This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in phil...

This volume offers English translations of three early works by Ernst Schröder (1841-1902), a mathematician and logician whose philosophical ruminations and pathbreaking contributions to algebraic logic attracted the admiration and ire of figures such as Dedekind, Frege, Husserl, and C. S. Peirce. Today he still engages the sympathetic interest of logicians and philosophers.The works translated record Schröder’s journey out of algebra into algebraic logic and document his transform...

Sometimes it is not big events or great men or women that change history. Often, an apparently trivial occasion or insignificant decision changes everything.Stephen Pollard's alternative history of the past sixty years examines ten such crucial days in our history. None of them are obviously historic. But each of them changed the country - some for good, others for ill.Combining history, analysis, humour and polemic, this incisive look at events stretched across six decades...

Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert’s program of consistency proofs; he ex...

Löwenheim's theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory--that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with Löwenheim's theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem--a logician raised in the ...

A History of Modern Britain by Andrew Marr confronts head-on the victory of shopping over politics. This edition also includes an extra chapter charting the course from Blair to Brexit.It tells the story of how the great political visions of New Jerusalem or a second Elizabethan Age, rival idealisms, came to be defeated by a culture of consumerism, celebrity and self-gratification. In each decade, political leaders think they know what the...

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method...

'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.' – The GuardianIn 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The impo...

A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. The text will be of particular interest to teachers and future teachers as it lin...

As a small boy in Epping Forest, Jack Straw could never have imagined that one day he would become Britain's Lord Chancellor. As one of five children of divorced parents, he was bright enough to get a scholarship to a direct-grant school, but spent his holidays as a plumbers' mate for his uncles to bring in some much-needed extra income. Yet he spent 13 years and 11 days in government, including long and influential spells as Home Secretary and Foreign Secretary. This is the story of how h...

The main purpose of this book is the development of a new method for the semantical analysis of meaning, that is, a new method for analyzing and describing the meanings of linguistic expressions. This method, called the method of extension and intension, is developed by modifying and extending certain customary concepts, especially those of class and property. The method will be contrasted with various other semantical methods used in traditional philosophy or by contemporary authors. Thes...

The British campaign in Northern Ireland remains one of the most controversial actions in recent history.This book by Aaron Edwards considers the strategic, operational and tactical aspects of what become the longest ever campaign embarked upon by British troops. The 38-year campaign, codenamed Operation Banner, went through a number of phases. It began as a peacekeeping operation, morphed into a counter-insurgency operation and ending as a policing and counter-ter...