The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection
Author: Thomas Bedürftig,Roman Murawski
Publisher: Walter de Gruyter GmbH & Co KG
Author: Alfred North Whitehead,Bertrand Russell
Category: Logic, Symbolic and mathematical
MasterClass in Mathematics Education provides accessible links between theory and practice and encourages readers to reflect on their own understanding of their teaching context. Each chapter, written by an internationally respected authority, explores the key concepts within the selected area of the field, drawing directly on published research to encourage readers to reflect on the content, ideas and ongoing debates. Using international case studies, each chapter will encourage readers to think about ways that the teaching and learning of mathematics reflect different cultural traditions and expectations and enable them to evaluate effective strategies for their own contexts.
International Perspectives on Teaching and Learning
Author: Paul Andrews,Tim Rowland
Publisher: Bloomsbury Publishing
Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition.
Author: Raymond L. Wilder
Publisher: Courier Corporation
Stimulating, thought-provoking text by one of the 20th century's most creative philosophers makes accessible such topics as probability, measurement and quantitative language, causality and determinism, theoretical laws and concepts, more.
Author: Rudolf Carnap
Publisher: Courier Corporation
Mathematik durchdringt den ganzen Kosmos. Das weiß jeder, doch nur die wenigsten verstehen die Zusammenhänge wirklich. Steven Strogatz nimmt uns bei der Hand und spaziert mit uns durch diese Welt der Weisheit, Klarheit und Eleganz. Als Reiseleiter geht er neue, erfrischende Wege, deutet auf Besonderheiten, schildert Hintergründe und erklärt die unsichtbaren Mechanismen. Wir erfahren unter anderem von dem Wunder des Zählens, der genialen Einfachheit der Algebra, dem ewigen Erbe Newtons, dem Tango mit Quadraten, der Zweisamkeit von Primzahlen und der Macht des Unendlichen. Mit all seiner Begeisterung, seinem Scharfblick und seinem leichtem Ton hat Steven Strogatz ein herrliches Buch für alle geschrieben, die ihr Verständnis von Mathematik auf eine neue Art vertiefen möchten.
Die Schönheit der Mathematik
Author: Steven Strogatz
Publisher: Kein & Aber AG
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
Author: Reuben Hersh
Publisher: Springer Science & Business Media
Written for any readers interested in better harnessing philosophy’s real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic, Neil Tennant helps any student in pursuit of a ‘big picture’ to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all students to employ the concepts and techniques of this millennia-old discipline throughout their college careers – and beyond. KEY FEATURES AND BENEFITS: -- Investigates the philosophy of various subjects (psychology, language, biology, math), helping students contextualize philosophy and view it as an interdisciplinary pursuit; also helps students with majors outside of philosophy to see the relationship between philosophy and their own focused academic pursuits -- Author comes from a distinguished background in Logic and Philosophy of Language, which gives the book a level of rigor, balance, and analytic focus sometimes missing from primers to philosophy -- Introduces students to various important philosophical distinctions (e.g. fact vs. value, descriptive vs. prescriptive, norms vs. laws of nature, analytic vs. synthetic, inductive vs. deductive, a priori vs. a posteriori) providing skills that are important for undergraduates to develop in order to inform their study at higher levels. They are essential for further work in philosophy but they are also very beneficial for students pursuing most other disciplines -- Is much more methodologically comprehensive than competing introductions, giving the student the ability to address a wide range of philosophical problems – and not just the ones reviewed in the book -- Offers a companion website with links to apt primary sources, organized chapter-by-chapter, making unnecessary a separate Reader/Anthology of primary sources – thus providing students with all reading material necessary for the course -- Provides five to ten discussion questions for each chapter, helping instructors and students better interact with the ideas and concepts in the text
God, Mind, World, and Logic
Author: Neil Tennant
The Existence of God: A Philosophical Introduction assesses classical and contemporary arguments concerning the existence of God. It is an ideal introduction to philosophy of religion and an excellent starting point for anyone interested in arguments about the existence of God.
A Philosophical Introduction
Author: Yujin Nagasawa
Publisher: Taylor & Francis
The Greatest Puzzle of Science
Author: Spencer Scoular
In "The Concept of Model" Alain Badiou establishes a new logical 'concept of model'. Translated for the first time into English, the work is accompanied by an exclusive interview with Badiou in which he elaborates on the connections between his early and most recent work--for which the concept of model remains seminal.
An Introduction to the Materialist Epistemology of Mathematics
Author: Alain Badiou,Zachary Luke Fraser,Tzuchien Tho
Category: Electronic books
This ground-breaking book analyzes contemporary education discourse in the light of curriculum politics and popular culture, using sources ranging from academic scholarship to popular magazines, music video, film and television game shows. Mathematics is used as an “extreme case,” since it is a discipline so easily accepted as separable from politics, ethics or the social construction of knowledge. Appelbaum’s juxtaposition of popular culture, public debate and professional practice enables an examination of the production and mediation of “common sense” distinctions between school mathematics and the world outside of schools. Terrain ordinarily displaced or excluded by traditional education literature becomes the pendulum for a new conversation which merges research and practice while discarding pre-conceived categories of understanding The book also serves as an entertaining introduction to emerging theories in cultural studies, progressively illustrating the uses of discourse analysis for comprehending ideology, the implications of power/knowledge links, professional practice as a technology of power, and curriculum as at once commodities and cultural resources. In this way, Appelbaum effectively reveals a direction for teachers, students and researchers to cooperatively form a community attentive to the politics of curriculum and popular culture
Author: Peter M. Appelbaum
Publisher: SUNY Press
Mecca is the heart of Islam. It is the birthplace of Muhammad, the direction towards which Muslims turn when they pray and the site of pilgrimage which annually draws some three million Muslims from all corners of the world. Yet Mecca's importance goes beyond religion. What happens in Mecca and how Muslims think about the political and cultural history of Mecca has had and continues to have a profound influence on world events to this day. In this captivating book, Ziauddin Sardar unravels the significance of Mecca. Tracing its history, from its origins as a 'barren valley' in the desert to its evolution as a trading town and sudden emergence as the religious centre of a world empire, Sardar examines the religious struggles and rebellions in Mecca that have powerfully shaped Muslim culture. Interweaving stories of his own pilgrimages to Mecca with those of others, Sardar offers a unique insight into not just the spiritual aspects of Mecca – the passion, ecstasy and longing it evokes – but also the conflict between heritage and modernity that has characterised its history. He unpeels the physical, social and cultural dimensions that have helped transform the city and also, though accounts of such Orientalist travellers as Richard Burton and Charles Doughty, the strange fascination that Mecca has long inspired in the Western imagination. And, ultimately, he explores what this tension could mean for Mecca's future. An illuminative, lyrical and witty blend of history, reportage and memoir, this outstanding book reflects all that is profound, enlightening and curious about one of the most important religious sites in the world.
The Sacred City
Author: Ziauddin Sardar
Publisher: Bloomsbury Publishing
This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.
Author: Ian Hacking
Publisher: Cambridge University Press
In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He establishes that both platonism and anti-platonism are defensible views and introduces a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, proceeding to defend anti-platonism (in particular, mathematical fictionalism) against various attacks--most notably the Quine-Putnam indispensability attack.
Author: Mark Balaguer
Publisher: Oxford University Press on Demand
An Introduction to His Thought
Author: W. Mays
Publisher: Springer Science & Business Media
The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.
Author: Paolo Mancosu
Publisher: Oxford University Press