Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

In their target article, Rips et al. have presented the view that there is no necessary dependency between natural numbers and internal magnitude. However, they do not give enough weight to neuroimaging and neuropsychological studies. We provide evidence demonstrating that the acquisition of natural numbers depends on magnitude representation and that natural numbers develop from a general magnitude mechanism in the parietal lobes.

Here we argue that a multidisciplinary research approach, such as currently practised in the field of developmental cognitive neuroscience, is key to maintaining current momentum and to future-proof the field of microbiome-gut-brain research. Moreover, such a comprehensive approach will also bring us closer to our aims of translation and targeted intervention approaches to improve mental health and well-being.

Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).

The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.

The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological (...) and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex. (shrink)

The Darwinian theory of evolution is itself evolving and this book presents the details of the core of modern Darwinism and its latest developmental directions. The authors present current scientific work addressing theoretical problems and challenges in four sections, beginning with the concepts of evolution theory, its processes of variation, heredity, selection, adaptation and function, and its patterns of character, species, descent and life. The second part of this book scrutinizes Darwinism in the philosophy of science and its usefulness in (...) understanding ecosystems, whilst the third section deals with its application in disciplines beyond the biological sciences, including evolutionary psychology and evolutionary economics, Darwinian morality and phylolinguistics. The final section addresses anti-Darwinism, the creationist view and issues around teaching evolution in secondary schools. The reader learns how current experimental biology is opening important perspectives on the sources of variation, and thus of the very power of natural selection. This work examines numerous examples of the extension of the principle of natural selection and provides the opportunity to critically reflect on a rich theory, on the methodological rigour that presides in its extensions and exportations, and on the necessity to measure its advantages and also its limits. Scholars interested in modern Darwinism and scientific research, its concepts, research programs and controversies will find this book an excellent read, and those considering how Darwinism might evolve, how it can apply to the human sciences and other disciplines beyond its origins will find it particularly valuable. Originally produced in French (Les Mondes Darwiniens), the scope and usefulness of the book have led to the production of this English text, to reach a wider audience. This book is a milestone in the impressive penetration by Francophone scholars into the world of Darwinian science, its historiography and philosophy over the last two decades. Alex Rosenberg, R. Taylor Cole Professor of Philosophy, Duke University Until now this useful and comprehensive handbook has only been available to francophones. Thanks to this invaluable new translation, this collection of insightful and original essays can reach the global audience it deserves. Tim Lewens, University of Cambridge. (shrink)