Current approaches to mathematical cognition divide into two major camps. Cognitive studies try to render mathematical intuition—the faculty that gives us immediate and authoritative knowledge of mathematics—respectable on scientific grounds. Cultural studies, on the other hand, regard mathematics as a form of cultural achievement, like literature or architecture. Both positions have their own shortcomings. While cognitive approaches are limited in scope and fail to account for complex mathematical developments, cultural approaches are short of detailed answers as to what enables us (...) to participate in a common mathematical practice. This situation evinces a need to balance a cognitive perspective on mathematical culture against a cultural perspective on mathematical cognition. According to current behavioral and neuropsychological evidence, the complex, uniquely human, culture-specific mathematical skills exhibited by human adults rest on a set of psychological and neural mechanisms that (a) are shared by other animals, and (b) emerge early in human development, continue to function throughout the lifespan, and thus are common to infants, children and adults. It has been proposed that these common and evolutionary ancient mechanisms account for humans’ basic “number sense” and form the building blocks for the development of more sophisticated numerical skills. Indeed, infants leave animals far behind in their numerical sophistication. What boosts this developmental difference? How do human beings acquire mathematical concepts such as the concept of natural number? First, I specify the representations that are the building blocks for the target concepts. Second, I describe how the target concepts differ from these basic representations. And finally, I characterize the learning mechanisms that enable the construction of the target concepts out of those prior representations. I argue that the power of the resulting conceptual system derives from the combination and integration of previously distinct representational systems, capitalizing on the human capacity for creating and using external symbols: human beings can only develop their distinct conceptual abilities due to their original embeddedness in both the physical world and, most importantly, in a rich milieu of cultural resources. Thus, an important developmental source of number representations, in addition to the preverbal systems mentioned above, is the representation of numbers within natural language. (shrink)

In this paper, we will elaborate on a pronouncement that should be at the onset of any study in epistemology and ethnomathematics, namely, we will argue that learners do think mathematically and it is our responsibility as educators to recognize and appreciate their modes of mathematical reasoning. We will conduct our study in five parts. Following a brief introduction, in the second part, we will briefly discuss some of the critical tenets of epistemology especially as it applies to mathematics. The (...) third part will be devoted to elucidating the basic nomenclature and hypotheses associated with ethnomathematics. In the fourth part we will expound on the organic and intrinsic relationship between these two fields. Lastly, we will propose some changes in the way academic mathematicians regard philosophy and pedagogy of mathematics that, in our opinion, will facilitate students’ understanding of the cultural aspects of mathematics. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

A theme like “town and science” invites to comparative analysis, and suggests questions like these: Is the urban context a particularly fertile soil for the development of scientific thinking? Or rather the contrary? Is it fertile or barren under specific circumstances? Or does it favour a particular kind of scientific activity?General answers to such questions can hardly be found; still, they may provide case studies with a guiding perspective. Case studies, on the other hand, may lead to better understanding of (...) the implications of the general questions if not to answers. In the following I therefore try to read the emergence of Mesopotamian mathematics into the context of the “urban revolution” and the rise of states (c. 3300 to c. 2300 B.C.).Mesopotamian city states appeared in the late fourth millennium B.C., and were eventually absorbed in larger territorial states a thousand years later. Contemporary with the incipient state formation was the creation of a script and of genuine, coherent mathematics. Both were based on earlier accounting techniques, both were used by Temple officials for administrative purposes, and both were marked decisively by this application. At the same time, both presuppose modes of thought which are not produced directly by their application but only as mediated by a school system.In the mid‐third millennium, an autonomous scribal profession appears for the first time. At the same time emerge the first literary texts and the first “pure” mathematical problems (i.e., problems not directed at or following the pattern of genuine application). It seems as if the scribes, as soon as they were emancipated as a craft, tried out the range of their professional tools.Neither the use of writing for literary purposes nor the possibility to use numbers for “higher” computation were forgotten when the city states were absorbed in larger social systems. Both, however, were subsumed by the interest of the new territorial state, the former as a means to propagate royal ideology, the latter in the royal administration.The final section of the article compares this early Mesopotamian development with other instances of the “urban revolution” and with the development of scientific thought in the Ancient Greek city state and in the context of Medieval city communes. (shrink)