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.

"Circularity" is the story of a Janus-face conceptual structure, that on the one hand led to deep scientific discoveries, and on the other hand is used to trick the mind into believing the impossible. Alongside mathematical revolutions that eventually led to the invention of the computer, the book describes ancient paradoxes that arise from circular thinking. Another aspect of Circularity, its ability to entertain, leads to a surprising insight on the time old question " What is humor."

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.

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout. Alfred Tarski, one of the greatest logicians of all time, is widely thought of as 'the man who defined truth'. His work on the concepts of truth and logical consequence as defined by mathematical theory are cornerstones of modern logic, influencing developments in mathematics, philosophy, linguistics, and computer science. His teaching on logic and mathematics culminated in the 1941 classic INTRODUCTION TO LOGIC, which uses the method of deduction and explores logic and methodology as it pertains to creating mathematical theories.

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

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.

2014 Reprint of 1962 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. In "Algebraic Logic" Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. This book addresses some of the problems of mathematical logic and the theory of polyadic Boolean algebras in particular. It is intended to be an efficient way of treating algebraic logic in a unified manner.

Geared toward college undergraduates new to the subject, this concise introduction to formal logic was written by Alice Ambrose and Morris Lazerowitz, a pair of noted scholars and prolific authors in this field. A preliminary section opens the subject under the heading of truth-functions. Two subsequent parts on quantification and classes, each subdivided into numerous brief specifics, complete the overview. Suitable for students of philosophy as well as mathematics, the three-part treatment begins with the intuitive development of the standard theory of sentential connectives (called "operators"). The theory is further developed with the assistance of truth-tables and ultimately as a logistic system. Part II explores first-order quantification theory. In addition to examining most of the familiar laws that can be expressed by monadic formulas, the text addresses polyadic principles and the theories of identity and descriptions. Part III focuses on elementary concepts of classes, from class membership and class inclusion to the algebra of classes. Each part concludes with a series of exercises.

After a series of light logic puzzles, the author presents a bombshell puzzle so startling that it seems incredible that there could be any solution at all! But there is indeed a solution - moreover, one that requires a chain of lesser puzzles to be solved first. The reader is thus journeyed through a maze of subsidiary problems that has all the earmarks of an entertaining detective story. Of the author's 20 published books, this is the 11th whose purpose is to lead the unwary reader into deep logical waters through seductively entertaining logic puzzles. The "deep waters" of this book is boolean algebra with such weird looking equations as 1+1=0 - a subject which today plays a vital role, not only in mathematical systems, but also in computer science and artificial intelligence.

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.

Logic is of course a general resource for reasoning at large. But in the first half of the twentieth century, it developed particularity with a view to mathematical applications, and the field of mathematical logic came into being and flourished. In the second half of the century, much the same happened with regard to philosophical applications. Hence philosophical logic. The deliberations of this book cover a varied but interrelated array of key issues in the field. They address the representation of information in linguistic formulation, and modes of cogent demonstration in logic, mathematics, and empirical investigation, as well as the role of logic in philosophical deliberations. Overall, the book seeks to demonstrate and illustrate the utility of logic as a productive resource for rational inquiry at large.

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

Formalization plays an important role in semantics. Doing semantics and following the literature requires considerable technical sophistica tion and acquaintance with quite advanced mathematical techniques and structures. But semantics isn't mathematics. These techniques and structures are tools that help us build semantic theories. Our real aim is to understand semantic phenomena and we need the technique to make our understanding of these phenomena precise. The problems in semantics are most often too hard and slippery, to completely trust our informal understanding of them. This should not be taken as an attack on informal reasoning in semantics. On the contrary, in my view, very often the essential insight in a diagnosis of what is going on in a certain semantic phenomenon takes place at the informal level. It is very easy, however, to be misled into thinking that a certain informal insight provides a satisfying analysis of a certain problem; it will often turn out that there is a fundamental unclarity about what the informal insight actually is. Formalization helps to sharpen those insights and put them to the test.