Visualization, Explanation and Reasoning Styles in MathematicsPaolo Mancosu, Klaus Frovin Jørgensen, S.A. Pedersen 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. |
Contents
II | 1 |
III | 11 |
IV | 13 |
V | 17 |
VI | 21 |
VIII | 26 |
IX | 27 |
X | 28 |
XLIII | 109 |
XLIV | 113 |
XLV | 118 |
XLVI | 123 |
XLVII | 125 |
XLVIII | 126 |
XLIX | 131 |
L | 132 |
XI | 31 |
XII | 39 |
XIII | 44 |
XIV | 46 |
XV | 50 |
XVI | 51 |
XVII | 53 |
XVIII | 57 |
XIX | 59 |
XX | 60 |
XXI | 62 |
XXII | 64 |
XXIII | 66 |
XXIV | 67 |
XXV | 70 |
XXVI | 71 |
XXVII | 72 |
XXVIII | 75 |
XXIX | 76 |
XXX | 77 |
XXXI | 81 |
XXXII | 84 |
XXXIII | 86 |
XXXV | 87 |
XXXVI | 89 |
XXXVII | 91 |
XXXIX | 92 |
XL | 103 |
XLI | 105 |
XLII | 107 |
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Common terms and phrases
aesthetic algebraic geometry algorithm an+1 angles approach Archimedes argument Artin axioms Babylonian mathematics basic beauty belief Boston Studies century characterizing property Chemla circle cognitive commentary complex analysis computation concept for squares context converges definition Desargues Desargues theorem description set diagrams discussion epistemic epistemology Essays Euclid's Euclid's Elements example explanatory fact figure formulation framework Friedman function Giaquinto graphic statics Greek mathematics Høyrup Hui's intuition ISBN justification Kitcher Kummer's test Liu Hui Logic Mancosu mathe mathematical activity mathematical beauty mathematical explanation mathematical practice mathematical proof mathematical texts mathematicians matical n-gon narrative natural objects Old Babylonian patterns perceived perception perceptual concept perfectly square philosophy of mathematics Philosophy of Science physical polygons Pringsheim's proof problem projective geometry propositions rectangle reference relation representation Riemann role sense sequence side Steiner Steiner's theory structure style symmetry theorem tion triangle understanding unification visual
References to this book
The Architecture of Modern Mathematics:Essays in History and Philosophy ... J. Ferreiros,J. J. Gray No preview available - 2006 |