The Philosophy of Mathematical PracticePaolo Mancosu 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. |
Contents
| 1 | |
1 Visualizing in Mathematics | 22 |
2 Cognition of Structure | 43 |
3 DiagramBased Geometric Practice | 65 |
4 The Euclidean Diagram 1995 | 80 |
Why it Matters | 134 |
6 Beyond Unification | 151 |
7 Purity as an Ideal of Proof | 179 |
Fruitfulness and Naturalness | 276 |
11 Computers in Mathematical Inquiry | 302 |
12 Understanding Proofs | 317 |
13 What Structuralism Achieves | 354 |
Visual and Structural Geometry in Arithmetic | 370 |
15 The Boundary Between Mathematics and Physics | 407 |
Strategies of Assimilation | 417 |
| 441 | |
8 Reflections on the Purity of Method in Hilberts Grundlagen der Geometrie | 198 |
9 Mathematical Concepts and Definitions | 256 |
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Common terms and phrases
algebraic analysis analytic angles argument pattern arithmetic attribution axiomatic Bolzano calculation category specification category theory circle claim co-exact cohomology complex numbers concepts Congruence Axiom conjectures construction context Dedekind defined definition diagram diagram-based discursive text discussion distinction elementary elements equation Euclid Euclidean geometry example explanatory fact finite formal formal verification foundational function Gauss given Grundlagen Hallett and Majer Hilbert induction inferences integers intersection intuition investigation isomorphism Kitcher lectures Legendre symbol logical Mancosu mathematical explanation mathematical practice mathematicians morphisms natural numbers notion number theory objects Oxford philosophical philosophy of mathematics physicists physics polynomials prime number problem Proclus proof properties proposition prove purity of method quadratic reciprocity question real closed fields real numbers reasoning reductio representation result ring ring morphism role schemes segment semi-algebraic set sense set theory solution space straight line systematization topology traditional geometrical triangle unification University Press visual thinking