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Searching... | 30000010124530 | QA8.6 V57 2005 | Open Access Book | Book | Searching... |
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Summary
Summary
In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert's program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.
Table of Contents
Contributing Authors |
IntroductionP. Mancosu and K.P. Jorgensen and S.A. Pedersen |
Part I Mathematical Reasoning And Visualization |
Visualization in Logic and MathematicsP. Mancosu |
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 |
From Symmetry Perception to Basic GeometryM. Giaquinto |
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 |
Naturalism, Pictures, and Platonic IntuitionsJ.R. Brown |
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 |
Mathematical ActivityM. Giaquinto |
1 Discovery |
2 Explanation |
3 Justification |
4 Refining and Extending the List of Activities |
5 Concluding Remarks |
Notes |
References |
Part II Mathematical Explanation and Proof Styles |
Tertium Non Datur: On Reasoning Styles in Early MathematicsJ. Hoyrup |
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 |
The Interplay Between Proof and AIgorithm in 3rd Century ChinaK. Chemla |
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 |
Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom ChoiceJ. Tappenden |
1 Introduction - a "New Riddle" of Deduction |
2 Understanding and Explanation in Mathematical Methodology: The Target |
3 Understanding, Unification and Explanation - Friedman |
4 Pattems of ArgumentKitcher |
5 Artin and Axiom Choice: "Visual Reasoning" Without Vision |
6 Summary - the "new Riddle of Deduction". Notes |
References |
The Varieties of Mathematical ExplanationsJ. Hafner and P. Mancosu |
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 |
The Aesthetics of Mathematics: A StudyR. Netz |
1 The Problem Motivated |
2 Sources of Beauty in Mathematics |
3 Conclusion |
Notes |
References |
Index |