Randolph C. Grace

Theories of numeracy assume that logical concepts which enable formal definitions of numbers (cardinality, one-to-one correspondence, successor function) play key roles in ontogenesis, but empirical support is lacking. These concepts may not be suitable targets for theory because such definitions are not explanatory of what numbers are. We propose a correspondence between formal ideas that are generative for the natural numbers and children’s cognitive representations prior to numeracy (‘precursor system’). Building on our account of arithmetic foundations (Grice et al., 2024), we give a new construction of the natural numbers – the first to not rely on the Peano axioms – and corresponding theory of numeracy. We show that numbers can be derived from linear order, monotonicity and convexity – evolved principles of perceptual organization. A key concept is the ℕ-chain, the limit of finite linear orders, which we show is equivalent to the successor function but can be approximated in finite terms. The precursor system is based on linear orders – number word sequences, play schemas – and their relationships in the count routine by order embedding (structure mapping). It develops gradually, with order isomorphisms as proto-cardinality, reaching a representation analogous to an ℕ-chain when the child realizes that numbers never end. Ordinality precedes cardinality, one-to-one correspondence results from minimal (and therefore convex) embeddings, and numbers and arithmetic emerge as a common structure. Our theory shows that numeracy is both culturally-mediated and biologically-based, although ‘core number systems’ do not play a direct role. The dawning of numeracy in a child’s mind is both a reinvention and rediscovery of mathematics itself.

High-impact climate actions, such as sustainable transport and dietary choices, play a crucial role in mitigating the climate crisis, yet people’s perceptions of and responses to climate change may vary substantially across different social and cultural contexts. Drawing on the theory of basic human values, appraisal theory of emotion and warm-glow/cold-prickle frameworks, this study examined how biospheric and altruistic values, together with anticipated emotions (pride and guilt), shape intentions to engage in high-impact climate actions among young adults (aged 18–30) in New Zealand (n = 514) and China (n = 520). Multi-group path analyses were used to test the direct and indirect effects of values on climate-friendly transportation and dietary intentions via anticipated emotions. Results showed that biospheric values and anticipated emotions were significant predictors of both types of climate action intentions. After controlling for biospheric values, altruistic values no longer predicted transportation intentions in either country and predicted dietary intentions only in China, where the effect was small. Anticipated emotions mediated the effects of biospheric values on both behavioural intentions. Cross-national comparisons revealed that biospheric values were a weaker predictor of climate action intentions in China than in New Zealand, and their associations with two anticipated emotions were stronger in the latter. Anticipated guilt exerted a stronger effect on sustainable transport intentions in China, whereas the effects of anticipated pride did not differ significantly across countries. These findings advance understanding of the value–emotion–behaviour pathways regarding the cross-national validity and highlight the need for culturally sensitive climate communication strategies that integrate both individual values and emotional engagement.

Grace and colleagues (Chen et al., 2020; Grace et al., 2018) found that participants learned to respond accurately to both ratios and differences of intensive (brightness) and extensive (line length) modalities in a non-symbolic ‘artificial algebra’ task by feedback and without explicit instruction. However, Masin and colleagues (Masin et al., 2019) found that participants were unable to estimate brightness ratios in a task with explicit instructions (magnitude estimation). These contrasting results suggest that use of numbers and explicit cognition may impede the ability to estimate brightness ratios. To test this hypothesis, we conducted several experiments where participants responded to brightness or line length pairs in Grace et al.'s task to estimate ratios or differences. Those in Explicit groups were told specifically to respond based on differences or ratios, and given numeric feedback after each response, whereas those in Implicit groups were trained on Grace et al.’s original non-symbolic task. Experiments 1a (n = 23) and 1b (n = 40) (Line lengths) found that the Explicit group was more accurate than the Implicit group when estimating ratios of line lengths. In contrast, Experiments 2a and 2b found that the Explicit group was less accurate in estimating brightness ratios than the Implicit group. Specific instructions and numeric feedback had no effect on accuracy for participants trained on differences. These results resolve the inconsistency between Grace et al.’s and Masin’s results, suggesting that although the perceptual system can compute brightness ratios, participants have difficulty associating these perceived ratios with numbers.

How do children learn to count? Acquisition of number words and their representation of quantity (Cardinality Principle; CP) is commonly assessed with Wynn’s (1990; 1992) give-a-number (Give-N) task. However, results and methods are inconsistent across studies with this task, contributing to theoretical debates about how children acquire number knowledge. The primary aim of this project was to review these debates, and to evaluate systematically a graphical knower-level model of number word acquisition (Lee & Sarnecka, 2010), by analysing both the model fit and predictive error across seven Give-N datasets. Data were sourced from studies with French-, Hungarian-, and English-speaking children (N = 823) aged 24-86 months (M = 49.7). The model described most children’s number word acquisition with a high degree of confidence, and reported 29.6% of participants had applied the CP to their entire counting list (CP-knowers). Fits to data appeared to be reasonably good, but analyses revealed systematic error between model predictions and data. Most notably, predictions for children who had only applied the CP to a subset of their counting list (subset-knowers/N-knowers) were significantly higher than performance in trials for N + 1 and N + 2 (where N is the capacity of this subset). Additionally, performance error was reviewed for each knower-level individually. Ultimately, our results suggested that Lee and Sarnecka’s model did not accurately account for children’s behaviour on the Give-N task. Moreover, this highlights that the knower-level account of children’s number word acquisition may not effectively describe this process.

Pigeons show a suboptimal choice for probabilistic reinforcement in the presence of informative stimuli, which has been proposed as an animal model of gambling. However, this behaviour has been difficult to replicate in rats. In our study, rats responded on alternatives that provided 50% or 100% reinforcement with half of the sessions reliably signalled and half un-signalled. Contrary to expectation, rats preferred the optimal alternative to a greater extent in the signalled sessions compared to the un-signalled sessions. Acute methamphetamine increased preference for the 50% alternative exclusively in signalled sessions. These results suggest that rats, unlike pigeons, are able to use the information provided by the stimulus correlated with non-reinforcement to shift preference away from the suboptimal alternative, and that the influence of methamphetamine on rats’ probabilistic choice is context dependent

The ratio of two magnitudes can take one of two values depending on the order they are operated on: a ‘big’ ratio of the larger to smaller magnitude, or a ‘small’ ratio of the smaller to larger. Although big and small ratio scales have different metric properties and carry divergent predictions for perceptual comparison tasks, no psychophysical studies have directly compared them. Two experiments are reported in which subjects implicitly learned to compare pairs of brightnesses and line lengths by non-symbolic feedback based on the scaled big ratio, small ratio or difference of the magnitudes presented. Results of Experiment 1 showed all three operations were learned quickly and estimated with a high degree of accuracy that did not significantly differ across groups or between intensive and extensive modalities, though regressions on individual data suggested an overall predisposition towards differences. Experiment 2 tested whether subjects learned to estimate the operation trained or to associate stimulus pairs with correct responses. For each operation, Gaussian noise was added to the feedback that was constant for repetitions of each pair. For all subjects, coefficients for the added noise component were negative when entered in a regression model alongside the trained differences or ratios, and were statistically significant in 80% of individual cases. Thus, subjects learned to estimate the comparative operations and effectively ignored or suppressed the added noise. These results suggest the perceptual system is highly flexible in its capacity for non-symbolic computation, which may reflect a deeper connection between perceptual structure and mathematics.

Human infants have ‘core knowledge systems’ that support basic intuitions about the world including objects and their motion, space, number, and time. What is the origin of these systems, and what is their nature? Although often regarded as separate, domain-specific modules, evidence for similar abilities across many nonhuman species suggests that core systems might be integrated, consistent with views of modularity in evolutionary-developmental biology. Here we propose that core knowledge systems are based on an ability to form representations of the environment with algebraic structure – that is, on implicit computation. Algebraic groups encode symmetries, with computation inherent in the structure – a view that complements an understanding of computation as action or function. Our proposal is related to previous applications of group theory in perception and computational-representational accounts of learning (Gallistel, 1990), but suggests for the first time a common basis for core knowledge across humans and nonhumans. Implicit computation can be studied experimentally with an ‘artificial algebra’ task in which adults learn to respond based on arithmetic combinations of stimulus magnitudes, by feedback and without explicit instruction. Asking why organisms have a capacity for implicit computation suggests two possibilities: Either the geometric invariants of the world have been internalized in perceptual systems by natural selection (Shepard, 1994), or mathematical structure is intrinsic to the mind. Understood more broadly in a framework offered by Penrose (2004), implicit computation is a linchpin with potential to unlock some of the most fundamental questions about relationships between the mind, mathematics, and the world.

Grace et al. (2018) developed an ‘artificial algebra’ task in which participants learn to make an analogue response based on a combination of non-symbolic magnitudes by feedback and without explicit instruction. Here we tested if participants could learn to add stimulus magnitudes in this task in accord with the properties of an algebraic group. Three pairs of experiments tested the group properties of commutativity (Experiments 1a-b), identity and inverse existence (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback, r = .97 and .96 for Experiments 1a-b, r = .97 and .96 for Experiments 2a-b, and ranged between r = .97 and .99 for Experiment 3a and between r = .82 and .95 for Experiment 3b. Responding on transfer trials was accurate and provided strong support for commutativity, identity and inverse existence, and associativity with line length, and for commutativity and identity and inverse existence with brightness. Deviations from associativity in Experiment 3b suggested that participants were averaging rather than adding brightness magnitudes. Our results confirm that the artificial algebra task can be used to study implicit computation and suggest that representations of magnitudes may have a structure similar to an algebraic group.