Similarity and generalization are often assumed to reflect distance in an underlying representational space. In semantic networks, however, distance can be defined by both shortest paths or by multiple indirect pathways. Previous research has shown that non-shortest, higher-order paths influence similarity judgements. Here, we extend these findings by re-analyzing a publicly available dataset to compare shortest-path models with models integrating over discounted higher-order paths. The latter better accounted for similarity judgements; error analyses indicate that this advantage arose from integrating multiple indirect paths rather than relying on shortest connections. These random-walk models implement the same core computation as the successor representation (SR), which has been proposed to support human generalization. Consistently, an SR-based model outperformed alternatives in accounting for inductive generalization judgements from the same dataset. These findings suggest that similarity and generalization are both shaped by higher-order connectivity between concepts.

Abstract One of the most astonishing abilities documented in human newborns is that they can abstract numerosity – the number of items in sets - across sensory modalities. Its underlying neural mechanisms remain unknown. Using high-density EEG and a frequency-tagging paradigm, we measured neural entrainment to periodically presented visual arrays that were either numerically congruent or incongruent with previously familiarized and concurrently presented auditory sequences in 21 newborns (0–3 days old). The amplitude of neural entrainment to the visual arrays provided a robust index of cross-modal numerical congruency: it was significantly reduced for numerically congruent relative to incongruent stimuli, consistent with a cross-modal numerosity repetition-suppression mechanism. These findings identify a candidate neural mechanism supporting newborns’ ability to encode numerosity in an abstract supramodal format, and reinforce the view that number constitutes a foundational dimension of human perception since birth.

Adults tend to systematically associate numerical information with spatial positions, such as associating smaller numbers with the left and larger numbers with the right side of space. These spatial–numerical associations were initially attributed to cultural factors such as reading direction, but evidence from non-human animals suggests a biological foundation. In this Review, we examine behavioural evidence to clarify the origins of spatial–numerical associations and how cultural and biological factors shape their existence and direction. First, we discuss how specific tasks and stimulus types constrain the implications of spatial–numerical association studies. Then we review findings from non-human animals and human infants that suggest early biological predispositions for a left-to-right mapping of numerical information onto space. Next, we consider findings from adults with different written language experience, which highlight the role of cultural experience. Finally, we speculate on possible mechanisms underlying an implicit form of non-symbolic spatial–numerical associations and propose directions for future research. Adults tend to systematically associate numerical information with space, such as small numbers with the left. In this Review, Eccher and colleagues discuss how cultural and biological factors shape the existence and direction of these spatial–numerical associations.

Commutativity—the principle that operand order does not affect the result of an operation—is a core feature of two foundational operations of arithmetic: addition and multiplication. Recent theories propose that such compositional structures emerge from an innate “language of thought” that enables abstract principles to apply flexibly across symbolic and non-symbolic formats. While children possess early intuitions about additive commutativity, less is known about their grasp of the commutativity of multiplication. Here, we ask whether children’s understanding of the commutative principle of multiplication stems from pre-existing intuitions that the order in which sets are grouped does not matter, or whether it instead emerges from reasoning about structural relations among numerical symbols. We tested 8- to 9-year-old children in a number comparison game, probing their understanding of multiplicative commutativity in both symbolic expressions (3x2 = 2x3) and in non-symbolic sets of grouped dots. On average, children performed better with symbolic expressions than with non-symbolic arrays. Moreover, their performance on the arrays of grouped dots was directly linked to their mastery of commutativity in symbolic expressions, and independent of their intuitive, non-symbolic numerosity estimation skills. To probe children’s developing mastery of the commutativity of multiplication, a subset of the children were given brief training on commutativity and played the same game in a second session, 2 to 14 days later. Children who, in the first session, did not master the commutative principle with symbolic stimuli showed improvement on symbolic trials, while improvement on non-symbolic trials lagged for the entire group. These findings suggest that symbolic learning of the principle of commutativity of multiplication comes first, before children can apply it to concrete contexts. Moreover, they provide evidence for format- and operation-specific limits to children’s mastery of the compositional structure of the natural number system.

Learning is a multifaceted process that continues across the lifespan, with variation in how different types of knowledge are acquired and retained. One distinction exists between the cognitive skills learned through memory-based associations and those learned through rule-based (strategy) execution. While both forms are central to arithmetic acquisition, little is known about their dynamics in adulthood and their cognitive underpinnings. Addressing this gap is essential for developing effective re-education programs, especially in arithmetic, where learning difficulties can impact personal, academic, and socio-economic outcomes. Here, healthy adults with varying ages and cognitive profiles learned to solve a limited set of novel arithmetic problems using either associative (memory-based) or procedural (strategy-based) methods. Group-level analyses revealed that both learning conditions reached comparable performance. Individual learning trajectories, modeled using a composite efficiency index (inverse efficiency score), revealed similar efficiency gain but distinct dynamics: associative learning showed a steep, abrupt gain, whereas procedural learning progressed gradually along a smoother trajectory. Regression analyses indicated that verbal long-term memory predicted associative learning, while arithmetic abilities specifically predicted procedural learning. Notably, efficiency gains in one condition did not predict gains in the other, further supporting that, notwithstanding their similarities, the two learning trajectories are different in nature. Error analysis further revealed that performance in the procedural condition shifted from calculation-based to retrieval-based processes with practice. These findings demonstrate that associative and procedural arithmetic learning are initially supported by distinct cognitive mechanisms but converge over time, underscoring the importance of considering individual cognitive profiles when designing tailored educational/rehabilitation interventions.

Humans can estimate the number of objects in a scene within a fraction of a second, suggesting that numerosity is encoded rapidly and directly by the visual system. Yet how this encoding unfolds over time and interacts with other visual features remains unclear. Here, we combined magnetoencephalography (MEG) with time-resolved representational similarity analysis (RSA) and MEG–fMRI fusion to track how numerosity is represented in the brain over time. We also used multidimensional scaling (MDS) to visualize the evolving patterns of neural activity. Two main findings emerged. First, numerosity exhibited the hallmarks of a primary perceptual attribute: its neural signature appeared rapidly after stimulus onset, preceding the encoding of non-numeric features that could otherwise define number. Second, Visualization of the neural patterns using MDS suggested a temporal transformation in representational geometry, reflecting the engagement of two distinct coding schemes- an early, linear number line, consistent with a “summation code”, dominating activity in occipital regions, and a later, curved number line, consistent with “numerosity-tuned code”, emerging more strongly in associative areas along the dorsal stream. Together, these findings demonstrate that numerosity processing is encoded directly from the visual image and unfolds through a rapid hierarchical transformation, from a broad quantity signal to a finely tuned, number-specific code, linking perceptual encoding to higher-level numerical abstraction. ### Competing Interest Statement The authors have declared no competing interest.

Number and space are intertwined in human and non-human cognition. A substantial body of research has shown that numerical magnitudes are mentally represented along a spatial continuum, akin to a “mental number line”. Some suggested that its directionality is determined by culture and context. Nevertheless, evidence from preverbal infants and non-human animals indicates a consistent left-to-right directional mapping of numerosities, suggesting a biologically predisposed Spatial-Numerical Association (SNA) that may precede cultural factors. A recent study has shown that an implicit association between “left” and “small” emerges not only in literate adults, but also in unschooled indigenous populations and preschool Western children. This finding suggests that SNAs may originate from universal innate mechanisms rather than being solely a by-product of cultural learning. However, while the study reported a strong association between “left” and “decreasing” numerosity, there was only a very weak association between “right” and “increasing” numerosity. This asymmetry was not predicted and needs further investigations to be understood. Here we further investigated the number/space association in implicit tasks in educated Western adults by using more variable and better controlled stimuli compared to the ones used in the previous study, and also manipulating stimulus format, using both dot patterns and symbolic numbers. Fifty-one adult participants performed a numerical comparison task within a Go-No-Go paradigm on subsequent pairs of visual stimuli (with ratios spanning from 0.75 to 0.94) that could appear on the left or on the right of a fixation point and completed two different tasks: “press when more” and “press when less”. Results revealed distinct response patterns depending on the symbolic/non-symbolic nature of the stimuli. When non-symbolic stimuli were used, a consistent association between small numerosities and the left side and large numerosities and the right side was observed. When symbolic stimuli were used, only an association between large numerosities and the right side was observed. These findings support the hypothesis that SNAs may reflect a biological predisposition associated with brain asymmetry, and that task demands may interact with the underlying hemispheric specializations. ### Competing Interest Statement The authors have declared no competing interest. European Council, 833504 SPANUMBRA

Visual numerosity, traditionally linked to the parietal cortex, is now thought to be represented across a broader cortical network, including early visual and associative areas in both streams. However, how numerosity is encoded relative to other visual features remains unclear. We conducted a whole-brain functional magnetic resonance imaging (fMRI) study with thirty-one adults performing a numerosity estimation task on visual sets varying in number, item size, total item area, field area, and density, ensuring tight stimulus control. Using model-based representational similarity analyses, we found numerosity represented independently of other visual properties in early visual areas and amplified in retinotopic and non-retinotopic associative regions across both streams. Dimensionality reduction of BOLD patterns revealed distinct geometries: a one-dimensional representation of numerical rank in early visual and ventral retinotopic areas, and a curved structure encoding rank and distance-to-endpoints in associative dorsal and ventral regions. These results demonstrate distinct neural coding schemes for numerosity across cortical regions.

Number and space are inherently related. Previous research has provided evidence that numbers are aligned to a so-called "mental number line", which is malleable and affected by cultural factors mostly linked to literacy-related habits. However, preverbal humans and non-human animals also map numerosities into space, in a consistent left-to-right direction. These contrasting findings raise the question of whether Spatial Numerical Associations (SNA) are culturally or biologically determined. Here, we investigated Italian adults, Italian preschoolers, and Himba adults (an indigenous population with an oral cultural system) to examine whether cultural influences are necessary for SNA to emerge. We found that, when explicitly asked to order numerosities, only Italian adults showed a consistent left-to-right preference, while preschoolers and Himba adults did not have a consistent preference for one direction or the other. On the other hand, in a numerosity comparison task, all groups performed better when small numerosities were presented in the left hemispace. These results suggest that humans may display two forms of SNAs, one that emerges mostly in implicit tasks and is biologically determined, and one that emerges in explicit ordering tasks and is determined by cultural habits.

Understanding how brain activity evolves with repetitive cognitive load is critical for assessing neural adaptations related to learning, fatigue, and efficiency. This pilot study investigates electroencephalographic (EEG) changes over five days of cognitive training in young adults performing a memory task. The study included eight healthy participants, who completed EEG recordings during a cognitive load task. To assess brain activity, we computed absolute spectral power across theta, alpha, sub-alpha (lower, upper), beta, sub-beta (lower, mid, upper), and gamma frequency bands. Nonparametric statistical methods, including a cluster-based permutation test, were applied to assess differences between the first and fifth day. Results indicate a significant power increase in theta, particularly in the midline and parietal regions, suggesting enhanced cognitive control and memory retrieval. Alpha power also increased, reflecting improved processing efficiency with reduced cognitive effort. Furthermore, lower and mid-beta activity showed an increase over time, indicating sustained attentional engagement and optimized cognitive processing. A reduction in the beta/alpha ratio suggests a transition from effortful processing to more automatic retrieval. No significant changes were observed in gamma power, implying that cognitive load did not induce excessive strain. These findings highlight neuroplastic adaptations to cognitive training, supporting the role of EEG in tracking learning-related changes in brain activity.

Arithmetic competence is crucial for navigating modern society and maintaining independence. It relies on domain-general and domain-specific cognitive skills, as well as subjective factors. Given its importance, understanding how these factors shape adult arithmetic proficiency is essential. This study investigated demographic, cognitive, and subjective influences on various arithmetic skills throughout adulthood, including both younger and older individuals. In total, 134 adults aged 20-68 completed computerized tasks assessing simple calculations, exact and approximate complex calculations, and arithmetic principles, alongside neuropsychological testing and self-ratings on math anxiety, math self-concept, attitudes toward mathematics, and the frequency of engagement with numbers. The results indicate that accuracy varied by task, with approximate calculations being the most challenging. Self-ratings showed low math anxiety but moderate-to-high math self-concept, positive attitudes, and moderate engagement with numbers. Age correlated only with arithmetic principles; however, interference inhibition and engagement with numbers, not age, best predicted performance. Executive functions correlated solely with approximate calculations and arithmetic principles, while subjective measures were related to all arithmetic tasks. The regression analyses indicate strong interrelationships, particularly among calculation tasks. The findings highlight the multifaceted nature of arithmetic competence and suggest it remains stable in adulthood, with age-related declines only evident in arithmetic principles, likely due to declining executive functions.

To understand a visual scene, observers need to both recognize objects and encode relational structure. For example, a scene comprising three apples requires the observer to encode concepts of "apple" and "three." In the primate brain, these functions rely on dual (ventral and dorsal) processing streams. Object recognition in primates has been successfully modeled with deep neural networks, but how scene structure (including numerosity) is encoded remains poorly understood. Here, we built a deep learning model, based on the dual-stream architecture of the primate brain, which is able to count items "zero-shot"-even if the objects themselves are unfamiliar. Our dual-stream network forms spatial response fields and lognormal number codes that resemble those observed in the macaque posterior parietal cortex. The dual-stream network also makes successful predictions about human counting behavior. Our results provide evidence for an enactive theory of the role of the posterior parietal cortex in visual scene understanding.

Preverbal infants spontaneously represent the number of objects in collections. Is this 'sense of number' (also referred to as Approximate Number System, ANS) part of the cognitive foundations of mathematical skills? Multiple studies reported a correlation between the ANS and mathematical achievement in children. However, some have suggested that such correlation might be mediated by general-purpose inhibitory skills. We addressed the question using a longitudinal approach: we tested the ANS of 60 12 months old infants and, when they were 4 years old (final N = 40), their symbolic math achievement as well as general intelligence and inhibitory skills. Results showed that the ANS at 12 months is a specific predictor of later maths skills independent from general intelligence or inhibitory skills. The correlation between ANS and maths persists when both abilities are measured at four years. These results confirm that the ANS has an early, specific and longstanding relation with mathematical abilities in childhood. RESEARCH HIGHLIGHTS: In the literature there is a lively debate about the correlation between the ANS and maths skills. We longitudinally tested a sample of 60 preverbal infants at 12 months and rested them at 4 years (final sample of 40 infants). The ANS tested at 12 months predicted later symbolic mathematical skills at 4 years, even when controlling for inhibition, general intelligence and perceptual skills. The ANS tested at 4 years remained linked with symbolic maths skills, confirming this early and longstanding relation in childhood.

Children appear to have some arithmetic abilities before formal instruction in school, but the extent of these abilities as well as the mechanisms underlying them are poorly understood. Over two studies, an initial exploratory study of preschool children in the U.S. (N = 207; Age = 2.89-4.30 years) and a pre-registered replication of preschool children in Italy (N = 130; Age = 3-6.33 years), we documented some basic behavioral signatures of exact arithmetic using a non-symbolic subtraction task. Furthermore, we investigated the underlying mechanisms by analyzing the relationship between individual differences in exact subtraction and assessments of other numerical and non-numerical abilities. Across both studies, children performed above chance on the exact non-symbolic arithmetic task, generally showing better performance on problems involving smaller quantities compared to those involving larger quantities. Furthermore, individual differences in nonverbal approximate numerical abilities and exact cardinal number knowledge were related to different aspects of subtraction performance. Specifically, non-verbal approximate numerical abilities were related to subtraction performance in older but not younger children. Across both studies we found evidence that cardinal number knowledge was related to performance on subtraction problems where the answer was zero (i.e., subtractive negation problems). Moreover, subtractive negation problems were only solved above chance by children who had a basic understanding of cardinality. Together these finding suggest that core non-verbal numerical abilities, as well as emerging knowledge of symbolic numbers provide a basis for some, albeit limited, exact arithmetic abilities before formal schooling.

Number and space are inherently related. For decades, authors have collected evidence showing that numbers are aligned to a so-called “mental number line”, which is malleable and affected by cultural factors. However, preverbal human and non-human animals also map numerosities into space, in a consistent left to right direction. These contrasting pieces of evidence raise the question of whether Space Number Associations (SNA) are culturally or biologically determined. Here, we investigated Italian adults, Italian preschoolers and Himba adults to determine whether cultural influences are necessary for SNA to emerge. We found that, when explicitly asked to order numerosities, only Italian adults showed a consistent left to right preference, while preschoolers and Himba adults did not have a consistent preference for one direction or the other. On the other hand, in a numerosity comparison task, all groups performed better when small numerosities were presented in the left hemispace. These results suggest that SNA is not a unique phenomenon, but rather is dissociable in two components: a universal one, biologically predisposed and left-to-right oriented, and an acquired one, culturally dependent and not fixed in orientation.

Representing the number of items in sets (numerosity) is a core and evolutionary ancient ability. In a recent fMRI study Castaldi et al. (eLife 2019) showed that visual numerosity is represented independently from other visual features starting from early visual areas and progressively amplified along the dorsal stream hierarchy up to parietal areas. However, because in this study we recorded the brain activity from a restricted brain volume, and performed the analyses only in a series of retinotopically organized regions of interest along the dorsal stream, we still miss an exhaustive picture of the full network of regions encoding pure numerosity across the whole brain. In this research advance, we extend the findings of Castaldi et al. in two significant ways. First, we recorded the whole brain which allowed us to characterize the neural responses to numerosity beyond the dorsal stream using a searchlight analysis in addition to an ROIs approach. Second, in addition to the classical model-based analytical approach, such as RSA, we compared the neural representational geometries across the different regions using multi-dimensional scaling, an hypothesis-free approach capable of unveiling latent neural dimensions that might otherwise go unnoticed. Our results confirm that numerosity is represented over and above other visual features in early visual areas and progressively enhanced in associative areas of the dorsal stream but also, notably, of the ventral stream. We also found that numerosity representations in association areas differ substantially from those of early visual areas, and that in associative regions of both streams numerosity is represented on a similar manifold akin to a curved number line, a structure suggestive of their similar involvement in numerosity-based decision making. Taken together, these results call for an important revision of the existing neurocognitive models of number cognition.Impact statement Visual numerosity is represented in associative areas of dorsal and ventral streams on a neural manifold that also reflects decision variables, suggesting a common role in numerosity decision-making.### Competing Interest StatementThe authors have declared no competing interest.

Inter-individual differences in infants' numerosity processing have been assessed using a change detection paradigm, where participants were presented with two concurrent streams of images, one alternating between two numerosities and the other showing one constant numerosity. While most infants look longer at the changing stream in this paradigm, the reasons underlying these preferences have remained unclear. We suggest that, besides being attracted by numerosity changes, infants perhaps also respond to the alternating pattern of the changing stream. We conducted two experiments (N = 32) with 6-month-old infants to assess this hypothesis. In the first experiment, infants responded to changes in numerosity even when the changing stream showed numerosities in an unpredictable random order. In the second experiment, infants did not display any preference when an alternating stream was pitted against a random stream. These findings do not provide evidence that the alternating pattern of the changing stream contributes to drive infants' preferences. Instead, around the age of 6 months, infants' responses in the numerosity change detection paradigm appear to be mainly driven by changes in numerosity, with different levels of preference reflecting inter-individual difference in the acuity of numerosity perception.