Visualization, Explanation and Reasoning Styles in MathematicsP. Mancosu, Klaus Frovin Jørgensen, S.A. Pedersen Springer Science & Business Media, 30. mars 2006 - 300 sider 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. |
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Side v
... Diagrams and Images in the Late Nineteenth Century 13 2. The Return of the Visual as a Change in Mathematical Style ... 17 3. New Directions of Research and Foundations of Mathematics Acknowledgements 21 ... 26 Notes References 27 28 M ...
... Diagrams and Images in the Late Nineteenth Century 13 2. The Return of the Visual as a Change in Mathematical Style ... 17 3. New Directions of Research and Foundations of Mathematics Acknowledgements 21 ... 26 Notes References 27 28 M ...
Side 1
... diagrams etc. have become the subject of intense interest. These historians and philosophers agree that there is more to understanding mathematics than a study of its logical struc- ture and put much emphasis on mathematical activity as ...
... diagrams etc. have become the subject of intense interest. These historians and philosophers agree that there is more to understanding mathematics than a study of its logical struc- ture and put much emphasis on mathematical activity as ...
Side 2
... diagrams and visual imagery can provide us with mathe- matical knowledge. Most of the contributions in the book touch upon this question but the first part of the book is fully devoted to it. In “Visualization in Logic and Mathematics ...
... diagrams and visual imagery can provide us with mathe- matical knowledge. Most of the contributions in the book touch upon this question but the first part of the book is fully devoted to it. In “Visualization in Logic and Mathematics ...
Side 3
... diagrams . Using a simple example of a picture - proof , Brown claims that a diagram can function as a " telescope " that allows us to “see” into the Platonic realm. The INTRODUCTION 3.
... diagrams . Using a simple example of a picture - proof , Brown claims that a diagram can function as a " telescope " that allows us to “see” into the Platonic realm. The INTRODUCTION 3.
Side 4
... diagram displays only a specific case but through the diagram we are able to intuit a general truth and this intuition cannot be confused with the sensory information given by the diagram. Brown's position on intuition is then ...
... diagram displays only a specific case but through the diagram we are able to intuit a general truth and this intuition cannot be confused with the sensory information given by the diagram. Brown's position on intuition is then ...
Innhold
1 | |
13 | |
18 | |
GIAQUINTO From Symmetry Perception to Basic Geometry | 31 |
J R BROWNNaturalism Pictures and Platonic Intuitions 57 | 56 |
GIAQUINTO Mathematical Activity | 75 |
On Reasoning Styles in Early Mathema | 91 |
K CHEMLA The Interplay Between Proof and Algorithm in 3rd Century | 122 |
Understanding Unification and Explanation Friedman | 158 |
Patterns of Argument | 168 |
J HAFNER AND P MANCOSU The Varieties of Mathematical Explana | 215 |
Notes | 246 |
Sources of Beauty in Mathematics | 254 |
Notes | 289 |
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Visualization, Explanation and Reasoning Styles in Mathematics P. Mancosu,Klaus Frovin Jørgensen,S.A. Pedersen Ingen forhåndsvisning tilgjengelig - 2010 |
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aesthetic algebraic algebraic geometry algorithm analysis angles argument axes axioms axis Babylonian mathematics Barwise and Etchemendy basic belief belief-forming disposition Cambridge characterizing property Chemla circle claim cognitive commentary complex complex analysis concept for squares context definition description set diagrammatic reasoning diagrams diamond epistemic epistemology example experience explanatory fact figure formulation framework function geometrical concept geometry Giaquinto Greek mathematics Hilbert Høyrup intuition involved ISBN justification Kitcher knowledge Kummer's test Liu Hui Logic Mancosu mathe mathematical activity mathematical beauty mathematical explanation mathematical practice mathematical proof mathematical texts mathematicians matical narrative natural objects Old Babylonian parallel perceived perception perceptual concept perfectly square philosophy of mathematics Philosophy of Science problem propositions rectangle reference system reflection symmetry representation Riemann role sense sequence set for squares shape side Springer Steiner structure theorem theory tion triangles understanding unification University Press visual system width