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 2
... discussion of visualization and diagrammatic reasoning and their relevance for recent discussions in the philosophy of mathematics. Mancosu begins by outlining how visual intuition and diagrammatic reasoning were discred- ited in late ...
... discussion of visualization and diagrammatic reasoning and their relevance for recent discussions in the philosophy of mathematics. Mancosu begins by outlining how visual intuition and diagrammatic reasoning were discred- ited in late ...
Side 4
... discussion of discovery through visual imag- ing nicely ties up with the previous material on visualization and again Gi- aquinto points out that although we might reach knowledge by such means this need not be a proof. Explanation is ...
... discussion of discovery through visual imag- ing nicely ties up with the previous material on visualization and again Gi- aquinto points out that although we might reach knowledge by such means this need not be a proof. Explanation is ...
Side 5
... discuss why or under which conditions the operations performed are le- gitimate and lead to correct results”. The ... discussion and for why the procedure works. Thus, Old Babylonian mathematics displays its own char- acteristic style ...
... discuss why or under which conditions the operations performed are le- gitimate and lead to correct results”. The ... discussion and for why the procedure works. Thus, Old Babylonian mathematics displays its own char- acteristic style ...
Side 6
... discussion of 'fruitful' con- cepts. Fruitful concepts have unifying and explanatory roles but it is often difficult to say what makes a concept fruitful in mathematics. Tappenden mentions the unification of the theory of algebraic ...
... discussion of 'fruitful' con- cepts. Fruitful concepts have unifying and explanatory roles but it is often difficult to say what makes a concept fruitful in mathematics. Tappenden mentions the unification of the theory of algebraic ...
Side 8
... discussion of this difficult topic. Netz argues from the outset that every type of human expres- sion possesses an aesthetic dimension. Moreover, the aesthetic dimension is an objective fact, although a difficult one to analyze. Whereas ...
... discussion of this difficult topic. Netz argues from the outset that every type of human expres- sion possesses an aesthetic dimension. Moreover, the aesthetic dimension is an objective fact, although a difficult one to analyze. Whereas ...
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 |
Vanlige uttrykk og setninger
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