*The Ontology and Epistemology of the Exact Sciences*

**Author**: Jody Azzouni

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 249

**View:** 795

Jody Azzouni argues in this original and exciting study that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyzes the linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers.

Some of the world's specialists provide in this handbook essays about what kinds of things there are, in what ways they exist, and how they relate to each other. They give the word on such topics as identity, modality, time, causation, persons and minds, freedom, and vagueness.

This Festschrift contains numerous colorful and eclectic essays from well-known mathematicians, philosophers, logicians, and linguists celebrating the 90th birthday of Reuben Hersh. The essays offer, in part, attempts to answer the following questions set forth by Reuben himself as a focus for this volume: Can practicing mathematicians, as such, contribute anything to the philosophy of math? Can or should philosophers of math, as such, say anything to practicing mathematicians? Twenty or fifty years from now, what will be similar, and what will, or could, or should be altogether different: About the philosophy of math? About math education? About math research institutions? About data processing and scientific computing? The essays also offer glimpses into Reuben’s fertile mind and his lasting influence on the mathematical community, as well as revealing the diverse roots, obstacles and philosophical dispositions that characterize the working lives of mathematicians. With contributions from a veritable “who’s who” list of 20th century luminaries from mathematics and philosophy, as well as from Reuben himself, this volume will appeal to a wide variety of readers from curious undergraduates to prominent mathematicians.

This volume features essays about and by Paul Benacerraf, whose ideas have circulated in the philosophical community since the early nineteen sixties, shaping key areas in the philosophy of mathematics, the philosophy of language, the philosophy of logic, and epistemology. The book started as a workshop held in Paris at the Collège de France in May 2012 with the participation of Paul Benacerraf. The introduction addresses the methodological point of the legitimate use of so-called “Princess Margaret Premises” in drawing philosophical conclusions from Gödel’s first incompleteness theorem. The book is then divided into three sections. The first is devoted to an assessment of the improved version of the original dilemma of “Mathematical Truth” due to Hartry Field: the challenge to the platonist is now to explain the reliability of our mathematical beliefs given the very subject matter of mathematics, either pure or applied. The second addresses the issue of the ontological status of numbers: Frege’s logicism, fictionalism, structuralism, and Bourbaki’s theory of structures are called up for an appraisal of Benacerraf’s negative conclusions of “What Numbers Could Not Be.” The third is devoted to supertasks and bears witness to the unique standing of Benacerraf’s first publication: “Tasks, Super-Tasks, and Modern Eleatics” in debates on Zeno’s paradox and associated paradoxes, infinitary mathematics, and constructivism and finitism in the philosophy of mathematics. Two yet unpublished essays by Benacerraf have been included in the volume: an early version of “Mathematical Truth” from 1968 and an essay on “What Numbers Could Not Be” from the mid 1970’s. A complete chronological bibliography of Benacerraf’s work to 2016 is provided.Essays by Jody Azzouni, Paul Benacerraf, Justin Clarke-Doane, Sébastien Gandon, Brice Halimi, Jon Pérez Laraudogoitia, Mary Leng, Antonio León-Sánchez and Ana C. León-Mejía, Marco Panza, Fabrice Pataut, Philippe de Rouilhan, Andrea Sereni, and Stewart Shapiro.

Our experience of objects (and consequently our theorizing about them) is very rich. We perceive objects as possessing individuation conditions. They appear to have boundaries in space and time, for example, and they appear to move independently of a background of other objects or a landscape. In Ontology Without Boundaries Jody Azzouni undertakes an analysis of our concept of object, and shows what about that notion is truly due to the world and what about it is a projection onto the world of our senses and thinking. Location and individuation conditions are our product: there is no echo of them in the world. Features, the ways that objects seem to be, aren't projections. Azzouni shows how the resulting austere metaphysics tames a host of ancient philosophical problems about constitution ("Ship of Theseus," "Sorities"), as well as contemporary puzzles about reductionism. In addition, it's shown that the same sorts of individuation conditions for properties, which philosophers use to distinguish between various kinds of odd abstracta-universals, tropes, and so on, are also projections. Accompanying our notion of an object is a background logic that makes cogent ontological debate about anything from Platonic objects to Bigfoot. Contemporary views about this background logic ("quantifier variance") make ontological debate incoherent. Azzouni shows how a neutral interpretation of quantifiers and quantifier domains makes sense of both philosophical and pre-philosophical ontological debates. Azzouni also shows how the same apparatus makes sense of our speaking about a host of items--Mickey Mouse, unicorns, Martians--that nearly all of us deny exist. It's allowed by what Azzouni shows about the background logic of our ontological debates, as well as the semantics of the language of those debates that we can disagree over the existence of things, like unicorns, without that background logic and semantics forcing ontological commitments onto speakers that they don't have.

A Companion to Relativism presents original contributions from leading scholars that address the latest thinking on the role of relativism in the philosophy of language, epistemology, ethics, philosophy of science, logic, and metaphysics. Features original contributions from many of the leading figures working on various aspects of relativism Presents a substantial, broad range of current thinking about relativism Addresses relativism from many of the major subfields of philosophy, including philosophy of language, epistemology, ethics, philosophy of science, logic, and metaphysics

This book is a collection of papers presented at the conference New Trends in the History and Philosophy of Mathematics held at the University of Roskilde, Denmark, 6-8 August 1998. The purpose of the meeting was to present some of the new ideas on the study of mathematics, its character and the nature of its development. During the last decades work in history and philosophy of mathematics has led to several new original views on mathematics. Both new methods and angles of study have been introduced, and old views of, say, the nature of mathematical theories and proofs have been challenged. For instance, disciplines as etnohistorical studies of mathematics and the sociology of mathematics have resulted in several new insights, and classical historians of mathematics are also experimenting with new perspectives. In a similar way philosophy of mathematics has witnessed rather deep changes. Classical foundational studies have been challenged by new broader perspectives. The aim was to provide a forum within which historians of mathematics, philosophers, and mathematicians could exchange ideas and discuss different new approaches in the history and philosophy of mathematics. The book includes papers by Joan Richards, Henk J. M. Bos, Donald MacKenzie, Arthur Jaffe, Jody Azzouni and Paulus Gerdes. It also includes an extended introduction.

Mathematics education research routinely receives the attention of educators, mathematicians, linguists, psychologists, anthropologists, and others. In this volume, the induction of students into mathematical meaning-making is studied through the prism of these several disciplines. What unites all such approaches to pedagogy and to the assessment of pegagogy- and to the subject matter of mathematics itself - is semiotics. Myrdene Anderson teaches at Purdue University, Adalira Saenz-Ludlow teaches at the U of North Carolina, Shea Zetlweger is former chair at Mount Union College, Ohio, Victor V. Cifarelli teaches at the U. ol North Carolina.