Far from the thankless, pointless exercises they are often thought to be, mathematics and logic are indispensable guides to ridding ourselves of dominant opinions and making possible an access to truths, or to a human experience of the utmost value. That is why mathematics may well be the shortest path to the true life, which, when it exists, is characterized by an incomparable happiness.

    In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Godel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

    A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

    

    In Are Numbers Real?, Brian Clegg explores the way that math has become more and more detached from reality, and yet despite this is driving the development of modern physics. From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity, to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader’s eyes to the hidden reality of the strange yet familiar entities that are numbers.

    Yet despite logic's widely acknowledged importance, it remains an unbroken seal for many, due to its heavy use of jargon and mathematical symbolism.This book follows the historical development of logic, explains the symbols and methods involved and explores the philosophical issues surrounding the topic in an easy-to-follow and friendly manner. It will take you through the influence of logic on scientific method and the various sciences from physics to psychology, and will show you why computers and digital technology are just another case of logic in action.

    

    There could be many reasons for this—issues at work, unhappy family life, financial troubles or embarrassing social situations. Most of us could use a little advice in these circumstances. Chanakya Neeti provides precisely that guidance to face life’s many daunting challenges. Chanakya, the great thinker and teacher, is well-known for his insights into the needs of both the privileged and the masses. The original Chanakya Neeti was written over two thousand years ago, but its brilliant verses are still applicable today because the basic quests of man remain the same—peace, prosperity and happiness. In this volume, Radhakrishnan Pillai offers a modern interpretation of Chanakya’s crisp and practical maxims in his characteristic easy-to-follow and elegant prose. Imbibe Chanakya’s wisdom to break loose from the web of troubles and create the life you desire on your terms.Radhakrishnan Pillai is the bestselling author of Corporate Chanakya, Chanakya’s 7 Secrets of Leadership, Chanakya in You, Katha Chanakya and Thus Spoke Chanakya. He has a Master’s degree in Sanskrit and has done his PhD in Kautilya’s Arthashastra. A renowned management consultant and speaker, he is the Deputy Director of the Chanakya International Institute of Leadership Studies (CIILS) at the University of Mumbai. He tweets using the handle @rchanakyapillai and is also active on other major social media platforms.

    His philosophy directly inspired Napoleon Hill, James Allen, Wallace D. Wattles, and Elbert Hubbard. Today's inspirational leaders, like Tony Robbins, Wayne Dyer, Rhonda Byrne, and Oprah Winfrey, are equally indebted to Emerson. "We are the Builders of Our Fortunes" is the first-ever collection of Emerson's essential writings on success. In these ten essays, Emerson will teach you: How to develop your inner genius; How the power of thought shapes our world; How the universe responds to our every action; Why wealth is a spiritual phenomena; Why economies rise or fall; What defines human greatness; and much more. Includes an introduction by Charles Conrad, who also selected "Emerson's Keys to Success" after each chapter.

    Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

    However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.

    Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.

    Each chapter, or unit, is divided into easily comprehended small "bites" that enable learners to master the material step-by-step, rather than being overwhelmed by masses of information covered too quickly. The book provides extremely detailed explanations of procedures and techniques, and was written in the conviction that anyone can thoroughly master its content. A four-part organization covers sentential logic, monadic predicate logic, relational predicate logic, and extra credit units that glimpse into alternative methods of logic and more advanced topics. For individuals interested in the formal study of logic.