Mathematical Knowledge Objects And Applications

This book provides a survey of a number of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the papers in this volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike.

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field.

Like most areas of scholarship, mathematics is a cumulative discipline: new research is reliant on well-organized and well-curated literature. Because of the precise definitions and structures within mathematics, today's information technologies and machine learning tools provide an opportunity to further organize and enhance discoverability of the mathematics literature in new ways, with the potential to significantly facilitate mathematics research and learning. Opportunities exist to enhance discoverability directly via new technologies and also by using technology to capture important interactions between mathematicians and the literature for later sharing and reuse. Developing a 21st Century Global Library for Mathematics Research discusses how information about what the mathematical literature contains can be formalized and made easier to express, encode, and explore. Many of the tools necessary to make this information system a reality will require much more than indexing and will instead depend on community input paired with machine learning, where mathematicians' expertise can fill the gaps of automatization. This report proposes the establishment of an organization; the development of a set of platforms, tools, and services; the deployment of an ongoing applied research program to complement the development work; and the mobilization and coordination of the mathematical community to take the first steps toward these capabilities. The report recommends building on the extensive work done by many dedicated individuals under the rubric of the World Digital Mathematical Library, as well as many other community initiatives. Developing a 21st Century Global Library for Mathematics envisions a combination of machine learning methods and community-based editorial effort that makes a significantly greater portion of the information and knowledge in the global mathematical corpus available to researchers as linked open data through a central organizational entity-referred to in the report as the Digital Mathematics Library. This report describes how such a library might operate - discussing development and research needs, role in facilitating discover and interaction, and establishing partnerships with publishers.

Mathematics is becoming increasingly collaborative, but software does not sufficiently support that: Social Web applications do not currently make mathematical knowledge accessible to automated agents that have a deeper understanding of mathematical structures. Such agents exist but focus on individual research tasks, such as authoring, publishing, peer-review, or verification, instead of complex collaboration workflows. This work effectively enables their integration by bridging the document-oriented perspective of mathematical authoring and publishing, and the network perspective of threaded discussions and Web information retrieval. This is achieved by giving existing representations of mathematical and relevant related knowledge about applications, projects and people a common Semantic Web foundation. Service integration is addressed from the two perspectives of enriching published documents by embedding assistive services, and translating between different knowledge representations inside knowledge bases. A usability evaluation of a semantic wiki that coherently integrates knowledge production and consumption services points out the remaining challenges in making such heterogeneously integrated environments support realistic workflows. The results of this thesis will soon also enable collaborative acquisition of new mathematical knowledge, as well as the contributions of existing knowledge collections of the Web of Data.

First published in 1991. What is mathematics, and how can we account for its nature? What philosophies of mathematics have been developed? What philosophical assumptions, possibly implicit, underpin the learning of mathematics? What are the aims of mathematics education and are they valid? What philosophical assumptions, possibly implicit, does mathematics teaching rest on? This study looks at these questions on the philosophy of mathematics.

This book constitutes the refereed proceedings of the Third International Conference on Mathematical Knowledge Management, MKM 2004, held in Bialowieza, Poland, in September 2004. The 27 revised full papers presented were carefully reviewed and selected from 48 submissions. Among the topics addressed are mathematics retrieval, formalizing mathematics, formal mathematics, digital mathematical libraries, semantic Web, knowledge repositories, mathematical knowledge representation, theorem proving systems, OWL, proof verification, formal representation, mathematical formulae processing, and the OpenMath project.

Mathematics is generally considered as the only science where knowledge is uni form, universal, and free from contradictions. „Mathematics is a social product - a 'net of norms', as Wittgenstein writes. In contrast to other institutions - traffic rules, legal systems or table manners -, which are often internally contradictory and are hardly ever unrestrictedly accepted, mathematics is distinguished by coherence and consensus. Although mathematics is presumably the discipline, which is the most differentiated internally, the corpus of mathematical knowledge constitutes a coher ent whole. The consistency of mathematics cannot be proved, yet, so far, no contra dictions were found that would question the uniformity of mathematics" (Heintz, 2000, p. 11). The coherence of mathematical knowledge is closely related to the kind of pro fessional communication that research mathematicians hold about mathematical knowledge. In an extensive study, Bettina Heintz (Heintz 2000) proposed that the historical development of formal mathematical proof was, in fact, a means of estab lishing a communicable „code of conduct" which helped mathematicians make themselves understood in relation to the truth of mathematical statements in a co ordinated and unequivocal way.

Spark your students to actually want to learn through the creative application of technology! Type II applications in education make it possible to teach in new and more effective ways. Type II Uses of Technology in Education: Projects, Case Studies, and Software Applications clearly explains methods and strategies presently used by teachers to offer students a creative learning experience through the application of technology. Each chapter presents individual examples of how teachers have applied technology in schools and classrooms, illustrating through case studies, projects, and software applications how to effectively spark students’ interest and learning. Type II Uses of Technology in Education is the third in a series (Internet Applications of Type II Uses of Technology in Education and Classroom Integration of Type II Uses of Technology in Education, both from Haworth) that provides a clear view of the advantages—and challenges—involved in the use of technology to enhance and actively involve students in the learning process. The applications described and discussed at length here go beyond the mundane educational functions like grading or presenting drill and practice exercises to explore fresh ways of teaching and learning. Students can become involved and actually want to learn, all through the use of creative technology application. The book also includes tables and figures to enhance understanding of the material. Type II Uses of Technology in Education discusses: data collection, analysis, and communication in student research using pocket PCs and laptops the educational effect of using a learning object as a pedagogical model rather than simply being technological in nature examples of integrated Type II activities e-learning courses using interactive video, WebCT, and on-site discussion groups electronic discussion applications in a laptop university teacher education program challenges facing students using computers to enhance and express the extent of their learning information and communication technology (ICT) integration into schools—using three illustrative case studies forward planning needed to make the difficult change to technological application for learning a case study that used problem-based learning software with at-risk students using technology to reinforce visual learning strategies digital portfolio development as a Type II application interactive computer technology in art instruction on-demand help features for effective interactive learning experience Personal Educational Tools (PETs) Type II Uses of Technology in Education: Projects, Case Studies, and Software Applications provides numerous illustrations of technology learning in action and is perfect for educators and students in programs dealing with information technology in education, and for public school personnel with interests and responsibilities in using information technology in the classroom.