Varieties of Continua

From Regions to Points and Back

Author: Geoffrey Hellman,Stewart Shapiro

Publisher: Oxford University Press

ISBN: 0191021350

Category: Philosophy

Page: 224

View: 3273


Varieties of Continua explores the development of the idea of the continuous. Hellman and Shapiro begin with two historical episodes. The first is the remarkably rapid transition in the course of the nineteenth century from the ancient Aristotelian view, that a true continuum cannot be composed of points, to the now standard, point-based frameworks for analysis and geometry found in modern mainstream mathematics (stemming from the work of Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, et al.). The second is the mid-tolate-twentieth century revival of pre-limit methods in analysis and geometry using infinitesimals including non-standard analysis (due to Abraham Robinson), and the more radical smooth infinitesimal analysis that uses intuitionistic logic. Hellman and Shapiro present a systematic comparison of these and related alternatives (including constructivist and predicative conceptions), weighing various trade-offs, helping articulate a modern pluralist perspective, and articulate a modern pluralist perspective on continuity. The main creative work of the book is the development of rigorous regions-based theories of classical continua, including Euclidean and non-Euclidean geometries, that are mathematically equivalent (inter-reducible) to the currently standard, point-based accounts in mainstream mathematics.

Feferman on Foundations

Logic, Mathematics, Philosophy

Author: Gerhard Jäger,Wilfried Sieg

Publisher: Springer

ISBN: 3319633341

Category: Mathematics

Page: 551

View: 8403


This volume honours the life and work of Solomon Feferman, one of the most prominent mathematical logicians of the latter half of the 20th century. In the collection of essays presented here, researchers examine Feferman’s work on mathematical as well as specific methodological and philosophical issues that tie into mathematics. Feferman’s work was largely based in mathematical logic (namely model theory, set theory, proof theory and computability theory), but also branched out into methodological and philosophical issues, making it well known beyond the borders of the mathematics community. With regard to methodological issues, Feferman supported concrete projects. On the one hand, these projects calibrate the proof theoretic strength of subsystems of analysis and set theory and provide ways of overcoming the limitations imposed by Gödel’s incompleteness theorems through appropriate conceptual expansions. On the other, they seek to identify novel axiomatic foundations for mathematical practice, truth theories, and category theory. In his philosophical research, Feferman explored questions such as “What is logic?” and proposed particular positions regarding the foundations of mathematics including, for example, his “conceptual structuralism.” The contributing authors of the volume examine all of the above issues. Their papers are accompanied by an autobiography presented by Feferman that reflects on the evolution and intellectual contexts of his work. The contributing authors critically examine Feferman’s work and, in part, actively expand on his concrete mathematical projects. The volume illuminates Feferman’s distinctive work and, in the process, provides an enlightening perspective on the foundations of mathematics and logic.

Transcultural Performance

Negotiating Globalized Indigenous Identities

Author: Michele Back

Publisher: Springer

ISBN: 1137412437

Category: Social Science

Page: 160

View: 6139


Featuring interviews, conversations and observations from a multi-sited ethnography of Ecuadorean musicians and their families, this book offers an innovative response to previous analyses of globalization and indigenous languages, demonstrating how transcultural practices can enhance the use and maintenance of indigenous and minority languages.

Discourse on a New Method

Reinvigorating the Marriage of History and Philosophy of Science

Author: Mary Domski,Michael Dickson

Publisher: Open Court Publishing

ISBN: 081269662X

Category: Philosophy

Page: 852

View: 8733


Addressing a wide range of topics, from Newton to Post-Kuhnian philosophy of science, these essays critically examine themes that have been central to the influential work of philosopher Michael Friedman. Special focus is given to Friedman's revealing study of both history of science and philosophy in his work on Kant, Newton, Einstein, and other major figures. This interaction of history and philosophy is the subject of the editors' "manifesto" and serves to both explain and promote the essential ties between two disciplines usually regarded as unrelated.


Author: Illinois State Horticultural Society

Publisher: N.A


Category: Horticulture

Page: N.A

View: 8770