18 Unconventional Essays On The Nature Of Mathematics
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18 Unconventional Essays on the Nature of Mathematics
Author: Reuben Hersh
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-01-16
Collection of the most interesting recent writings on the philosophy of mathematics written by highly respected researchers from philosophy, mathematics, physics, and chemistry Interdisciplinary book that will be useful in several fields—with a cross-disciplinary subject area, and contributions from researchers of various disciplines
Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method
Author: Carlo Cellucci
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-10-09
This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. trigger
Emerging Perspectives on Gesture and Embodiment in Mathematics
The purpose of the book is to establish a common language for, and understanding of, embodiment as it applies to mathematical thinking, and to link mathematics education research to recent work in gesture studies, cognitive linguistics and the theory of embodied cognition. Just as in past decades, mathematics education experienced a "turn to the social" in which socio-cultural factors were explored, in recent years there has been a nascent "turn to the body." An increasing number of researchers and theorists in mathematics education have become interested in the fact that, although mathematics may be socially constructed, this construction is not arbitrary or unconstrained, but rather is rooted in, and shaped by, the body. All those who engage with mathematics, whether at an elementary or advanced level, share the same basic biological and cognitive capabilities, as well as certain common physical experiences that come with being humans living in a material world. In addition, the doing and communicating of mathematics is never a purely intellectual activity: it involves a wide range of bodily actions, from committing inscriptions to paper or whiteboard, to speaking, listening, gesturing and gazing. This volume will present recent research on gesture and mathematics, within a framework that addresses several levels of mathematical development. The chapters will begin with contributions that examine early mathematical and proto-mathematical knowledge, for example, the conservation of volume and counting. The role of gesture in teaching and learning arithmetic procedures will be addressed. Core concepts and tools from secondary level mathematics will be investigated, including algebra, functions and graphing. And finally, research into the embodied understanding of advanced topics in geometry and calculus will be presented. The overall goal for the volume is to acknowledge the multimodal nature of mathematical knowing, and to contribute to the creation of a model of the interactions and mutual influences of bodily motion, spatial thinking, gesture, speech and external inscriptions on mathematical thinking, communication and learning. The intended audience is researchers and theorists in mathematics education as well as graduate students in the field.