General theories of explanation: buyer beware

We argue that there is no general theory of explanation that spans the sciences, mathematics, and ethics, etc. More specifically, there is no good reason to believe that substantive and domain-invariant constraints on explanatory information exist. Using Nickel (Noûs 44(2):305–328, 2010) as an exemplar of the contrary, generalist position, we first show that Nickel’s arguments rest on several ambiguities, and then show that even when these ambiguities are charitably corrected, Nickel’s defense of general theories of explanation is inadequate along several different dimensions. Specifically, we argue that Nickel’s argument has three fatal flaws. First, he has not provided any compelling illustrations of domain-invariant constraints on explanation. Second, in order to fend off the most vehement skeptics of domain-invariant theories of explanation, Nickel must beg all of the important questions. Third, Nickel’s examples of explanations from different domains with common explanatory structure rely on incorrect formulations of the explanations under consideration, circular justifications, and/or a mischaracterization of the position Nickel intends to critique. Given that the best and most elaborate defense of the generalist position fails in so many ways, we conclude that the standard practice in philosophy (and in philosophy of science in particular), which is to develop theories of explanation that are tailored to specific domains, still is justified. For those who want to buy into a more ambitious project: beware of the costs!     

William stanley jevons and the extent of meaning in logic and economics

This paper shows that William Stanley Jevons was not precursor of logical positivism despite his attempt to build up a unified science. His mechanical reductionism was directed towards this project, and Jevons tried to found mathematics on logic through the development of a theory of number. We show that his attempts were unsuccessful, and that his errors remain visible within the totality of his mechanical system, including his economics. We argue that both his logic and his economics are comprehensible only when interpreted in terms of extent of meaning, and that Jevons’ system gives rise to difficulties when interpreted in terms of intent of meaning. We argue that Jevons’ methodological recommendations were intended to bridge the gap between extent and intent of meaning. Although Jevons did not succeed in establishing a unified science, his flawed methodology resulted in one of the first applications of statistics to the social sciences     

Associations of non‐symbolic and symbolic numerical magnitude processing with mathematical competence: a meta‐analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non‐symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta‐analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two‐level random‐effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non‐symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non‐symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school‐aged children and for adults.     

The role of general and number-specific order processing in adults’ arithmetic performance

Digit order processing is highly related to individual differences in arithmetic performance. To examine whether serial scanning or associative mechanisms underlie order processing, order tasks (i.e. deciding whether three digits were presented in an order or not) were administered in two experiments. In the first experiment, digits were presented in different directions namely ascending, descending and non-ordered. For each direction, close and far distance sequences were presented. Results revealed reversed distance effects for ordered sequences, but ascending sequences elicited faster performance and stronger reversed distance effects than descending sequences, suggesting that associative mechanisms underlie order processing. In the second experiment, it was examined to which extent the relation between order processing and arithmetic is number-specific by presenting order tasks with digits, letters and months. In all order tasks similar distance effects were observed and similar relations with arithmetic were found, suggesting that both general associative mechanisms and number-specific mechanisms contribute to arithmetic.     

The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming

The development of number processing is generally studied by examining the performance on basic number tasks (comparison task, same-different judgment, and priming task). Using these tasks, so-called numerical distance effects are obtained. All these effects are generally explained by assuming a magnitude representation related to a mental number line: magnitudes are represented from left to right with partially overlapping representations for nearby numbers. In this study, we compared the performance of adults on these different tasks using non-symbolic stimuli. First, we investigated whether the effects obtained in these behavioral tasks are reliable. Second, we examined the relation between the three different effects. The results showed that the observed effects in the case of the comparison task and the same-different task proved to be reliable. The numerical distance effect obtained in the priming task, however, was not reliable. In addition, a correlation was found between the distance effects in the comparison task and the same-different task. The priming distance effect did not correlate with the other two effects. These results suggest important differences between distance effects obtained under automatic and intentional task instructions regarding the use of them as indices of mathematical ability.     

Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties

This study examined numerical magnitude processing in first graders with severe and mild forms of mathematical difficulties, children with mathematics learning disabilities (MLD) and children with low achievement (LA) in mathematics, respectively. In total, 20 children with MLD, 21 children with LA, and 41 regular achievers completed a numerical magnitude comparison task and an approximate addition task, which were presented in a symbolic and a nonsymbolic (dot arrays) format. Children with MLD and LA were impaired on tasks that involved the access of numerical magnitude information from symbolic representations, with the LA children showing a less severe performance pattern than children with MLD. They showed no deficits in accessing magnitude from underlying nonsymbolic magnitude representations. Our findings indicate that this performance pattern occurs in children from first grade onward and generalizes beyond numerical magnitude comparison tasks. These findings shed light on the types of intervention that may help children who struggle with learning mathematics.     

The weight of time: affordances for an integrated magnitude system

In five experiments we explored the effects of weight on time in different action contexts to test the hypothesis that an integrated magnitude system is tuned to affordances. Larger magnitudes generally seem longer; however, Lu and colleagues (2009) found that if numbers were presented as weights in a range heavy enough to affect lifting, the "larger seems longer" effect was enhanced, but it was eliminated with weights too light to affect lifting. Experiments 1 and 2 revealed that actually lifting kilogram and gram weights had effects parallel to symbolized weights, suggesting that Lu et al.'s task implicitly evoked a lifting context. Experiments 3 and 4 showed that weights too heavy (e.g., tons) or too light to be discriminated by lifting, but relevant to other affordances (e.g., grams of a toxin) had effects on time as large or larger than for kilograms. Experiment 5 showed that the effect for grams in a toxicology context did not generalize to the lifting task of Experiment 2. Weight appears to integrate with other magnitudes when it is relevant to meaningful actions, including but not limited to lifting.     

Attitude similarity and familiarity and their links to mental health: An examination of potential interpersonal mediators

Similarity and familiarity with partner’s attitudes are linked to positive relationship outcomes, while interpersonal variables have been linked to mental health. Using multilevel models (MLMs), we modeled the associations between these attitudinal variables and mental health outcomes in 74 married couples. We found that higher levels of attitude similarity in couples were linked to lower depression, while higher levels of attitude familiarity in couples were associated with greater satisfaction with life. Mediational analyses indicated marital satisfaction and interpersonal stress mediated the link between attitude similarity and depression. Marital satisfaction also mediated the link between familiarity and satisfaction with life. This study is the first linking attitude familiarity to mental health and provides evidence that familiarity and similarity have mental health effects partly due to their interpersonal consequences.