This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century.

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge.

A sophisticated, original introduction to the philosophy of mathematics from one of its leading thinkers Mathematics is a model of precision and objectivity, but it appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic, accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Øystein Linnebo, one of the world's leading scholars on the subject, introduces all of the classical approaches to the field as well as more specialized issues, including mathematical intuition, potential infinity, and the search for new mathematical axioms. Sophisticated but clear and approachable, this is an essential book for all students and teachers of philosophy and of mathematics.

Original anthology features less-technical essays discussing logic, topology, abstract algebra, relativity theory, and the works of David Hilbert. Most have been long unavailable or previously unpublished in book form. 2012 edition.

How philosophy transformed human knowledge and the world we live in Philosophical investigation is the root of all human knowledge. Developing new concepts, reinterpreting old truths, and reconceptualizing fundamental questions, philosophy has progressed—and driven human progress—for more than two millennia. In short, we live in a world philosophy made. In this concise history of philosophy's world-shaping impact, Scott Soames demonstrates that the modern world—including its science, technology, and politics—simply would not be possible without the accomplishments of philosophy. Firmly rebutting the misconception of philosophy as ivory-tower thinking, Soames traces its essential contributions to fields as diverse as law and logic, psychology and economics, relativity and rational decision theory. Beginning with the giants of ancient Greek philosophy, The World Philosophy Made chronicles the achievements of the great thinkers, from the medieval and early modern eras to the present. It explores how philosophy has shaped our language, science, mathematics, religion, culture, morality, education, and politics, as well as our understanding of ourselves. Philosophy's idea of rational inquiry as the key to theoretical knowledge and practical wisdom has transformed the world in which we live. From the laws that govern society to the digital technology that permeates modern life, philosophy has opened up new possibilities and set us on more productive paths. The World Philosophy Made explains and illuminates as never before the inexhaustible richness of philosophy and its influence on our individual and collective lives.