Includes bibliographical references.

Introduction by Reuben Hersh.- A. Renyi: Socratic Dialogue.- C. Celluci: Filosofia e Matematica, introduction.- W. Thurston: On Proof and Progress in Mathematics.- A. Aberdein: The Informal Logic of Mathematical Proof.- Y. Rav: Philosophical Problems of Mathematics in Light of Evolutionary Epistemology.- B. Rotman: Towards a Semiotics of Mathematics.- D. Mackenzie: Computers and the Sociology of Mathematical Proof.- T. Stanway: From G.H.H. and Littlewood to XML and Maple: Changing Needs and Expectations in Mathematical Knowledge Management.- R. Nunez: Do Numbers Really Move?- T. Gowers: Does Mathematics Need a Philosophy?- J. Azzouni: How and Why Mathematics is a Social Practice.- G.C. Rota: The Pernicious Influence of Mathematics Upon Philosophy.- J. Schwartz: The Pernicious Influence of Mathematics on Science.- Alfonso Avila del Palacio: What is Philosophy of Mathematics Looking For?.- A. Pickering: Concepts and the Mangle of Practice: Constructing Quaternions.- E. Glas: Mathematics as Objective Knowledge and as Human Practice.- L. White: The Locus of Mathematical Reality: An Anthropological Footnote.- R. Hersh: Inner Vision, Outer Truth.

This book collects some of the most interesting recent writings that are tackling, from various points of view, the problem of giving an accounting of the nature, purpose, and justification of real mathematical practice - mathematics as actually done by real live mathematicians. What is the nature of the objects being studied? What determines the directions and styles in which mathematics progresses (or, perhaps, degenerates)? What certifies its claim to certainty, or to a priori status, to independence of experience? Why is mathematics the same for all times and places, or is it really the same, or in what senses is it the same and in what senses different? Many of these writings were read at conferences in Europe and America under the heading of history or cultural studies as well as philosophy. It is the editor's hope to help foster healthy interdisciplinary mutual aid in this young and fertile area.'I was pleasantly surprised to find that this book does not treat mathematics as dessicated formal logic but as a living organism, immediately recognizable to any working mathematician' - Sir Michael Atiyah, University of Edinburgh. 'A wonderful collection of essays on the philosophy of mathematics, some by mathematicians, others by philosophers, and all having significant things to say. Most readers will be informed, some will be infuriated, but all will be stimulated' - John H. Conway, John von Neumann Distinguished Professor of Mathematics, Princeton University.