This fascinating collection of essays is a must-have for those who are interested in the history and philosophy of mathematics but are tired of the usual "foundational" discussion of formalism versus platonism. As the editors point out in their introduction, there are other ways to think about what mathematics is. In this case, the focus is on understanding what mathematicians actually do (or have done).
Part I of the book focuses on the role of visualization in mathematics, and especially on how visualization interacts with mathematical reasoning. Anyone who is familiar with "Proofs without Words" knows that there is something here to study. The essays in this section are not light reading, but they raise and explore interesting issues. Mancosu's essay, for example, discusses the "return of the visual" in recent mathematics, contrasting 19th century attitudes ("see, no pictures!") with recent work (for example, on fractals).
The essays in part II are about mathematical explanation and styles of reasoning. They include an essay by Reviel Netz on the aesthetics of mathematics, which tries to think through what mathematicians mean when they say an argument is beautiful, two essays on styles of reasoning in other mathematical cultures (Høyrup on ancient Mesopotamia, Chemla on China), and two more straightforwardly philosophical essays.
All in all, this is a book that libraries will want to have, particularly if they strive to have good collections on the history and philosophy of mathematics.
Fernando Q. Gouvêa is Professor of Mathematics and Colby College and the co-author of Math through the Ages: A Gentle History of Mathematics for Teachers and Others.
P. Mancosu, K.P. Jørgensen and S.A. Pedersen: Introduction.
P. Mancosu: Visualization in Logic and Mathematics. 1. Diagrams and Images in the Late Nineteenth Century. 2. The Return of the Visual as a Change in Mathematical Style. 3. New Directions of Research and Foundations of Mathematics. Acknowledgements. Notes. References.
M. Giaquinto: From Symmetry Perception to Basic Geometry. Introduction. 1. Perceiving a Figure as a Square. 2. A Geometrical Concept for Squares. 3. Getting the Belief. 4. Is It Knowledge? 5. Summary. Notes. References.
J.R. Brown: Naturalism, Pictures, and Platonic Intuitions. 1. Naturalism. 2. Platonism. 3. Godel's Platonism. 4. The Concept of Observable. 5. Proofs and Intuitions. 6. Maddy's Naturalism. 7. Refuting the Continuum Hypothesis. Acknowledgements. Appendix: Freiling's "Philosophical" Refutation of CH. References.
M. Giaquinto: Mathematical Activity. 1. Discovery. 2. Explanation. 3. Justification. 4. Refining and Extending the List of Activities. 5. Conc1uding Remarks. Notes. References.
J. Høyrup: Tertium Non Datur: On Reasoning Styles in Early Mathematics. 1. Two Convenient Scapegoats. 2. Old Babylonian Geometric Proto-algebra. 3. Euc1idean Geometry. 4. Stations on the Road. 5. Other Greeks. 6. Proportionality - Reasoning and its Elimination. Notes. References.
K. Chemla: The Interplay Between Proof and AIgorithm in 3rd Century China: The Operation as Prescription of Computation and the Operation as Argument. 1. Elements of Context. 2. Sketch of the Proof. 3. First Remarks on the Proof. 4. The Operation as Relation of Transformation. 5. The Essential Link Between Proof and AIgorithm. 6. Conc1usion. Appendix. Notes. References.
J. Tappenden: Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom Choice. 1. Introduction - a "New Riddle" of Deduction. 2. Understanding and Explanation in Mathematical Methodology: The Target. 3. Understanding, Unification and Explanation - Friedman. 4. Kitcher: Pattems of Argument. 5. Artin and Axiom Choice: "Visual Reasoning" Without Vision. 6. Summary - the "new Riddle of Deduction". Notes. References.
J. Hafner and P. Mancosu: The Varieties of Mathematical Explanations. 1. Back to the Facts Themselves. 2. Mathematical Explanation or Explanation in Mathematics? 3. The Search for Explanation within Mathematics. 4. Some Methodological Comments on the General Project. 5. Mark Steiner on Mathematical Explanation. 6. Kummer's Convergence Test. 7. A Test Case for Steiner's Theory. Appendix. Notes. References.
R. Netz: The Aesthetics of Mathematics: A Study. 1. The Problem Motivated. 2. Sources of Beauty in Mathematics. 3. Conclusion. Notes. References.