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The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. 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 Gödel 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.
This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.
This classic book tells the story of the development of Income Tax from its beginning in 1799 to the present day and relates it to the social, economic and political history of the period. There have been studies of Income Tax at various stages in its growth; studies of finance and taxation in general in which Income Tax has been closely concerned; studies too of some of the Chancellors of the Exchequer who have made significant contributions to the Income Tax system; but this is the first time an attempt has been made to encompass the whole 160 years or so of its life in one volume. And a fascinating story it is too when set in perspective. The author shows how Income Tax was introduced to finance the Napoleonic Wars, how it was revived by Peel to pay for Free Trade, and how it underwrote Victorian prosperity and confidence. He then describes its immense expansion through two World Wars to its present position as a dominant feature of British finance. This book was first published in 1966.
Review: "Depth and breadth of coverage, clarity of presentation, impressive bibliographies, excellent use of cross references, and an extensive index combine to make this an impressive reference work. The contributors have addressed both current and past scholarship on world philosophy and religion and have produced a worthy successor to Macmillan's 1967 Encyclopedia of Philosophy. It will be read and understood by the educated public as well as scholars and will be a fine addition to academic and large public library reference collections."--"Outstanding Reference Sources : the 1999 Selection Sources Committee, RUSA, ALA
Shapiro argues that both realist and anti-realist accounts of mathematics are problematic. To resolve this dilemma, he articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

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