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The Prehistory Of Mathematical Structuralism
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Book Synopsis The Prehistory of Mathematical Structuralism by : Erich H. Reck
Download or read book The Prehistory of Mathematical Structuralism written by Erich H. Reck and published by Oxford University Press. This book was released on 2020 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.
Book Synopsis The Prehistory of Mathematical Structuralism by :
Download or read book The Prehistory of Mathematical Structuralism written by and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematical Structuralism by : Geoffrey Hellman
Download or read book Mathematical Structuralism written by Geoffrey Hellman and published by Cambridge University Press. This book was released on 2018-11-29 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the Element considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.
Book Synopsis One Hundred Years of Russell's Paradox by : Godehard Link
Download or read book One Hundred Years of Russell's Paradox written by Godehard Link and published by Walter de Gruyter. This book was released on 2004 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers collected in this volume represent the main body of research arising from the International Munich Centenary Conference in 2001, which commemorated the discovery of the famous Russell Paradox a hundred years ago. The 31 contributions and the introductory essay by the editor were (with two exceptions) all originally written for the volume. The volume serves a twofold purpose, historical and systematic. One focus is on Bertrand Russell's logic and logical philosophy, taking into account the rich sources of the Russell Archives, many of which have become available only recently. The second equally important aim is to present original research in the broad range of foundational studies that draws on both current conceptions and recent technical advances in the above-mentioned fields. The volume contributes therefore, to the well-established body of mathematical philosophy initiated to a large extent by Russell's work.
Book Synopsis Handbook of the History and Philosophy of Mathematical Practice by : Bharath Sriraman
Download or read book Handbook of the History and Philosophy of Mathematical Practice written by Bharath Sriraman and published by Springer Nature. This book was released on with total page 3221 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Richness of the History of Mathematics by : Karine Chemla
Download or read book The Richness of the History of Mathematics written by Karine Chemla and published by Springer Nature. This book was released on 2023-11-27 with total page 702 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, a tribute to historian of mathematics Jeremy Gray, offers an overview of the history of mathematics and its inseparable connection to philosophy and other disciplines. Many different approaches to the study of the history of mathematics have been developed. Understanding this diversity is central to learning about these fields, but very few books deal with their richness and concrete suggestions for the “what, why and how” of these domains of inquiry. The editors and authors approach the basic question of what the history of mathematics is by means of concrete examples. For the “how” question, basic methodological issues are addressed, from the different perspectives of mathematicians and historians. Containing essays by leading scholars, this book provides a multitude of perspectives on mathematics, its role in culture and development, and connections with other sciences, making it an important resource for students and academics in the history and philosophy of mathematics.
Book Synopsis Theories, Sites, Toposes by : Olivia Caramello
Download or read book Theories, Sites, Toposes written by Olivia Caramello and published by Oxford University Press. This book was released on 2018 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.
Book Synopsis Rigor and Structure by : John P. Burgess
Download or read book Rigor and Structure written by John P. Burgess and published by . This book was released on 2015 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.
Book Synopsis Morality and Mathematics by : Justin Clarke-Doane
Download or read book Morality and Mathematics written by Justin Clarke-Doane and published by Oxford University Press. This book was released on 2020-03-12 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the genealogy of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together — and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not, and the sense in which they are objective can only be explained by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of factual areas like logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage.
Book Synopsis Platonism and Anti-Platonism in Mathematics by : Mark Balaguer
Download or read book Platonism and Anti-Platonism in Mathematics written by Mark Balaguer and published by . This book was released on 2001 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible. (Philosophy)
Book Synopsis Truth and Realism by : Patrick Greenough
Download or read book Truth and Realism written by Patrick Greenough and published by Oxford University Press. This book was released on 2006 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is truth objective or relative? What exists independently of our minds? This book is about these two questions. The essays in its pages variously defend and critique answers to each, grapple over the proper methodology for addressing them, and wonder whether either question is worth pursuing. In so doing, they carry on a long and esteemed tradition - for our two questions are among the oldest of philosophical issues, and have vexed almost every major philosopher, from Plato, to Kant to Wittgenstein. Fifteen eminent contributors bring fresh perspectives, renewed energy and original answers to debates which have been the focus of a tremendous amount of interest in the last three decades both within philosophy and the culture at large.
Book Synopsis In the Light of Logic by : Solomon Feferman
Download or read book In the Light of Logic written by Solomon Feferman and published by Oxford University Press, USA. This book was released on 1998 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects.
Book Synopsis Logic, Epistemology, and Scientific Theories - From Peano to the Vienna Circle by : Paola Cantù
Download or read book Logic, Epistemology, and Scientific Theories - From Peano to the Vienna Circle written by Paola Cantù and published by Springer Nature. This book was released on 2024-01-23 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a collection of chapters on the development of scientific philosophy and symbolic logic in the early twentieth century. The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The scholarship contained provides a rich historical and philosophical understanding of these groups and research areas. Specifically, the contributions focus on a detailed investigation of the relation between structuralism and modern mathematics. In addition, this book provides a closer understanding of the relation between symbolic logic and previous traditions such as syllogistics. This volume also informs the reader on the relation between logic, the history and didactics in the Peano School. This edition appeals to students and researchers working in the history of philosophy and of logic, philosophy of science, as well as to researchers on the Vienna Circle and the Peano School.
Book Synopsis The Logical Syntax of Greek Mathematics by : Fabio Acerbi
Download or read book The Logical Syntax of Greek Mathematics written by Fabio Acerbi and published by Springer. This book was released on 2021-06-23 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this monograph is to describe Greek mathematics as a literary product, studying its style from a logico-syntactic point of view and setting parallels with logical and grammatical doctrines developed in antiquity. In this way, major philosophical themes such as the expression of mathematical generality and the selection of criteria of validity for arguments can be treated without anachronism. Thus, the book is of interest for both historians of ancient philosophy and specialists in Ancient Greek, in addition to historians of mathematics. This volume is divided into five parts, ordered in decreasing size of the linguistic units involved. The first part describes the three stylistic codes of Greek mathematics; the second expounds in detail the mechanism of "validation"; the third deals with the status of mathematical objects and the problem of mathematical generality; the fourth analyzes the main features of the "deductive machine," i.e. the suprasentential logical system dictated by the traditional division of a mathematical proposition into enunciation, setting-out, construction, and proof; and the fifth deals with the sentential logical system of a mathematical proposition, with special emphasis on quantification, modalities, and connectors. A number of complementary appendices are included as well.
Book Synopsis Women in the History of Analytic Philosophy by : Jeanne Peijnenburg
Download or read book Women in the History of Analytic Philosophy written by Jeanne Peijnenburg and published by Springer Nature. This book was released on 2023-01-01 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains a selection of papers from the workshop Women in the History of Analytic Philosophy held in October 2019 in Tilburg, the Netherlands. It is the first volume devoted to the role of women in early analytic philosophy. It discusses the ideas of ten female philosophers and covers a period of over a hundred years, beginning with the contribution to the Significs Movement by Victoria, Lady Welby in the second half of the nineteenth century, and ending with Ruth Barcan Marcus’s celebrated version of quantified modal logic after the Second World War. The book makes clear that women contributed substantially to the development of analytic philosophy in all areas of philosophy, from logic, epistemology, and philosophy of science, to ethics, metaphysics, and philosophy of language. It illustrates that although women's voices were no different from men's as regards their scope and versatility, they had a much harder time being heard. The book is aimed at historians of philosophy and scholars in gender studies
Book Synopsis Logic for Philosophy by : Theodore Sider
Download or read book Logic for Philosophy written by Theodore Sider and published by Oxford University Press. This book was released on 2010-01-07 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Logic for Philosophy is an introduction to logic for students of contemporary philosophy. It is suitable both for advanced undergraduates and for beginning graduate students in philosophy. It covers (i) basic approaches to logic, including proof theory and especially model theory, (ii) extensions of standard logic that are important in philosophy, and (iii) some elementary philosophy of logic. It emphasizes breadth rather than depth. For example, it discusses modal logic and counterfactuals, but does not prove the central metalogical results for predicate logic (completeness, undecidability, etc.) Its goal is to introduce students to the logic they need to know in order to read contemporary philosophical work. It is very user-friendly for students without an extensive background in mathematics. In short, this book gives you the understanding of logic that you need to do philosophy.
Book Synopsis An Aristotelian Realist Philosophy of Mathematics by : J. Franklin
Download or read book An Aristotelian Realist Philosophy of Mathematics written by J. Franklin and published by Springer. This book was released on 2014-04-09 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.
