A Socratic Dialogue on Mathematics -- "Introduction" to Filosofia e matematica -- On Proof and Progress in Mathematics -- The Informal Logic of Mathematical Proof -- Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology -- Towards a Semiotics of Mathematics -- Computers and the Sociology of Mathematical Proof -- From G.H.H. and Littlewood to XML and Maple: Changing Needs and Expectations in Mathematical Knowledge Management -- Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics -- Does Mathematics Need a Philosophy? -- How and Why Mathematics Is Unique as a Social Practice -- The Pernicious Influence of Mathematics upon Philosophy -- The Pernicious Influence of Mathematics on Science -- What Is Philosophy of Mathematics Looking for? -- Concepts and the Mangle of Practice Constructing Quaternions -- Mathematics as Objective Knowledge and as Human Practice -- The Locus of Mathematical Reality: An Anthropological Footnote -- Inner Vision, Outer Truth.

Advance praise for 18 Unconventional Essays on the Nature of Mathematics: "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 This startling new collection of essays edited by Reuben Hersh contains frank facts and opinions from leading mathematicians, philosophers, sociologists, cognitive scientists, and even an anthropologist. Each essay provides a challenging and thought-provoking look at recent advances in the philosophy of mathematics, demonstrating the possibilities of thinking fresh, sticking close to actual practice, and fearlessly letting go of standard shibboleths. The following essays are included: * Alfred Renyi: Socratic Dialogue * Carlo Cellucci: Filosofia e Matematica, introduction * William Thurston: On Proof and Progress in Mathematics * Andrew Aberdein: The Informal Logic of Mathematical Proof * Yehuda Rav: Philosophical Problems of Mathematics in Light of Evolutionary Epistemology * Brian Rotman: Towards a Semiotics of Mathematics * Donald Mackenzie: Computers and the Sociology of Mathematical Proof * Terry Stanway: From G.H.H. and Littlewood to XML and Maple: Changing Needs and Expectations in Mathematical Knowledge Management * Rafael Nunez: Do Numbers Really Move? * Timothy Gowers: Does Mathematics Need a Philosophy? * Jody Azzouni: How and Why Mathematics is a Social Practice * Gian-Carlo Rota: The Pernicious Influence of Mathematics Upon Philosophy * Jack Schwartz: The Pernicious Influence of Mathematics on Science * Alfonso Avila del Palacio: What is Philosophy of Mathematics Looking For? * Andrew Pickering: Concepts and the Mangle of Practice: Constructing Quaternions * Eduard Glas: Mathematics as Objective Knowledge and as Human Practice * Leslie White: The Locus of Mathematical Reality: An Anthropological Footnote * Reuben Hersh: Inner Vision, Outer Truth