Proof and Other Dilemmas: Mathematics and Philosophy

doi:10.5948/UPO9781614445050

Has the advent of computers changed the nature of mathematical knowledge? Should it? Is the importance of proof decreasing? Is there an empirical aspect to mathematics after all? To what extent is mathematics socially constructed? Is mathematics the "science of patterns?" Recently emerging questions like these are discussed in this book along with some recent thinking about classical questions. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers are not yet discussing. The other half, written by philosophers of mathematics, summarize the discussion in that community during the last 35 years. In each case, a connection is made (in the article itself, or in its introduction) to issues relevant to the teaching of mathematics.

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Proof and Other Dilemmas

Mathematics and Philosophy

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Contents

Frontmatter
pp i-viii

Contents
pp ix-x

Acknowledgments
pp xi-xii

Introduction
pp xiii-xxxii

I - Proof and How it is Changing
pp 1-2

1 - Proof: Its Nature and Significance
pp 3-32

2 - Implications of Experimental Mathematics for the Philosophy of Mathematics
pp 33-60

3 - On the Roles of Proof in Mathematics
pp 61-78

II - Social Constructivist Views of Mathematics
pp 79-80

4 - When Is a Problem Solved?
pp 81-94

5 - Mathematical Practice as a Scientific Problem
pp 95-108

6 - Mathematical Domains: Social Constructs?
pp 109-128

III - The Nature of Mathematical Objects and Mathematical Knowledge
pp 129-130

7 - The Existence of Mathematical Objects
pp 131-156

8 - Mathematical Objects
pp 157-178

9 - Mathematical Platonism
pp 179-204

10 - The Nature of Mathematical Objects
pp 205-220

11 - When is One Thing Equal to Some Other Thing?
pp 221-242

IV - The Nature of Mathematics and its Applications
pp 243-244

12 - Extreme Science: Mathematics as the Science of Relations as Such
pp 245-264

13 - What is Mathematics? A Pedagogical Answer to a Philosophical Question
pp 265-290

14 - What Will Count as Mathematics in 2100?
pp 291-312

15 - Mathematics Applied: The Case of Addition
pp 313-322

16 - Probability—A Philosophical Overview
pp 323-340

Glossary of Common Philosophical Terms
pp 341-344

About the Editors
pp 345-346

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Proof and Other Dilemmas

Mathematics and Philosophy

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