Recent Approaches in Mathematics and Natural Science, Livre de Lyon

This book identifies the organizing concepts of physical and biological phenomena by an analysis of the foundations of mathematics and physics. Our aim is to propose a dialog between different conceptual universes and thus to provide a unification of phenomena. The role of “order” and symmetries in the foundations of mathematics is linked to the main invariants and principles, among them the geodesic principle (a consequence of symmetries), which govern and confer unity to various physical theories. Moreover, an attempt is made to understand causal structures, a central element of physical intelligibility, in terms of both symmetries and symmetry breakings. A distinction between the principles of (conceptual) construction and of proofs, both in physics and in mathematics, guides most of the work.The importance of mathematical tools is also highlighted to clarify differences in the models for physics and biology that are proposed by continuous and discrete mathematics, such as computational simulations.Since biology is particularly complex and not as well understood at a theoretical level, we propose a “unification by concepts” which in any case should precede mathematization. This constitutes an outline for unification also based on highlighting conceptual differences, complex points of passage and technical irreducibilities of one field to another. Indeed, we suppose here a very common monist point of view, namely the view that living objects are “big bags of molecules”. The main question though is to understand which “theory” can help better understand these bags of molecules. They are, indeed, rather “singular”, from the physical point of view. Technically, we express this singularity through the concept of “extended criticality”, which provides a logical extension of the critical transitions that are known in physics. The presentation is mostly kept at an informal and conceptual level./a

History of mathematics.

This book covers a course of mathematics designed primarily for physics and engineering students. It includes all the essential material on mathematical methods, presented in a form accessible to physics students, avoiding precise mathematical jargon and proofs which are comprehensible only to mathematicians. Instead, all proofs are given in a form that is clear and convincing enough for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each section of the book. Mathematics for Natural Scientists: Fundamentals and Basics is the first of two volumes. Advanced topics and their applications in physics are covered in the second volume.

"This book addresses the challenges that face science and mathematics education if it is to be relevant to 21st century citizens, as well as the ways that outstanding specialists from several countries around the world think it should deal with those challenges. Starting with the issue of science and mathematics teacher education in a changing world, it moves on to deal with innovative approaches to teaching science and mathematics. It then discusses contemporary issues related to the role played by technology in science and mathematics education, the challenges of the STEM agenda, and ways of making science and mathematics education more inclusive. Finally, it focuses on assessment issues, as the success of science and mathematics education depends at least in part on the purposes for which, and ways in which, students' learning is assessed. There is a worldwide trend towards providing meaningful science and mathematics education to all children for the sake of literacy and numeracy development and a need to produce enough science and technology specialists. This trend and need, coupled with the concern raised by students' disengagement in these two knowledge areas and the role that technology may play in countering it, put increasingly high demands on teachers. As shown in this book, science and mathematics education may offer a unique contribution in developing responsible citizens by fostering skills required in order to assume wider responsibilities and roles, focusing on personal, social and environmental dimensions. For instance, it offers unique insights into how teachers can build on students' complicated and interconnected real-worlds to help them learn authentic and relevant science and mathematics. Additionally, the book highlights potential positive relationships between science and mathematics, which are often envisaged as having a conflicting relationship in school curricula. By uncovering the similarities between them, and by providing evidence that both areas deal with issues that are relevant for citizens' daily lives, the book explores ways of linking and giving coherence to science and mathematics knowledge as components of everyday life settings. It also provides directions for future research on the educational potential of interconnecting science and mathematics at the different educational levels. Therefore, this is a worthwhile book for researchers, teacher educators and schoolteachers. It covers theoretical perspectives, research-based approaches and practical applications that may make a difference in education that is relevant and inclusive for citizens in the 21st century"--

Teachers often want to learn new ideas and approaches to improve their teaching, but their efforts are often blocked by structural constraints in their districts and schools. How can schools overcome these barriers to provide more supportive environments for change? The authors answer this question through the study of six cases of schools and districts where teachers and researchers collaborated to develop teaching for understanding in math and science. This new book features: a new conceptual model of how school resources relate to teaching and learning, focusing not only on material resources such as time and money but also on human and social resources; methods that administrators can use to support teachers who want to improve their teaching of math and science; elements that professional developers should look for in a school environment when they are considering working with staff on teaching improvements; and answers to important questions, including how schools operate as organizations, how they control work, how they respond to changes in their environment, and how they improve classroom teaching and learning.

This book provides a variety of methods required for the analysis and solution of equations which arise in the modeling of phenomena from the natural and engineering sciences. It can be used productively by both undergraduate and graduate students, as well as others who need to learn and understand these techniques. A detailed discussion is also presented for several topics that are usually not included in standard textbooks at this level: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations, and various perturbation methods. Each chapter contains a large number of worked examples and provides references to the appropriate literature.

This book addresses the construction, analysis, and intepretation of mathematical models that shed light on significant problems in the physical sciences, with exercises that reinforce, test and extend the reader's understanding. It may be used as an upper level undergraduate or graduate textbook as well as a reference for researchers.

The advancement of a scientific discipline depends not only on the "big heroes" of a discipline, but also on a community’s ability to reflect on what has been done in the past and what should be done in the future. This volume combines perspectives on both. It celebrates the merits of Michael Otte as one of the most important founding fathers of mathematics education by bringing together all the new and fascinating perspectives created through his career as a bridge builder in the field of interdisciplinary research and cooperation. The perspectives elaborated here are for the greatest part motivated by the impressing variety of Otte’s thoughts; however, the idea is not to look back, but to find out where the research agenda might lead us in the future. This volume provides new sources of knowledge based on Michael Otte’s fundamental insight that understanding the problems of mathematics education – how to teach, how to learn, how to communicate, how to do, and how to represent mathematics – depends on means, mainly philosophical and semiotic, that have to be created first of all, and to be reflected from the perspectives of a multitude of diverse disciplines.

This work presents a series of dramatic discoveries never before made public. Starting from a collection of simple computer experiments---illustrated in the book by striking computer graphics---Wolfram shows how their unexpected results force a whole new way of looking at the operation of our universe. Wolfram uses his approach to tackle a remarkable array of fundamental problems in science: from the origin of the Second Law of thermodynamics, to the development of complexity in biology, the computational limitations of mathematics, the possibility of a truly fundamental theory of physics, and the interplay between free will and determinism.