Historical objections against the number line

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations

Iconic and non-iconic stages in number development : the role of language

Is language the key to number? This article argues that the human language faculty provides the cognitive equipment that enables humans to develop a systematic number concept. Crucially, this concept is based on non-iconic representations that involve relations between relations: relations between numbers are linked with relations between objects. In contrast to this, language-independent numerosity concepts provide only iconic representations. The pattern of forming relations between relations lies at the heart of our language faculty, suggesting that it is language that enables humans to make the step from these iconic representations, which we share with other species, to a generalised concept of number

Kurt Gödel and Computability Theory

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s

The calculative reproduction of social structures : The field of gem mining in Sri Lanka

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Andrew Carnegie, World Making and the Logic of Contemporary Entrepreneurial Philanthropy

This paper focuses on the relationship between the business and philanthropic endeavours of world-making entrepreneurs; asking why, how and to what ends these individuals seek to extend their reach in society beyond business. We present an original model of entrepreneurial philanthropy which demonstrates how investment in philanthropic projects can yield positive returns in cultural, social and symbolic capital, which in turn may lead to growth in economic capital. The interpretive power of the model is demonstrated through analysis of the career of Andrew Carnegie, whose story, far from reducing to one of earning a fortune then giving it away, is revealed as more complex and more unified. His philanthropy raised his stock within the field of power, extending his influence and helping convert surplus funds into social networks, high social standing and intellectual currency, enabling him to engage in world making on a grand scale

American Orientalism and Cosmopolitan Mixed Race: Early Asian American Mixed Race in the American Literary Imagination

This paper offers and initial examination of representations of Asian American mixed race during the long period of Asian exclusion and enforced anti-miscegenation when racial mixing was legally proscribed. I argue that this literature must be understood through two categories Orientalist texts that produce and reproduce dominant discourses about Asian American mixed race and cosmopolitan texts written by mixed-race figures circling in vanguard cliques of art and literature. After tracking the representations of early Asian American mixed race in U.S. literature, this paper offers an analysis of the writing of Asian American mixed-race authors themselves in order to expand and push current theorizations

"I would like to see that one is able to say I'm proud of being a citizen of the EU…" – the way Hungarian people see Europe and the European Union

This paper explores the concepts of Europe, Europeanism and European Union, their meaning to Hungarians, how people define them and how they relate to these concepts through the analysis of qualitative in-depth interviews. The main question is whether the discourse, expressing attitudes towards Europe and the European Union, are of symbolic or utilitarian character. The symbolic way to relate to the EU is based on principles, an ideological or an emotional approach of the subject, while the pragmatic or utilitarian logic is based on rational cost-benefit analysis. The main argument of this current paper is that the way Hungarians tend to relate to the EU is rather utilitarian and it is the utilitarian logic that represents the relevant frame to understand people’s attitudes on the subject

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page