show that the mean value of the Weber fraction (w), the minimal amount of change in magnitude to detect a difference, is very heterogeneous across the literature. Within young adults, w might range from < 10 to more than 60, which is critical for its validity for research and diagnostic purposes. We illustrate here the concern that different methods controlling for non-numerical dimensions lead to substantially variable performance. Nevertheless, studies that referred to the exact same method (e.g., Panamath) showed high consistency among them, which is reassuring. We are thus encouraging researchers only to compare what is comparable and to avoid considering the Weber fraction as an abstract parameter independent from the context. Eventually, we observed that all reported correlation coefficients between the value of w and general accuracy were very high. Such result calls into question the relevance of computing and reporting at all the Weber fraction. We are thus in disfavor of the systematic use of the Weber fraction, to discourage any temptation to compare given data to some values of w reported from different tasks and generation algorithms.