This book is pretty useless. A chapter conclusion summarises well the entire book: the philosophy of mathematical practice "offers a vast and virgin territory to exploration" and "Most of the work remains to be done" (p. 148). Of course no potential reader would disagree with these trivialities. It is a pity, however, that the entire book is little but a constant reiteration of this call to arms, while the work that everyone keeps pointing to remains virtually untouched. I offer a partial diagnosis as to the reasons for book's failure: the formalistic point of view has not been rejected with sufficient zeal and conviction. The authors in this volume agree that formalism is not everything but their alternatives are meant to be modest and polite complements rather than serious alternatives. Look at the history of philosophy of mathematics: Plato, Pascal, Kepler, Descartes, Leibniz, Kant, Poincaré, Brouwer, Weyl, etc. All of these people would have said that logic and symbols and whatever formal systems you can come up with just fundamentally miss the point: what drives mathematics is the human cognitive capacities, innate intuitions, etc. This point of view predicted the shortcomings of the formalistic fad to the letter. Yet the authors in this volume refuse to draw the obvious conclusions. The cognitive perspective is condemned for no good reason for example by Avigad in his chapter on understanding: "We have all had such 'Aha!' moments and the deep sense of satisfaction that comes with them. But surely the philosophy of mathematics is not supposed to explain this sense of satisfaction, any more than economics is supposed to explain the feeling of elation that comes when we find a $20 bill lying on the sidewalk." (p. 322). All the above thinkers would have agreed that "Aha" experiences are to be explained in terms of the nature of the human cognitive endowment. To say that "surely" this would be stupid, on the basis of nothing more than a pathetic parallel to economics, is to dismiss the major historical alternative to formalism all too easily. One further example: Tappenden argues in chapter 9 "that mathematical defining is a more intricate activity, with deeper connections to explanation, fruitfulness of research, etc. than is sometimes realised" (p. 272). Like so many other "theses" presented in this book, this is a complete triviality from the cognitive point of view. In the mind there are ideas. When these are projected onto a formal presentation it happens that some ideas map to definitions, other to theorems, others to proofs, etc. Only someone who thought that this projection onto the formal representation was not a projection but a mere equation would think that properties of the images of ideas was an accurate description of the ideas themselves, and so commit the error that Tappenden says occurs "sometimes."

lgrsnf/D:\!genesis\library.nu\ad\_192343.adf02ff37415e571aa05c21193c8eed2.pdf

lgli/D:\!genesis\library.nu\ad\_192343.adf02ff37415e571aa05c21193c8eed2.pdf

Oxford Institute for Energy Studies

German Historical Institute London

Ebsco Publishing

OUP Oxford

United Kingdom and Ireland, United Kingdom

Oxford, New York, England, 2008

Reprint, 2008

Reprint, 2012

1, 2008

2011

до 2011-01

lg507754

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Includes bibliographical references (p. 438-440) and index.

<br>
Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment--such as visualization, explanation, and understanding--can nonetheless be subjected to philosophical analysis.
<p><em>The Philosophy of Mathematical Practice</em> comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of a short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representational systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.</p>

Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.

Contemporary philosophy of mathematics offers us an embarrassment of riches. But anyone familiar with this area will be aware of the need for new approaches that will pay closer attention to mathematical practice. This book provides a unified presentation of this new wave of work in philosophy of mathematics. This new approach is innovative in at least two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics that escape purely formal logical treatment such as visualization, explanation, and understanding can be nonetheless be subjected to philosophical analysis.
The book comprises an introduction and eight sections. Each section consists of a short introduction outlining the general topic followed by a related research article. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: visualization, diagrammatic reasoning and representational systems, mathematical explanation, purity of methods, mathematical concepts, philosophical relevance of category theory, philosophical aspects of computer science in mathematics, philosophical impact of recent developments in mathematical physics.

Contents......Page 8
Biographies......Page 9
Introduction......Page 14
1. Visualizing in Mathematics......Page 35
2. Cognition of Structure......Page 56
3. Diagram-Based Geometric Practice......Page 78
4. The Euclidean Diagram (1995)......Page 93
5. Mathematical Explanation: Why it Matters......Page 147
6. Beyond Unification......Page 164
7. Purity as an Ideal of Proof......Page 192
8. Reflections on the Purity of Method in Hilbert’s Grundlagen der Geometrie......Page 211
9. Mathematical Concepts and Definitions......Page 269
10. Mathematical Concepts: Fruitfulness and Naturalness......Page 289
11. Computers in Mathematical Inquiry......Page 315
12. Understanding Proofs......Page 330
13. What Structuralism Achieves......Page 367
14. ‘There is No Ontology Here’: Visual and Structural Geometry in Arithmetic......Page 383
15. The Boundary Between Mathematics and Physics......Page 420
16. Mathematics and Physics: Strategies of Assimilation......Page 430
C......Page 454
F......Page 455
J......Page 456
M......Page 457
R......Page 458
W......Page 459
Z......Page 460

This Title Offers Philosophical Analyses Of Important Characteristics Of Contemporary Mathematics And Of Many Aspects Of Mathematical Activity Which Escape Purely Formal Logical Treatment. Visualizing In Mathematics -- Cognition Of Structure -- Diagram-based Geometric Practice -- The Eulidean Diagram (1995) -- Mathematical Explanation: Why It Matters -- Beyond Unification -- Purity As An Ideal Proof -- Reflections On The Purity Of Method In Hilbert's Grundlagen Der Geometrie -- Mathematical Comcepts And Definitions -- Mathematical Concept: Fruitfulness And Naturalness -- Computers In Mathematical Inquiries -- Understanding Proofs -- What Structuralism Achieves -- 'there Is No Ontology Here': Visual And Structural Geometry In Arithmetic -- The Boundary Between Mathematics And Physics -- Mathematics And Physics: Strategies Of Assimilation. Paolo Mancosu. Includes Bibliographical References (p. 438-440) And Index.

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment. - ;Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments

2011-06-04