Author: Mary Leng

Publisher: Oxford University Press

ISBN: 0199280797

Category: Mathematics

Page: 278

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### Mathematics and Reality

Mary Leng defends a philosophical account of the nature of mathematics which views it as a kind of fiction (albeit an extremely useful fiction). On this view, the claims of our ordinary mathematical theories are more closely analogous to utterances made in the context of storytelling than to utterances whose aim is to assert literal truths.

### Einstein, mathematics and reality

### Mathematics and the Real World

### Mathematics and Its Applications

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies.

### The Philosophy of Mathematics Education Today

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future.

### Between Logic and Reality

Is reality logical and is logic real? What is the origin of logical intuitions? What is the role of logical structures in the operations of an intelligent mind and in communication? Is the function of logical structure regulative or constitutive or both in concept formation? This volume provides analyses of the logic-reality relationship from different approaches and perspectives. The point of convergence lies in the exploration of the connections between reality – social, natural or ideal – and logical structures employed in describing or discovering it. Moreover, the book connects logical theory with more concrete issues of rationality, normativity and understanding, thus pointing to a wide range of potential applications. The papers collected in this volume address cutting-edge topics in contemporary discussions amongst specialists. Some essays focus on the role of indispensability considerations in the justification of logical competence, and the wide range of challenges within the philosophy of mathematics. Others present advances in dynamic logical analysis such as extension of game semantics to non-logical part of vocabulary and development of models of contractive speech act.

### Sociocultural Research on Mathematics Education

This volume--the first to bring together research on sociocultural aspects of mathematics education--presents contemporary and international perspectives on social justice and equity issues that impact mathematics education. In particular, it highlights the importance of three interacting and powerful factors--gender, social, and cultural dimensions. Sociocultural Research on Mathematics Education: An International Perspective is distinguished in several ways: * It is research based. Chapters report on significant research projects; present a comprehensive and critical summary of the research findings; and offer a critical discussion of research methods and theoretical perspectives undertaken in the area. * It is future oriented, presenting recommendations for practice and policy and identifying areas for further research. * It deals with all aspects of formal and informal mathematics education and applications and all levels of formal schooling. As the context of mathematics education rapidly changes-- with an increased demand for mathematically literate citizenship; an increased awareness of issues of equity, inclusivity, and accountability; and increased efforts for globalization of curriculum development and research-- questions are being raised more than ever before about the problems of teaching and learning mathematics from a non-cognitive science perspective. This book contributes significantly to addressing such issues and answering such questions. It is especially relevant for researchers, graduate students, and policymakers in the field of mathematics education.

### Converging Realities

The mysterious beauty, harmony, and consistency of mathematics once caused philosopher Hilary Putnam to term its existence a "miracle." Now, advances in the understanding of physics suggest that the foundations of mathematics are encompassed by the laws of nature, an idea that sheds new light on both mathematics and physics. The philosophical relationship between mathematics and the natural sciences is the subject of Converging Realities, the latest work by one of the leading thinkers on the subject. Based on a simple but powerful idea, it shows that the axioms needed for the mathematics used in physics can also generate practically every field of contemporary pure mathematics. It also provides a foundation for current investigations in string theory and other areas of physics. This approach to the nature of mathematics is not really new, but it became overshadowed by formalism near the end of nineteenth century. The debate turned eventually into an exclusive dialogue between mathematicians and philosophers, as if physics and nature did not exist. This unsatisfactory situation was enforced by the uncertain standing of physical reality in quantum mechanics. The recent advances in the interpretation of quantum mechanics (as described in Quantum Philosophy, also by Omnès) have now reconciled the foundations of physics with objectivity and common sense. In Converging Realities, Roland Omnès is among the first scholars to consider the connection of natural laws with mathematics.

### The Relativistic Deduction

When the author of Identity and Reality accepted Langevin's suggestion that Meyerson "identify the thought processes" of Einstein's relativity theory, he turned from his assured perspective as historian of the sciences to the risky bias of contemporary philosophical critic. But Emile Meyerson, the epis temologist as historian, could not find a more rigorous test of his conclusions from historical learning than the interpretation of Einstein's work, unless perhaps he were to turn from the classical revolution of Einstein's relativity to the non-classical quantum theory. Meyerson captures our sympathy in all his writings: " . . . the role of the epistemologist is . . . in following the development of science" (250); the study of the evolution of reason leads us to see that "man does not experience himself reasoning . . . which is carried on unconsciously," and as the summation of his empirical studies of the works and practices of scientists, "reason . . . behaves in an altogether predict able way: . . . first by making the consequent equivalent to the antecedent, and then by actually denying all diversity in space" (202). If logic - and to Meyerson the epistemologist is logician - is to understand reason, then "logic proceeds a posteriori. " And so we are faced with an empirically based Par menides, and, as we shall see, with an ineliminable 'irrational' within science. Meyerson's story, written in 1924, is still exciting, 60 years later.

### Critical Mathematics Education

Mathematics is traditionally seen as the most neutral of disciplines, the furthest removed from the arguments and controversy of politics and social life. However, critical mathematics challenges these assumptions and actively attacks the idea that mathematics is pure, objective, and value?neutral. It argues that history, society, and politics have shaped mathematics—not only through its applications and uses but also through molding its concepts, methods, and even mathematical truth and proof, the very means of establishing truth. Critical mathematics education also attacks the neutrality of the teaching and learning of mathematics, showing how these are value?laden activities indissolubly linked to social and political life. Instead, it argues that the values of openness, dialogicality, criticality towards received opinion, empowerment of the learner, and social/political engagement and citizenship are necessary dimensions of the teaching and learning of mathematics, if it is to contribute towards democracy and social justice. This book draws together critical theoretic contributions on mathematics and mathematics education from leading researchers in the field. Recurring themes include: The natures of mathematics and critical mathematics education, issues of epistemology and ethics; Ideology, the hegemony of mathematics, ethnomathematics, and real?life education; Capitalism, globalization, politics, social class, habitus, citizenship and equity. The book demonstrates the links between these themes and the discipline of mathematics, and its critical teaching and learning. The outcome is a groundbreaking collection unified by a shared concern with critical perspectives of mathematics and education, and of the ways they impact on practice.

### Order and Organism

What is now needed is a way of thinking about the physical that is realistic in outlook but which departs radically from the mechanistic post-Galilean tradition. Since it seems clear that we can no longer take for granted the certainty and absolute objectivity of scientific knowledge, any alternative view must be able to do full justice to subjective modes of knowing. Order and Organism shows how Alfred North Whitehead's thought can reconcile some of the most insistent demands of common sense with the esoteric results of modern physics and mathematics. Whitehead shows a way to resolve the perennial puzzle of why mathematics works. Under his view, it is possible to account for the necessity and uniqueness of mathematical theories without denying the fact that such theories often arise from the mathematician's essentially aesthetic interest in various kinds of pattern.

### MATHEMATICAL REALITY

A thing is complex, and hybrid with other things sometimes. Then, what is the reality of a thing? The reality of a thing is its state of existed, exists, or will exist in the world, independent on the understanding of human beings, which implies that the reality holds on by human beings maybe local or gradual, not the reality of a thing. Hence, to hold on the reality of things is the main objective of science in the history of human development.

### Epistemology & Methodology III: Philosophy of Science and Technology Part I: Formal and Physical Sciences

The aims of this Introduction are to characterize the philosophy of science and technology, henceforth PS & T, to locate it on the map ofiearning, and to propose criteria for evaluating work in this field. 1. THE CHASM BETWEEN S & T AND THE HUMANITIES It has become commonplace to note that contemporary culture is split into two unrelated fields: science and the rest, to deplore this split - and to do is some truth in the two cultures thesis, and even nothing about it. There greater truth in the statement that there are literally thousands of fields of knowledge, each of them cultivated by specialists who are in most cases indifferent to what happens in the other fields. But it is equally true that all fields of knowledge are united, though in some cases by weak links, forming the system of human knowledge. Because of these links, what advances, remains stagnant, or declines, is the entire system of S & T. Throughout this book we shall distinguish the main fields of scientific and technological knowledge while at the same time noting the links that unite them.

### Mathematics and Modern Art

The link between mathematics and art remains as strong today as it was in the earliest instances of decorative and ritual art. Arts, architecture, music and painting have for a long time been sources of new developments in mathematics, and vice versa. Many great painters have seen no contradiction between artistic and mathematical endeavors, contributing to the progress of both, using mathematical principles to guide their visual creativity, enriching their visual environment with the new objects created by the mathematical science. Owing to the recent development of the so nice techniques for visualization, while mathematicians can better explore these new mathematical objects, artists can use them to emphasize their intrinsic beauty, and create quite new sceneries. This volume, the content of the first conference of the European Society for Mathematics and the Arts (ESMA), held in Paris in 2010, gives an overview on some significant and beautiful recent works where maths and art, including architecture and music, are interwoven. The book includes a wealth of mathematical illustrations from several basic mathematical fields including classical geometry, topology, differential geometry, dynamical systems. Here, artists and mathematicians alike elucidate the thought processes and the tools used to create their work

### Concepts and Approaches in Evolutionary Epistemology

The present volume brings together current interdisciplinary research which adds up to an evolutionary theory of human knowledge, Le. evolutionary epistemology. It comprises ten papers, dealing with the basic concepts, approaches and data in evolutionary epistemology and discussing some of their most important consequences. Because I am convinced that criticism, if not confused with mere polemics, is apt to stimulate the maturation of a scientific or philosophical theory, I invited Reinhard Low to present his critical view of evolutionary epistemology and to indicate some limits of our evolutionary conceptions. The main purpose of this book is to meet the urgent need of both science and philosophy for a comprehensive up-to-date approach to the problem of knowledge, going beyond the traditional disciplinary boundaries of scientific and philosophical thought. Evolutionary epistemology has emerged as a naturalistic and science-oriented view of knowledge taking cognizance of, and compatible with, results of biological, psychological, anthropological and linguistic inquiries concerning the structure and development of man's cognitive apparatus. Thus, evolutionary epistemology serves as a frame work for many contemporary discussions of the age-old problem of human knowledge.

### Technology and Reality

In the following pages I have endeavored to show the impact on philosophy of tech nology and science; more specifically, I have tried to make up for the neglect by the classical philosophers of the historic role of technology and also to suggest what positive effects on philosophy the ahnost daily advances in the physical sciences might have. Above all, I wanted to remind the ontologist of his debt to the artificer: tech nology with its recent gigantic achievements has introduced a new ingredient into the world, and so is sure to influence our knowledge of what there is. This book, then, could as well have been called 'Ethnotechnology: An Explanation of Human Behavior by Means of Material Culture', but the picture is a complex one, and there are many more special problems that need to be prominently featured in the discussion. Human culture never goes forward on all fronts at the same time. In our era it is unquestionably not only technology but also the sciences which are making the most rapid progress. Philosophy has not been very successful at keeping up with them. As a consequence there is an 'enormous gulf between scientists and philosophers today, a gulf which is as large as it has ever been. ' (1) I can see that with science moving so rapidly, its current lessons for philosophy might well be outmoded tomorrow.

### Unser mathematisches Universum

„Max Tegmark, Prophet der Parallelwelten, flirtet mit der Unendlichkeit.“ ULF VON RAUCHHAUPT, FRANKFURTER ALLGEMEINE SONNTAGSZEITUNG WORUM GEHT ES? Max Tegmark entwickelt eine neue Theorie des Kosmos: Das Universum selbst ist reine Mathematik. In diesem Buch geht es um die physikalische Realität des Kosmos, um den Urknall und die „Zeit davor“ und um die Evolution des Weltalls. Welche Rollen spielen wir dabei – die Wesen, die klug genug sind, das alles verstehen zu wollen? Tegmark findet, dieses Terrain sollte nicht länger den Philosophen überlassen bleiben. Denn die Physiker von heute haben die besseren Antworten auf die ewigen Fragen. WAS IST BESONDERS? „Eine hinreißende Expedition, die jenseits des konventionellen Denkens nach der wahren Bedeutung von Realität sucht.“ BBC „Tegmark behandelt die großen Fragen der Kosmologie und der Teilchenphysik weitaus verständlicher als Stephen Hawking.“ THE TIMES WER LIEST? • Jeder, der das Universum verstehen will • Die Leser von Richard Dawkins und Markus Gabriel

### Science in search of truth and reality

### Mathematics, Ideas and the Physical Real

Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. Mathematics, Ideas and the Physical Real presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather than being preoccupied with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy.

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Author: Mary Leng

Publisher: Oxford University Press

ISBN: 0199280797

Category: Mathematics

Page: 278

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