While mathematically impaired individuals have been shown to have deficits in all kinds of basic numerical representations, among them spatial-numerical associations, little is known about individuals with exceptionally high math expertise. They might have a more abstract magnitude representation or more flexible spatial associations, so that no automatic left/small and right/large spatial-numerical association is elicited. To pursue this question, we examined the Spatial Numerical Association of Response Codes (SNARC) effect in professional mathematicians which was compared to two control groups: Professionals who use advanced math in their work but are not mathematicians (mostly engineers), and matched controls. Contrarily to both control groups, Mathematicians did not reveal a SNARC effect. The group differences could not be accounted for by differences in mean response speed, response variance or intelligence or a general tendency not to show spatial-numerical associations. We propose that professional mathematicians possess more abstract and/or spatially very flexible numerical representations and therefore do not exhibit or do have a largely reduced default left-to-right spatial-numerical orientation as indexed by the SNARC effect, but we also discuss other possible accounts. We argue that this comparison with professional mathematicians also tells us about the nature of spatial-numerical associations in persons with much less mathematical expertise or knowledge.

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relationship has only been studied in participants with normal or poor mathematical skills, not in mathematicians. Furthermore, the prevalence of these effects at the individual level is not known. Here we compared professional mathematicians, engineers, social scientists, and a reference group using the symbolic magnitude classification task with single-digit Arabic numbers. The groups did not differ with respect to symbolic numerical distance and size effects in either frequentist or Bayesian analyses. Moreover, we looked at their prevalence at the individual level using the bootstrapping method: while a reliable numerical distance effect was present in almost all participants, the prevalence of a reliable numerical size effect was much lower. Again, prevalence did not differ between groups. In summary, the phenomena were neither more pronounced nor more prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences for analogue processing of symbolic numerical magnitudes. Numerical knowledge is encoded in multiple formats serving specific functions 1-3. The first kind of code contains the analogue representation of number magnitude; the second one encompasses the visual form of numbers; and the third one stores linguistic representations of numbers. Regarding the first code, namely, analogue magnitude, there is a large body of evidence for shared behavioural characteristics of comparative judgements on symbolic numbers, e.g., Arabic 4-6 , non-symbolic numerals, e.g., sets of dots 7,8 , and other continua including line length 9 , angle 10 , physical object size 11,12 , luminance 13,14 , and non-directly perceivable properties like intelligence 15,16. Walsh 17 proposed the "theory of magnitude" (ATOM) for the processing of these and other continua, which can be thought of in terms of classification of "more or less than. " characteristics of analogue numerical magnitude processing. Analogue magnitude comparisons have been studied in different human cultures, languages and notations 7,18,19 , as well as age groups 20,21. Moreover, there is an extensive knowledge base regarding magnitude comparisons in various non-human animal species 22-24 , from insects 25 , through fish 26 , amphibians 27 , and birds 28 , up to monkeys 29 and apes 30. Taken together, these studies suggest presence of the analogue numerical magnitude representation among human beings and its deep evolutionary origins 31. On the other hand, studying numerical magnitude comparisons in animals is basically limited to non-symbolic material. Analogue magnitude comparisons, both involving symbolic and non-symbolic numerical instances, are often assum...

The paper concerns the problem of the legal responsibility of autonomous machines. In our opinion it boils down to the question of whether such machines can be seen as real agents through the prism of folk-psychology. We argue that autonomous machines cannot be granted the status of legal agents. Although this is quite possible from purely technical point of view, since the law is a conventional tool of regulating social interactions and as such can accommodate various legislative constructs, including legal responsibility of autonomous artificial agents, we believe that it would remain a mere 'law in books', never materializing as 'law in action'. It is not impossible to imagine that the evolution of our conceptual apparatus will reach a stage, when autonomous robots become full-blooded moral and legal agents. However, today at least, we seem to be far from this point.

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Cumulative transmission and innovation are the hallmark properties of the cultural achievements of human beings. Cognitive scientists have traditionally explained these properties in terms of social learning and creativity. The non-social cognitive dimension of cumulative culture, the so-called technical reasoning, has also been accounted for recently. These explanatory perspectives are methodologically individualistic since they frame cumulative and innovative culture in terms of the processing of inner cognitive representations. Here we show that going beyond methodological individualism could facilitate an understanding of why some inventions are disseminated in a stable form and constitute the basis for further modifications. Drawing on three cases of cognitive history of prominent achievements of Antiquity, i.e., Homerian epics, Euclidean geometry, and Roman law, we investigate which properties of cognitive artifacts shaped cognitive niches for modifying original cognitive tasks or developing new ones. These niches both constrained and enabled the cognitive skills of humans to promote cumulative culture and further innovations. At the same time, we claim that “wide cognition,” incorporating both intracranial resources and external cognitive representations, constitutes a platform for building explanations of cognitive phenomena developing over a historical time scale.

This article outlines the contributions of the Kraków School to the field of science and religion. The Kraków School is a group of philosophers, scientists, and theologians who belong to the milieu of the Copernicus Center for Interdisciplinary Studies. The members of the group are engaged in inquiries pertaining to the relationship between theology and various sciences, in particular cosmology, evolutionary theory, and neuroscience. The article includes a presentation of the historical background of the School, as well as its main original contributions pertaining to the history of the interactions between science and religion, the rationality and mathematicity of the universe, theology of science, and the role of logic in theology.