Honey bees zero in on the empty set

The origins of number sense: a commentary on "Is there an innate sense of number in the brain?"

The question of whether a "sense of number" is innate has been posed in a new article by Lorenzi et al. (2025). The article explores the behavioral and neurobiological evidence from newborn animals to delve into the evolutionary origins of a sense of number. Lorenzi et al.(2025) raises new questions, interpretations, and ideas for future work to understand how number sense has evolved in humans and nonhuman animals. In this commentary, I discuss the arguments for an innate number sense, evaluate the implications for numerical cognition, and suggest how future work could fill the current gaps in our knowledge.

Cichlids and stingrays can add and subtract 'one' in the number space from one to five

The numerical understanding of cichlids and stingrays was examined regarding addition and subtraction abilities within the number space of one to five. Experiments were conducted as two-alternative forced-choice experiments, using a delayed matching to sample technique. On each trial, fish had to perform either an addition or subtraction, based on the presentation of two-dimensional objects in two distinct colors, with the color signaling a particular arithmetic process. Six cichlids and four stingrays successfully completed training and recognized specific colors as symbols for addition and subtraction. Cichlids needed more sessions than stingrays to reach the learning criterion. Transfer tests showed that learning was independent of straightforward symbol memorization. Individuals did not just learn to pick the highest or lowest number presented based on the respective color; instead, learning was specific to adding or subtracting ‘one’. Although group results were significant for both species in all tests, individual results varied. Addition was learned more easily than subtraction by both species. While cichlids learned faster than stingrays, and more cichlids than stingrays learned the task, individual performance of stingrays exceeded that of cichlids. Previous studies have provided ample evidence that fish have numerical abilities on par with those of other vertebrate and invertebrate species tested, a result that is further supported by the findings of the current study.

Nonsymbolic and symbolic representations of null numerosity

Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1-9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

As intuições matemáticas primordiais

Resumo Este artigo apresenta um estudo exploratório sobre os aspectos quantitativos e qualitativos do número na obra de Jung. Assim, foi efetuado levantamento de aspectos do número, desde os testes de associação de palavras até o conceito de sincronicidade. O artigo foi dividido em três partes: apresentação de aspectos quantitativos (tempo de reação no teste de associação de palavras e conceito de energia psíquica); apresentação de aspectos qualitativos (número como conteúdo psíquico, número como elemento organizador da psique, caráter numinoso do número, número como grandeza imprevisível entre mito e realidade, a relação do número com os eventos sincronísticos e o número como arquétipo da ordem que se tornou consciente); e apresentação das intuições matemáticas primordiais (a série infinita de números naturais e a ideia de continuum). Ao enfatizar os aspectos qualitativos, Jung aponta para a base arquetípica do número como fator de ordenação e de articulação entre psique e matéria.

Different mechanisms underlie implicit visual statistical learning in honey bees and humans

The ability of developing complex internal representations of the environment is considered a crucial antecedent to the emergence of humans' higher cognitive functions. Yet it is an open question whether there is any fundamental difference in how humans and other good visual learner species naturally encode aspects of novel visual scenes. Using the same modified visual statistical learning paradigm and multielement stimuli, we investigated how human adults and honey bees (Apis mellifera) encode spontaneously, without dedicated training, various statistical properties of novel visual scenes. We found that, similarly to humans, honey bees automatically develop a complex internal representation of their visual environment that evolves with accumulation of new evidence even without a targeted reinforcement. In particular, with more experience, they shift from being sensitive to statistics of only elemental features of the scenes to relying on co-occurrence frequencies of elements while losing their sensitivity to elemental frequencies, but they never encode automatically the predictivity of elements. In contrast, humans involuntarily develop an internal representation that includes single-element and co-occurrence statistics, as well as information about the predictivity between elements. Importantly, capturing human visual learning results requires a probabilistic chunk-learning model, whereas a simple fragment-based memory-trace model that counts occurrence summary statistics is sufficient to replicate honey bees' learning behavior. Thus, humans' sophisticated encoding of sensory stimuli that provides intrinsic sensitivity to predictive information might be one of the fundamental prerequisites of developing higher cognitive abilities.

Symbolic representation of numerosity by honeybees ( Apis mellifera): matching characters to small quantities

The assignment of a symbolic representation to a specific numerosity is a fundamental requirement for humans solving complex mathematical calculations used in diverse applications such as algebra, accounting, physics and everyday commerce. Here we show that honeybees are able to learn to match a sign to a numerosity, or a numerosity to a sign, and subsequently transfer this knowledge to novel numerosity stimuli changed in colour properties, shape and configuration. While honeybees learned the associations between two quantities (two; three) and two signs (N-shape; inverted T-shape), they failed at reversing their specific task of sign-to-numerosity matching to numerosity-to-sign matching and vice versa (i.e. a honeybee that learned to match a sign to a number of elements was not able to invert this learning to match the numerosity of elements to a sign). Thus, while bees could learn the association between a symbol and numerosity, it was linked to the specific task and bees could not spontaneously extrapolate the association to a novel, reversed task. Our study therefore reveals that the basic requirement for numerical symbolic representation can be fulfilled by an insect brain, suggesting that the absence of its spontaneous emergence in animals is not due to cognitive limitation.

Numerical cognition in honeybees enables addition and subtraction

Many animals understand numbers at a basic level for use in essential tasks such as foraging, shoaling, and resource management. However, complex arithmetic operations, such as addition and subtraction, using symbols and/or labeling have only been demonstrated in a limited number of nonhuman vertebrates. We show that honeybees, with a miniature brain, can learn to use blue and yellow as symbolic representations for addition or subtraction. In a free-flying environment, individual bees used this information to solve unfamiliar problems involving adding or subtracting one element from a group of elements. This display of numerosity requires bees to acquire long-term rules and use short-term working memory. Given that honeybees and humans are separated by over 400 million years of evolution, our findings suggest that advanced numerical cognition may be more accessible to nonhuman animals than previously suspected.