The Search For Certainty A Philosophical Account Of Foundations Of Mathematics

The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all G--ouml--;del's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.

While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). ? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.

The philosophy of mathematics is an exciting subject. Philosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic. Most books on philosophy of mathematics have little to no focus on the effects of philosophical views on mathematical practice, and no concern on giving crucial mathematical results and their philosophical relevance, consequences, reasons, etc. This book fills this gap. The book can be used as a textbook for a one-semester or even one-year course on philosophy of mathematics. "Other textbooks on the philosophy of mathematics are aimed at philosophers. This book is aimed at mathematicians. Since the author is a mathematician, it is a valuable addition to the literature." - Mark Balaguer, California State University, Los Angeles "There are not many such texts available for mathematics students. I applaud efforts to foster the dialogue between mathematics and philosophy." - Michele Friend, George Washington University and CNRS, Lille, France

One main interest of philosophy is to become clear about the assumptions, premisses and inconsistencies of our thoughts and theories. And even for a formal language like mathematics it is controversial if consistency is acheivable or necessary like the articles in the firt part of the publication show. Also the role of formal derivations, the role of the concept of apriority, and the intuitions of mathematical principles and properties need to be discussed. The second part is a contribution on nominalistic and platonistic views in mathematics, like the "indispensability argument" of W. v. O. Quine H. Putnam and the "makes no difference argument" of A. Baker. Not only in retrospect, the third part shows the problems of Mill, Frege's and the unity of mathematics and Descartes's contradictional conception of mathematical essences. Together, these articles give us a hint into the relationship between mathematics and world, that is, one of the central problems in philosophy of mathematics and philosophy of science.

This book is about philosophy, mathematics and logic, giving a philosophical account of Pluralism which is a family of positions in the philosophy of mathematics. There are four parts to this book, beginning with a look at motivations for Pluralism by way of Realism, Maddy’s Naturalism, Shapiro’s Structuralism and Formalism. In the second part of this book the author covers: the philosophical presentation of Pluralism; using a formal theory of logic metaphorically; rigour and proof for the Pluralist; and mathematical fixtures. In the third part the author goes on to focus on the transcendental presentation of Pluralism, and in part four looks at applications of Pluralism, such as a Pluralist approach to proof in mathematics and how Pluralism works in regard to together-inconsistent philosophies of mathematics. The book finishes with suggestions for further Pluralist enquiry. In this work the author takes a deeply radical approach in developing a new position that will either convert readers, or act as a strong warning to treat the word ‘pluralism’ with care.

Columbia Companion to Twentieth-Century Philosophies is the first guide to cover both the Anglo-American analytic and European Continental traditions. The first section features Nicholas Rescher writing on neoidealism, Josephine Donovan commenting on feminist philosophy, Tyler Burge discussing the philosophy of language and mind, and Robert Hanna reflecting on Kant's legacy. The second section presents Jean Grondin on hermeneutics, Leonard Lawlor on phenomenology, Charles Scott on postmodernism, and Babette Babich on the philosophy of science. The volume also covers logical positivism, naturalism, pragmatism, aesthetics, existentialism, Marxism, the Frankfurt School, structuralism, psychoanalysis, political philosophy, ethics, and the philosophy of religion. The final section addresses concurrent trends in Indian, Chinese, Japanese, and African philosophy, and a comprehensive introduction by Constantin V. Boundas not only provides a thorough outline of the problems and issues of the analytic and Continental traditions but also boldly challenges the conviction that the two approaches must be rivals. Offering an unusually panoramic perspective, the Columbia Companion to Twentieth-Century Philosophies enables readers to encounter foundational materials on their own terms.

Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.