Download or read book The Methodological Roles of Tolerance and Conventionalism in the Philosophy of Mathematics written by Emerson P. Doyle and published by . This book was released on 2013 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science--as a part of the scientific enterprise--so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of the concepts of science (including mathematics) through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts (1992). The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language(s), and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful.

Download or read book Epistemic Relativism and Scepticism written by Steven Bland and published by Springer. This book was released on 2018-07-21 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book confronts the threats of epistemic relativism and Pyrrhonian scepticism to analytic philosophy. Epistemic relativists reject absolute notions of knowledge and justification, while sceptics claim that knowledge and justification of any kind are unattainable. If either of these views is correct, then there can be no objective basis for thinking that one set of methods does a better job of delivering accurate information than any other set of methods. Philosophers have generally sought to resist these threats by responding to the argument that seems to motivate both positions: the Agrippan trilemma. Steven Bland argues that this is a mistaken strategy. He surveys the most influential responses to the Agrippan trilemma, and shows that none of them succeeds in undermining epistemic relativism. Bland also offers a new, dialectical strategy of challenging epistemic relativism by uncovering how epistemic methods depend on one another for their applications. By means of this novel analysis, the book concludes that there are principled reasons to prefer naturalistic to non-naturalistic methods, even if these reasons do little to ease the threat of scepticism.

Download or read book Shadows of Syntax written by Jared Warren and published by Oxford University Press, USA. This book was released on 2020 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: What is the source of logical and mathematical truth? This volume revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. In Shadows of Syntax, Jared Warren offers the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. He argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims. In Part I, Warren explains exactly what conventionalism amounts to and what linguistic conventions are. Part II develops an unrestricted inferentialist theory of the meanings of logical constants that leads to logical conventionalism. This conventionalist theory is elaborated in discussions of logical pluralism, the epistemology of logic, and of the influential objections that led to the historical demise of conventionalism. Part III aims to extend conventionalism from logic to mathematics. Unlike logic, mathematics involves both ontological commitments and a rich notion of truth that cannot be generated by any algorithmic process. To address these issues Warren develops conventionalist-friendly but independently plausible theories of both metaontology and mathematical truth. Finally, Part IV steps back to address big picture worries and meta-worries about conventionalism. This book develops and defends a unified theory of logic and mathematics according to which logical and mathematical truths are reflections of our linguistic rules, mere shadows of syntax.

Download or read book Representation and Productive Ambiguity in Mathematics and the Sciences written by Emily R. Grosholz and published by Clarendon Press. This book was released on 2007-08-30 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian 'analysis' - the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience. Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable for mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are sublty altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

Download or read book The Methodological Unity of Science written by Mario BUNGE and published by Springer. This book was released on 1973-11-30 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume collects some of the talks given at the Bertrand Russell Colloquium on Exact Philosophy, attached to the McGill University Foundations and Philosophy of Science Unit. It also includes a paper, on Bertrand Russell's method of philosophizing, read at the memorial symposium held at Sir Gorge Williams University shortly after the philosopher's death. All the papers appear here for the first time. Unlike many a philosophy of science anthology, this one is not center ed on the philosophy of physics. In fact the papers deal with conceptual and, in particular, philosophical problems that pop up in almost every one of the provinces of the vast territory constituted by the foundations, meth odology and philosophy of science. A couple of border territories which are in the process of being infiltrated have been added for good measure. The inclusion of papers in the philosophy of formal science and in the philosophies of physics and of biology, in a volume belonging to a series devoted to the philosophy and methodology of the social and behavioral sciences, should raise no eyebrows. Because the sciences of man make use of logic and mathematics, they are interested in questions such as whether the formal sciences have anything to do with reality (rather than with our theories about reality) and whether or not logic has kept up with the practice of mathematicians. These two problems are tackled in Part II, on the philosophy of formal science.

Download or read book Philosophy of Mathematics written by Stewart Shapiro and published by Oxford University Press. This book was released on 1997-08-07 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

Download or read book Taking Wittgenstein at His Word written by Robert J. Fogelin and published by Princeton University Press. This book was released on 2020-04-28 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Taking Wittgenstein at His Word is an experiment in reading organized around a central question: What kind of interpretation of Wittgenstein's later philosophy emerges if we adhere strictly to his claims that he is not in the business of presenting and defending philosophical theses and that his only aim is to expose persistent conceptual misunderstandings that lead to deep philosophical perplexities? Robert Fogelin draws out the therapeutic aspects of Wittgenstein's later work by closely examining his account of rule-following and how he applies the idea in the philosophy of mathematics. The first of the book's two parts focuses on rule-following, Wittgenstein's "paradox of interpretation," and his naturalistic response to this paradox, all of which are persistent and crucial features of his later philosophy. Fogelin offers a corrective to the frequent misunderstanding that the paradox of interpretation is a paradox about meaning, and he emphasizes the importance of Wittgenstein's often undervalued appeals to natural responses. The second half of the book examines how Wittgenstein applies his reflections on rule-following to the status of mathematical propositions, proofs, and objects, leading to remarkable, demystifying results. Taking Wittgenstein at His Word shows that what Wittgenstein claims to be doing and what he actually does are much closer than is often recognized. In doing so, the book underscores fundamental—but frequently underappreciated—insights about Wittgenstein's later philosophy.

Download or read book Meaning and Modality written by Casimir Lewy and published by CUP Archive. This book was released on 1976-12-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: A study of various central and connected topics in philosophical logic and the theory of meaning. There are important sections on the relation between linguistic and abstract entities, on necessity and convention, on meaning, sense and reference, and on entailment. Dr Lewy proposes a number of original solutions to problems which have been widely discussed in literature, and there is in particular a sharp and sustained criticism of conventionalism and reductionism. These are among the most difficult and intricate issues in contemporary philosophy, but Dr Lewy writes with great clarity and a minimum of technicality. Where his views are controversial they are explained and supported in a detail which makes it both possible and necessary for potential critics to state their disagreement precisely. The book should therefore be of value as an advanced textbook as well as an original contribution to philosophical logic.

Download or read book The Philosophy of Mathematics Education written by Paul Ernest and published by Springer. This book was released on 2016-07-15 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant modern movements in the philosophy of mathematics. A case study is provided of an emerging research tradition in one country. This is the Hermeneutic strand of research in the philosophy of mathematics education in Brazil. This illustrates one orientation towards research inquiry in the philosophy of mathematics education. It is part of a broader practice of ‘philosophical archaeology’: the uncovering of hidden assumptions and buried ideologies within the concepts and methods of research and practice in mathematics education. An extensive bibliography is also included.

Download or read book The Routledge Companion to Philosophy of Science written by Martin Curd and published by Routledge. This book was released on 2013-07-24 with total page 736 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Routledge Companion to Philosophy of Science is an indispensable reference source and guide to the major themes, debates, problems and topics in philosophy of science. It contains sixty-two specially commissioned entries by a leading team of international contributors. Organized into four parts it covers: historical and philosophical context debates concepts the individual sciences. The Routledge Companion to Philosophy of Science addresses all of the essential topics that students of philosophy of science need to know - from empiricism, explanation and experiment to causation, observation, prediction and more - and contains many helpful features including chapters on individual sciences (such as biology, chemistry, physics and psychology), further reading and cross-referencing at the end of each chapter. Expanded and revised throughout, this second edition includes new chapters on Conventionalism, Social Epistemology, Computer Simulation, Thought Experiments, Pseudoscience, Species and Taxonomy, and Cosmology.

Download or read book Criticism and the Growth of Knowledge Volume 4 written by Imre Lakatos and published by Cambridge University Press. This book was released on 1970-09-02 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two books have been particularly influential in contemporary philosophy of science: Karl R. Popper's Logic of Scientific Discovery, and Thomas S. Kuhn's Structure of Scientific Revolutions. Both agree upon the importance of revolutions in science, but differ about the role of criticism in science's revolutionary growth. This volume arose out of a symposium on Kuhn's work, with Popper in the chair, at an international colloquium held in London in 1965. The book begins with Kuhn's statement of his position followed by seven essays offering criticism and analysis, and finally by Kuhn's reply. The book will interest senior undergraduates and graduate students of the philosophy and history of science, as well as professional philosophers, philosophically inclined scientists, and some psychologists and sociologists.

Download or read book Philosophy of Mathematics written by David Bostock and published by John Wiley & Sons. This book was released on 2009-03-09 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: Philosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author's personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals

Download or read book Understanding Space Time written by Robert DiSalle and published by Cambridge University Press. This book was released on 2006-04-27 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time and motion, and the roles they play in our assumptions about physical objects and physical measurements. This way of thinking leads to interpretations of the work of Newton and Einstein and the connections between them. It also offers ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy.

Download or read book Science and Convention written by Jerzy Giedymin and published by Elsevier. This book was released on 2014-05-20 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Science and Convention: Essays on Henri Poincare's Philosophy of Science and The Conventionalist Tradition contains essays concerned with Henri Poincare's philosophy of science, physics in particular, and with the conventionalist tradition in philosophy that he revived and reshaped, simultaneously with, but independently of, Pierre Duhem. Separating five essays as chapters, the book discusses the main ideas of the philosophy (Essays 1 and 5), traces at least some of its historical background (Essays 1, 2, and 3), and provides some of its developments (Essays 2 and 4).

Download or read book The Art of the Intelligible written by J. Bell and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Gödel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th century mathematician Hermann Weyl. Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.

Download or read book Error and the Growth of Experimental Knowledge written by Deborah G. Mayo and published by University of Chicago Press. This book was released on 1996-07-15 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: Preface1: Learning from Error 2: Ducks, Rabbits, and Normal Science: Recasting the Kuhn's-Eye View of Popper 3: The New Experimentalism and the Bayesian Way 4: Duhem, Kuhn, and Bayes 5: Models of Experimental Inquiry 6: Severe Tests and Methodological Underdetermination7: The Experimental Basis from Which to Test Hypotheses: Brownian Motion8: Severe Tests and Novel Evidence 9: Hunting and Snooping: Understanding the Neyman-Pearson Predesignationist Stance10: Why You Cannot Be Just a Little Bit Bayesian 11: Why Pearson Rejected the Neyman-Pearson (Behavioristic) Philosophy and a Note on Objectivity in Statistics12: Error Statistics and Peircean Error Correction 13: Toward an Error-Statistical Philosophy of Science ReferencesIndex Copyright © Libri GmbH. All rights reserved.

Download or read book The Collapse of the Fact Value Dichotomy and Other Essays written by Hilary Putnam and published by Harvard University Press. This book was released on 2004-03-30 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: If philosophy has any business in the world, it is the clarification of our thinking and the clearing away of ideas that cloud the mind. In this book, one of the world's preeminent philosophers takes issue with an idea that has found an all-too-prominent place in popular culture and philosophical thought: the idea that while factual claims can be rationally established or refuted, claims about value are wholly subjective, not capable of being rationally argued for or against. Although it is on occasion important and useful to distinguish between factual claims and value judgments, the distinction becomes, Hilary Putnam argues, positively harmful when identified with a dichotomy between the objective and the purely "subjective." Putnam explores the arguments that led so much of the analytic philosophy of language, metaphysics, and epistemology to become openly hostile to the idea that talk of value and human flourishing can be right or wrong, rational or irrational; and by which, following philosophy, social sciences such as economics have fallen victim to the bankrupt metaphysics of Logical Positivism. Tracing the problem back to Hume's conception of a "matter of fact" as well as to Kant's distinction between "analytic" and "synthetic" judgments, Putnam identifies a path forward in the work of Amartya Sen. Lively, concise, and wise, his book prepares the way for a renewed mutual fruition of philosophy and the social sciences.