An entertaining look at the origins of mathematical symbols While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols , popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted. Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics. From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

First published originally in German in 1921, "Tractatus Logico-Philosophicus" is the only book-length philosophical work by Austrian philosopher Ludwig Wittgenstein published during his lifetime. Based upon notes made during his time serving in World War I, the treatise is composed of 526 numbered declarative statements that are presented without argument and were seen by Wittgenstein as being self-evident. Wittgenstein's style is succinct and sparse and his ideas are presented in a highly logical, organized, and scientific manner. Wittgenstein proposes that the solution to all major philosophical problems can be solved only with logic and a scientific analysis of language and its meaning. His idea that philosophy is not about metaphysics and mysticism, but about critical and rational thought led to a massive shift in philosophical theories after its publication. The treatise was influential with many of the "logical positivists" of the Vienna Circle, such as Rudolf Carnap and Friedrich Waismann, and to philosophers such as Bertrand Russell. Considered one of the most significant and influential philosophical works of the twentieth century, this treatise is an important read for any student of the history of language and ideas. This edition is printed on premium acid-free paper.

The continuing discovery of important Hegel manuscripts and advances in the criticism of Hegel's works have set the stage for a major reevaluation of one of the greatest philosophers of all time. This volume constitutes the comprehensive reinterpretation of Hegel that has long been needed. The first chapters are devoted to the influences of other German philosophers on Hegel, his early publication as they are relevant to his later writings, and his Phenomenology--in itself and as a key to understanding his terminology and dialectic. Examined next are the further elaboration of his thought in Logic; his famous system, as presented in various editions of the Encyclopedia; and his little-known views on history. A final chapter details in letters and contemporary reports Hegel's intellectual development.

This book is an informal though systematic series of lectures on Boolean algebras. It contains background chapters on topology and continuous functions and includes hundreds of exercises as well as a solutions manual.

This book is the first English version of Prolegomena zu einer kritischen Grammatik, published by Julius Springer, Vienna, 1935, as Volume 10 of the Vienna Circle's series Schriften zur wissenschaftlichen Weltauffassung. The prefatory remarks of both editor and author acknowledge the influence ofWittgenstein in a general way. However, in aim and approach, the work differs from Wittgenstein's Philosophische Grammatik (l969). This is indeed based on material going back to 1932, some of which Schachter must have known. On the other hand, the present Prolegomena not only explains the general, philosophical principles to be followed, but in the light of these proceeds to cover the entire range of conventional grammar, showing where that is uncritical. Whether Wittgenstein in his turn knew of Schachter's work has never been explored. Schachter's object is universal grammar. As is natural, the examples in the original are largely drawn from German grammar, with occasional minor excursions into other languages. For English readers, what matters are the general problems of grammar: there is no point in tying these to the linguistic peculiarities of German, let alone a local variety of it. One who can grasp German at that level might as well read the original. The translation is therefore twofold: the text as a whole has been rendered into English, and the entire apparatus of examples has been replaced, as far as this can be done, by illustrations from English grammar, chosen so as to bring out the same kinds of problem as in the original.

Tadeusz Kotarbinski is one of towering figures in contemporary Polish philosophy. He was a great thinker, a great teacher, a great organizer of philosophical and scientific life (he was, among others, the rector of the Uni versi ty of t6dz, the president of the Polish Academy of Sciences, and the president of the International Institute of Philosophy), and, last but not least, a great moral authority. He died at the age of 96 on October 3, 1981. Kotarbinski was active in almost all branches of philosophy. He made many significant contributions to logic, semantics, ontology, epistemology, history of philosophy, and ethics. He created a new field, namely praxiology. Thus, using an ancient distinction, he contributed to theoretical as well as practical philoso~hy. Kotarbinski regarded praxiology as his major philosophical "child". Doubtless, praxiology belongs to practical philosophy. This collection, howewer, is mainly devoted to Kotarbinski' s theoretical philosophy. Reism - Kotarbinski' s fundamental idea of ontology and semantics - is the central topic of most papers included here; even Pszczolowski' s essay on praxiology considers its ontological basis. ,Only two papers, namely that of Zarnecka-Bialy and that of Wolenski, are not linked with reism. However, both fall under the general label "Kotarbinski: logic, semantics and ontology". The collection partly consists of earlier published papers.

The aim of this book is to explain the shape of Greek mathematical thinking. It can be read on three levels: as a description of the practices of Greek mathematics; as a theory of the emergence of the deductive method; and as a case-study for a general view on the history of science. The starting point for the enquiry is geometry and the lettered diagram. Reviel Netz exploits the mathematicians' practices in the construction and lettering of their diagrams, and the continuing interaction between text and diagram in their proofs, to illuminate the underlying cognitive processes. A close examination of the mathematical use of language follows, especially mathematicians' use of repeated formulae. Two crucial chapters set out to show how mathematical proofs are structured and explain why Greek mathematical practice manages to be so satisfactory. A final chapter looks into the broader historical setting of Greek mathematical practice.

Tractatus Logico-Philosophicus: Large Print by Ludwig Wittgenstein Perhaps this book will be understood only by someone who has himself already had the thoughts that are expressed in it--or at least similar thoughts.--So it is not a textbook.--Its purpose would be achieved if it gave pleasure to one person who read and understood it.The book deals with the problems of philosophy, and shows, I believe, that the reason why these problems are posed is that the logic of our language is misunderstood. The whole sense of the book might be summed up the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence. We are delighted to publish this classic book as part of our extensive Classic Library collection. Many of the books in our collection have been out of print for decades, and therefore have not been accessible to the general public. The aim of our publishing program is to facilitate rapid access to this vast reservoir of literature, and our view is that this is a significant literary work, which deserves to be brought back into print after many decades. The contents of the vast majority of titles in the Classic Library have been scanned from the original works. To ensure a high quality product, each title has been meticulously hand curated by our staff. Our philosophy has been guided by a desire to provide the reader with a book that is as close as possible to ownership of the original work. We hope that you will enjoy this wonderful classic work, and that for you it becomes an enriching experience.

This volume is number five in the 11-volume Handbook of the History of Logic. It covers the first 50 years of the development of mathematical logic in the 20th century, and concentrates on the achievements of the great names of the period--Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts: set theory, model theory, proof theory and recursion theory. Collectively, this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. • The entire range of modal logic is covered• Serves as a singular contribution to the intellectual history of the 20th century• Contains the latest scholarly discoveries and interpretative insights

Nicholas Rescher presents ten essays that offer the thoughts of major Arabic philosophers in history and speak to their origins in Greek philosophy, as well as the subsequent influence of Arabic philosophy on the West. Much of the material is presented for the first time in print. Topics include: the concentric structure of the universe; the concept of existence; the Theory of Temporal Modalities; the Platonic Solids; and several essays on logic.

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.