Theories of early numeracy assume that the logical concepts which enable formal definitions of numbers (e.g., 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 reconstructions of numbers in formal language, but not explanatory of what numbers are. We propose that there should be a correspondence between those formal ideas that are generative for the natural numbers and those realized in children’s early learning. Building on our account of foundations of arithmetic (Grice et al., 2024, Psychological Review), we give a new explanatory construction of the natural numbers and a theory of early numeracy analogous to it. We show that numbers can be derived from linear order, monotonicity and convexity – principles of perceptual organization that are evolutionarily based. A key step in our construction is the ℕ-chain, the limit of finite linear orders, which we show is equivalent to the successor function but preferable as a target for psychological theory because it can be approximated in finite terms. In our theory, building blocks of early numeracy are linear orders - the number word sequence, object play schemas - and how they are related in the count routine by structure mapping (order embeddings). The number word sequence develops gradually, giving a representation analogous to an ℕ-chain when the child realizes that numbers can be generated without limit. Ordinality is prior to cardinality as the basis for numeracy, one-to-one correspondence results from convexity, and the natural numbers and arithmetic emerge jointly as a common structure. Our theory reconciles constructivist and nativist accounts because it shows that numeracy is both culturally mediated and biologically-based, although ‘core number systems’ do not play a role. The dawning of numeracy in a child’s mind is both a reinvention and rediscovery of mathematics itself.

An increase in political polarization in the USA has been reported by many researchers using different kinds of data (e.g., attitudinal, affective and behavioral measures). Here, we report the results of three incentivized experimental studies (with a total of 1842 participants) in which participants had to choose how to divide money given to them by a political opponent (decisions were made for actual money of up to $11). In Study 1, participants could choose whether to act trustworthily (i.e., to share the money evenly or keep it for themselves). In Study 2, participants had no financial incentives not to reciprocate the trust placed in them (i.e., they did not earn their own benefits from harming their interaction partner). In both Study 1 and Study 2, more participants actively harmed a political opponent than a political co-partisan or a person with an unknown political affiliation. Most strikingly, such discrimination was not only governed by disliking members of the other side, but also perceived as justified moral aggression (i.e., it was regarded as the behavior that should be chosen). In Study 3, a previously well-established intervention to mitigate affective political polarization increased the likability of opponents but did not reduce discriminatory reciprocity. In all three studies, when compared with an anonymous interaction partner participants only slightly favored political affiliates but strongly discriminated against political opponents. In this, our results were highly symmetrical: Democrats and Republicans did not systematically differ in their willingness to act fairly towards each other.

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.

Although there is substantial evidence for an innate ‘number sense’ that scaffolds learning about mathematics, whether the underlying representations are based on discrete or continuous perceptual magnitudes has been controversial. Yet the nature of the computations supported by these representations has been neglected in this debate. While basic computation of discrete non-symbolic quantities has been reliably demonstrated in adults, infants, and non-humans, far less consideration has been given to the capacity for computation of continuous perceptual magnitudes. Here we used a novel experimental task to ask if humans can learn to add non-symbolic, continuous magnitudes in accord with the properties of an algebraic group, by feedback and without explicit instruction. Three pairs of experiments tested perceptual addition under the group properties of commutativity (Experiments 1a-b), identity and inverses (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. Responding on transfer trials was accurate and provided strong support for addition under all of the group axioms with line length, and for all except associativity with brightness. Our results confirm that adult human subjects can implicitly add continuous quantities in a manner consistent with symbolic addition over the integers, and that an ‘artificial algebra’ task can be used to study implicit computation.

Where does arithmetic come from, and why are addition and multiplication its fundamental operations? Although we know that arithmetic is true, no explanation that meets standards of scientific rigor is available from philosophy, mathematical logic, or the cognitive sciences. We propose a new approach based on the assumption that arithmetic has a biological origin: Many examples of adaptive behavior such as spatial navigation suggest that organisms can perform arithmetic-like operations on represented magnitudes. If so, these operations – non-symbolic precursors of addition and multiplication – might be optimal due to evolution and thus identifiable according to an appropriate criterion. We frame this as a meta-mathematical question, and using an order-theoretic criterion, prove that four qualitative conditions – monotonicity, convexity, continuity, and isomorphism – are sufficient to identify addition and multiplication over the real numbers uniquely from the uncountably infinite class of possible operations. Our results show that numbers and algebraic structure emerge from purely qualitative conditions, and as a construction of arithmetic, provide a rigorous explanation for why addition and multiplication are its fundamental operations. We argue that these conditions are pre-verbal psychological intuitions or principles of perceptual organization that are biologically based and shape how humans and nonhumans alike perceive the world. This is a Kantian view, and suggests that arithmetic need not be regarded as an immutable truth of the universe, but rather as a natural consequence of our perception. Algebraic structure may be inherent in the representations of the world formed by our perceptual system.

Understanding the relationship between personality and income is a topic of interest across multiple disciplines. Correlations between people’s personalities and their incomes may arise because differences in stable personalities relate to income differences (between-person effects) or because changes in personality or income are later reflected in the other variable (within-person effects). The current research uses random-intercept cross-lagged panel models to disentangle the two sorts of effects to better understand the relationship between the six factors of personality and income. Using data from 6824 working-age adults in New Zealand across 4 years, we found between-person effects showing higher incomes were obtained by both men and women who were more extraverted, agreeable and open, and less neurotic. Within-person effects showed that earning a higher income was associated with higher neuroticism and lower extraversion over time, while higher extraversion was associated with a lower income over time.

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.

When people do wrong and then apologise for it, others may react by believing what the apologiser says (or not), forgiving them, thinking they were truly sorry and believing the apology helps the victim. In three scenario studies with a total of 1,669 respondents, we examined how ratings of the different reactions varied with the transgressor's profession, the nature of the victim and the perspective taken by the judge. These three factors all influenced reactions to the wrongdoing. In general, politicians were rated as being less believable, trustworthy, and less likely to be sorry than doctors; willingness to forgive decreased when the victim was junior; and respondents' attitudes towards the admission of an offence varied as a function of the specific perspective that they took. However, we also found that the different dimensions of the reactions varied differently with these factors and were not highly correlated.

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.

Sequential search is often costly and time-consuming. The time cost is usually unknown ex ante and its presence and duration must be inferred as the search progresses. We disentangle the effect of time cost on search behavior from people’s (in)ability to perceive time delay between offers. We find that people are able to infer the existence of the time cost, but their inference is imperfect. We also compare the effect of time cost with the effect of monetary cost and find that the time cost reduces the amount of exerted search, but not as much as the monetary cost does. Discriminating between the effects is critical for increasing the empirical validity of search models and designing mechanisms capable of improving the quality of decisions, especially in unfamiliar or infrequently encountered situations.

We tested the effectiveness of performance-based incentive structures using three incentive structures --- commission base, best only and flat fee --- and two levels of context --- no context and house-selling --- in an experiment in which participants made decisions in a variant of the secretary problem. Key measures of performance were the amount of search and the rounds in which the very best (optimal) offer was chosen. We found that having a commission-based proportional incentive did not produce better performance than having a flat payment for any of the performance measures considered. However, another performance-based incentive --- the best only --- increased the length of their searches and led to more optimal offers. These results applied both when there was no context and when the context was selling a house.

There is considerable evidence for computationally complex behavior, that is, behavior that appears to require the equivalent of mathematical calculation by the organism. Spatial navigation by path integration is perhaps the best example. The most influential account of such behavior has been Gallistel's (1990) computational-representational theory, which assumes that organisms represent key environmental variables such as direction and distance traveled as real numbers stored in engrams and are able to perform arithmetic computations on those representations. But how are these computations accomplished? A novel perspective is gained from the historical development of algebra. We propose that computationally complex behavior suggests that the perceptual system represents an algebraic field, which is a mathematical concept that expresses the structure underlying arithmetic. Our field representation hypothesis predicts that the perceptual system computes 2 operations on represented magnitudes, not 1. We review recent research in which human observers were trained to estimate differences and ratios of stimulus pairs in a nonsymbolic task without explicit instruction (Grace, Morton, Ward, Wilson, & Kemp, 2018). Results show that the perceptual system automatically computes two operations when comparing stimulus magnitudes. A field representation offers a resolution to longstanding controversies in psychophysics about which of 2 algebraic operations is fundamental (e.g., the Fechner-Stevens debate), overlooking the possibility that both might be. In terms of neural processes that might support computationally complex behavior, our hypothesis suggests that we should look for evidence of 2 operations and for symmetries corresponding to the additive and multiplicative groups. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

We present a model, related to an earlier model of Miller and Schwarz (2011), which shows that the probability of conceptual replication depends on the standard deviation of the true effect size, SD(δ), among other parameters. Computational analyses show that the dependency is more pronounced for smaller sample sizes. A sample of 50 meta-analyses is also investigated to obtain estimates of SD(δ), which turn out to be often in the range 0.2–0.5. Putting these two sets of results together indicates that the probability of conceptual replication is likely to vary for the range of SD(δ) often found in psychology, especially for smaller samples. The results can also be used to obtain approximate estimates of how the probability of replication is changed for typical SD(δ)in psychology.

This paper examined whether personal values and empathy and systemising traits relate to support levels for the Trans-Pacific Partnership Agreement (TPPA) in New Zealand. Two hundred and seventy respondents from New Zealand completed a 61-item questionnaire which measured self-rated knowledge and support for the TPPA, Schwartz Values, and Empathy and Systemising Quotients. Little to no self-rated knowledge of the TPPA strongly predicted neutrality in support levels for the TPPA. People who rated power and achievement highly tended to support the TPPA; those rating benevolence and universalism highly tended to oppose it. Higher levels of systemising were related to higher levels of self-rated TPPA knowledge, but empathy was only weakly associated with opposition to the TPPA. Overall, our results show that people's values are important in indicating support or opposition to trade deals, and perhaps need to be taken into account by those proposing or opposing the deals.

Why do New Zealanders invest overwhelmingly in housing and not in shares? This paper adopted a psychological approach to examine the question. Study 1 investigated whether the relatively high level of New Zealand ownership of housing might stem from risk aversion. A sample of New Zealanders was more likely to prefer housing investment than a Hong Kong sample, but there were no differences in Investment Risk Attitude although this variable was positively correlated with share investment in both samples. Study 2 looked at how New Zealanders perceived past rates of return on different investments and found a tendency to overestimate the returns from housing. They showed a similar pattern for their expectations of future returns. However, the estimates of returns were very variable and the estimates of returns from housing were not strongly related to investment choice.

This study explores the impact of post-earthquake images inserted in a vigilance task, in terms of performance, self-reports of task-focus, and cerebral activity using functional near-infrared spectroscopy (fNIRS). Vigilance tasks present a sequence of stimuli in which only a few are pre-designated critical or target stimuli requiring an overt response from the participant. Seventy-one residents participated (51 women, 20 men) by taking part in a vigilance task with task-irrelevant images inserted in the sequence. There were three conditions consisting positive (emotive inducing), negative (emotive inducing), and control (devoid of meaning) images embedded in the vigilance task to assess possible impacts on vigilance performance. The images were obtained through crowdsourcing and represented parts of the city 3–4 years post-earthquake. Task performance was assessed with signal detection theory metrics of sensitivity A ’ and bias β ’’. This enables the separation of an individual's ability to accurately discriminate critical signals from non-critical stimuli (sensitivity) and shifts in their willingness to respond to any stimuli whether critical or not (bias). Individuals viewing the positive images, relating to progress, rebuild, or aesthetic aspects within the city, had a more conservative response bias (they responded less to both rare critical and distractor stimuli) than those in the other conditions. These individuals also reported lower task-focus, as would be expected. However, contrary to expectations, indicators of cerebral activity (fNIRS) did not differ significantly between the experimental groups. These results, when combined, suggest that mind wandering events may be being generated when exposed to positive post-earthquake images.

OCCUPATIONAL APPLICATIONS Many occupations, including those of emergency responders and military personnel, require the person to maintain volitional control of movement speed while simultaneously engaging in cognitive tasks. In the present study, runners who ran over uneven and even natural terrain slowed their running speed when confronted with increased cognitive load. Runners had impaired cognitive task performance only when the cognitive load was high and they were running over uneven terrain. While more research is warranted, the present findings indicate that willed running or movement speed reduces with increasing cognitive load. The addition of cognitive load to people who have to cover terrain may impair their speed. In occupational settings requiring both movement and cognitive tasks, the interacting effect of these tasks needs to be carefully considered. For example, where speed is important any additional cognitive load should be reduced or eliminated if possible.TECHNICAL ABSTRACT Background: Many occupations require movement over natural terrain while the person is simultaneously performing cognitive tasks (communication, navigations, etc.), yet there is little known about how cognitive load impacts movement over natural (not artificial) terrain. Purpose: The present study was designed to examine the impact of cognitive load on volitionally controlled running speed over natural terrain. Method: We examined the performance of runners on even and uneven terrain in a dual running and tone-counting working memory task. The tone counting task was performed at both a low and a high workload. Participants performed the tone-counting tasks both while running and while seated. In addition, they ran without a cognitive task load. Results: Counting accuracy significantly decreased during the dual task trials only for the high workload task and only for the uneven terrain runners. For both terrain groups there was a linear trend observed; run distance decreased as cognitive load increased. Reports of workload, task-focus and feelings of being spent after running all increased with increased cognitive load regardless of terrain. Conclusion: These findings have important theoretical and practical implications, particularly where natural running is coupled with complex cognitive tasks.

Loyalty reward schemes often have their own currency, for example, frequent flyer miles, which is a form of near money or quasi money. In a variation of earlier work by Snelders et al. (1992), when examining both New Zealand (Study 1) and Hong Kong (Study 2) residents, respondents provided typicality ratings, similarity ratings, and answers and reaction times to the question “Is X a type of money?” for examples of money, near money and objects of value. The results from both studies showed that near money is conceptualised in a way that is like but distinct from legal tender. Two further studies investigated implications of this conceptualisation. Study 3 found that preferences for spending near money were influenced by the ostensible purpose of the currency, and Study 4 showed that near money seemed to be placed outside of regular legal tender mental accounts.