Mathematical Structuralism

Mathematical Structuralism
Title Mathematical Structuralism PDF eBook
Author Geoffrey Hellman
Publisher Cambridge University Press
Pages 167
Release 2018-11-29
Genre Science
ISBN 110863074X

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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.

The Prehistory of Mathematical Structuralism

The Prehistory of Mathematical Structuralism
Title The Prehistory of Mathematical Structuralism PDF eBook
Author Erich H. Reck
Publisher Oxford University Press
Pages 469
Release 2020
Genre Mathematics
ISBN 0190641223

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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.

Scientific Structuralism

Scientific Structuralism
Title Scientific Structuralism PDF eBook
Author Alisa Bokulich
Publisher Springer Science & Business Media
Pages 194
Release 2011-01-21
Genre Science
ISBN 9048195977

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Recently there has been a revival of interest in structuralist approaches to science. Taking their lead from scientific structuralists such as Henri Poincaré, Ernst Cassirer, and Bertrand Russell, some contemporary philosophers and scientists have argued that the most fruitful approach to solving many problems in the philosophy of science lies in focusing on the structural features of our scientific theories. Much of the work in scientific structuralism to date has been focused on the problem of scientific realism, where it has been argued that even in cases of radical theory change the most important structural features of predecessor theories are preserved. These structural realists argue that what our most successful theories get right about the world is these abstract structural features, rather than any particular ontological claims. More recently, philosophers of science have adopted structuralist approaches to many other issues in the philosophy of science, such as scientific explanation and intertheory relations. The nine articles collected in this volume, written by the leading researchers in scientific structuralism, represent some of the most important directions of research in this field. This book will be of particular interest to those philosophers, scientists, and mathematicians who are interested in the foundations of science.

Lectures on the Philosophy of Mathematics

Lectures on the Philosophy of Mathematics
Title Lectures on the Philosophy of Mathematics PDF eBook
Author Joel David Hamkins
Publisher MIT Press
Pages 350
Release 2021-03-09
Genre Mathematics
ISBN 0262542234

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An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Conceptions of Set and the Foundations of Mathematics

Conceptions of Set and the Foundations of Mathematics
Title Conceptions of Set and the Foundations of Mathematics PDF eBook
Author Luca Incurvati
Publisher Cambridge University Press
Pages 255
Release 2020-01-23
Genre Science
ISBN 1108758355

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Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.

Phenomenology and Mathematics

Phenomenology and Mathematics
Title Phenomenology and Mathematics PDF eBook
Author Mirja Hartimo
Publisher Springer Science & Business Media
Pages 244
Release 2010-03-11
Genre Philosophy
ISBN 9048137292

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During Edmund Husserl’s lifetime, modern logic and mathematics rapidly developed toward their current outlook and Husserl’s writings can be fruitfully compared and contrasted with both 19th century figures (Boole, Schröder, Weierstrass) as well as the 20th century characters (Heyting, Zermelo, Gödel). Besides the more historical studies, the internal ones on Husserl alone and the external ones attempting to clarify his role in the more general context of the developing mathematics and logic, Husserl’s phenomenology offers also a systematically rich but little researched area of investigation. This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics. It gathers the contributions of the main scholars of this emerging field into one publication for the first time. Combining both historical and systematic studies from various angles, the volume charts answers to the question "What kind of philosophy of mathematics is phenomenology?"

Constructibility and Mathematical Existence

Constructibility and Mathematical Existence
Title Constructibility and Mathematical Existence PDF eBook
Author Charles S. Chihara
Publisher Clarendon Press
Pages 298
Release 1990-04-12
Genre Mathematics
ISBN 0191520004

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This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach in an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.