Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom Choice

Synthese Library

Volumn 327, Issue , 2005, Pages 147-214

  Tappenden, Jamie ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Genus Lizati; Mathematical Practice; Nineteenth Centur; Proof Style

Indexed keywords

EID: 85144713598     PISSN: 01666991     EISSN: 25428292     Source Type: Book Series    
DOI: 10.1007/1-4020-3335-4_8     Document Type: Chapter

References (140)

1. Abhyankar, S. (1964). Local Analytic Geometry, Academic Press, New York.
2. Allwein, G. and Barwise, J. (1996). Logical Reasoning with Diagrams, Oxford University Press, Oxford.
3. Apostol, T. (1976). Introduction to Analytic Number Theory,Springer, Berlin.
4. Artin, E. (1957). Geometric Algebra, Wiley Interscience, New York.
5. Assis, A. (1994). Weber’s Electrodynamics, Kluwer, Dordrecht.
6. Atiyah, M. (1984). Interview with Michael Atiyah, Mathematical Intelligencer 6(1). Page references to the reprinting in (Atiyah, 1988).
7. Atiyah, M. (1988). Collected Worksvol.1,Oxford University Press, Oxford.
8. Babbage, C. (1827). On the Influence of Signs in Mathematical Reasoning, Transactions of theCambridge Philosophical Society II: 325 – 377.
9. Baker, H. (1929). Principles of Geometry vol. I Foundations, Cambridge University Press, Cambridge.
10. Batten, L. (1997). Combinatorics of Finite Geometries, 2nd edn, Cambridge University Press, Cambridge.
11. Batterman, R. (2000). A ’Modern’ (Victorian) Attitude Towards Scientific Understanding, The Monist 83(2): 228 – 257.
12. Batterman, R. (2002). The Devil is in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence, Oxford University Press, Oxford.
13. Benvenuto, E. (1991). An Introduction to the History of Structural Mechanics Part II: Vaulted Systems and Elastic Structures, Springer-Verlag, Berlin.
14. Block, N. (1981). Imagery, MIT Press, Cambridge Mass.
15. Blumenthal, L. M. (1961). A Modern View of Geometry, W. H. Freeman and Co., San Francisco.
16. Booth, J. (1873). A Treatise on Some NewGeometrical Methods, Vol. I and II, Longman Green, London.
17. Bottazini, U. (1986). The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer, Berlin. W. van Egmund (trans.).
18. Bottazini, U. (1994). Three Traditions in Complex Analysis: Cauchy, Riemann and Weierstrass, in I. Grattan-Guinness (ed.), Companion Encyclopedia ofthe History and Philosophy of the Mathematical Sciences,Vol.I,Routledge, London, pp. 419– 431.
19. Bower, G. (1972). Mental Imagery and Associative Learning, in L. Gregg (ed.), Cognition in Learning and Memory, Wiley, New York.
20. Bridgeman, P. (1938). The Logicof Modern Physics, McMillan, New York.
21. Butts, R. (1973). Whewell’s Logic of Induction, in R. Giere and R. Westfall (eds), Foundation of Scientific Method: Nineteenth Century, Indiana University Press, Bloomington, pp. 53–85.
22. Campbell, N. (1919/1957). Foundations of Science, Dover, New York. 1957reprint of 1919 original.
23. Carbone, A. and Semmes, S. (2000). A Graphic Apology for Symmetry and Explicitness, Oxford University Press, Oxford.
24. Carruthers, M. (1990). The Book of Memory: A Study of Memory in Medieval Culture, CambridgeUniversity Press, Cambridge.
25. Carruthers, M. (1998). The CraftofThought, Cambridge University Press, Cambridge.
26. Cartier, P. (2001). A Mad Day’s Work: From Grothendieck to Connes and Kontsevich, the Evolution of the Concepts of Space and Symmetry, Bulletin of the American Mathematical Society 38(4): 389 – 408.
27. Cartwright, N. (1983). How the Laws of Physics Lie, Clarendon Press, Oxford.
28. Casati, R.and Varzi, A. (1994). Holes and other Superficialities, MIT Press, Cambridge Mass.
29. Casorati, F. (1868). Teorica delle funzioni di variabili complexi, Pavia.
30. Charlton, T. (1982). A History of theTheory of Structures in the Nineteenth Century, Cambridge University Press, Cambridge.
31. Chasles, M. (1837). Aperçuc historique sur l’origine et le developpement´ des methodes´ en géometrie,´ particulierement` de celles qui se rapportent alag` eometrie´ moderne, suivi d’un memoire´ de géometrie´ sur deux principes génereaux´ de la science, la dualite´ et l’homographie,M.Hayez, Bruxelles.
32. Chevalley, C. (1951). Introduction to the Theory of Algebraic Functions of One Variable, American Mathematical Society, Reinhart and Wilson, New York.
33. Clarke, G. (1888). The Principles of Graphic Statics, 2nd edn, E. & F. N. Spon, London.
34. Clemens, C. (1980). A Scrapbook of Complex Curve Theory, Plenum Press, New York.
35. Coolidge, J. (1959). A Treatise on Algebraic Plane Curves, Dover, New York.
36. Courant, R. and Robbins, H. (1941). What is Mathematics?, Oxford University Press, Oxford.
37. Cox, D., Little, J. and O’Shea, D. (1992). Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry, Springer, Berlin.
38. Coxeter, H. (1992). The Real Projective Plane, 3rd edn, Springer, New York.
39. Crapo, H. and Whiteley, W. (1982). Statics of Frameworks and Motions of Panel Structures: A Projective Geometric Introduction, Structural Topology 6: 43 – 82.
40. Cremona, L. (1872). Le figure reciprochenella statica grafica,Hoepli,Milan.
41. Cremona, L. (1874). Elementi di calcolo grafico, Paravia, Torino.
42. Cremona, L. (1890). Graphical statics. Two treatises on the graphical calculus and reciprocal figures in graphical statics, Oxford University Press, Oxford. English translation of (Cremona, 1872) and (Cremona, 1874) by Thomas Beare, with anew preface by Cremona.
43. Crotti, F. (1888). La teoria dell’elasticita` ne’ suoi principi fondamentali e nelle sue applicazioni pratichealle costruzioni,Hoepli,Milan.
44. Culmann, K. (1865). Die Graphische Statik, Meyer and Zeller, Zürich.¨
45. Culmann, K. (1875). Die Graphische Statik, 2nd edn, Meyer and Zeller, Zürich.¨
46. Darrigol, O. (2000). Electrodynamics from Ampere` to Einstein, Oxford University Press, Oxford.
47. Dedekind, R.and Weber, H. (1882). Theorie der algebraischen Funktionen einer Veranderlichen,¨ Journal furf¨ Reine und Angewante Mathematik 92: 181–290. Dated 1880, page reference to the reprinting in Dedekind’s Werke vol. I 238-249.
48. Demopoulos, W. (ed.) (1995). Frege’s Philosophyof Mathematics, Harvard University Press, Cambridge.
49. Dieudonne,´ J. (1969). Linear Algebra and Geometry, Hermann, Paris.
50. Dieudonne,´ J. (1985). History of Algebraic Geometry, Wadsworth Books, Monterey. J.Sally, (trans.).
51. Dubois, A. (1877). The Elements of Graphical Statics and their Application to Framed Structures, Wiley and Sons, New York.
52. Dugac, P. (1976). Richard Dedekind et les Fondements des Mathematique´,Vrin, Paris.
53. Earman, J. (1992). Bayes or Bust: A Critical Examination of Bayesian Confirmation Theory, MIT press, Cambridge.
54. Edwards, H. (1996). Linear Algebra,Birkhäuser,¨ Boston.
55. Eisenbud, J.and Harris, J. (1992). Schemes: The Language of Modern Algebraic Geometry, Wadsworth and Cole, Pacific Grove, Ca.
56. Eisenbud, J. and Harris, J. (2000). The Geometry of Schemes, Springer, Berlin.
57. Ferguson, E. (1992). Engineering and the Mind’s Eye, MIT Press, Cambridge Mass.
58. Field, J. V. (1987). Linear perspective and theprojective geometry of Girard Desargues, Nuncius Ann. Storia Sci. 2(2): 3–40.
59. Field, J. V. (1988). Perspective and the Mathematicians: Alberti to Desargues, in C. Hay (ed.), Mathematics from Manuscript to Print, 1300–1600, Oxford Sci. Publ., Oxford Univ. Press, New York, pp. 236–263.
60. Field, J. V. (1997). The Invention of Infinity, Oxford University Press, Oxford.
61. Field, J. V. and Gray, J. J. (1987). The Geometrical Work of Girard Desargues, Springer-Verlag, New York.
62. Frege, G. (1903). Grundgesetze der Arithmetik, Vol. II, H. Pohle, Jena.
63. Frege, G. (1980). Philosophical and Mathematical Correspondence, University of Chicago Press, Chicago. Gabriel, G. Hermes, H. Kambartel, F. Thiel, C and Veraart, A. Abridged from the German edition by McGuinness, B. Translated by Kaal, H.
64. Friedman, M. (1974). Explanation and Scientific Understanding, Journal of Philosophy 73:250 –261.
65. Friedman, M. (1979). Truth and Confirmation, Journal of Philosophy 76:361–382.
66. Friedman, M. (2000). Geometry, Construction and Intuition in Kant and his Successors, in G. Sher and R. Tieszen (eds), Between Logic and Intuition, Cambridge University Press, Cambridge, pp. 186–218.
67. Gaukroger, S. (1995). Descartes: A Critical Biography, Oxford University Press, Oxford.
68. Geyer, W. (1981). Die Theorie der Algebraischen Funktionen einer Veränderlichen¨ nach Dedekind und Weber, in W. Scharlau (ed.), Richard Dedekind, 1831-1981: eine WurdigungW¨ zu seinem 150 Geburtstag, F. Vieweg, Braunschweig, Wiesbaden, pp. 109–133.
69. Glasgow, J., Narayanan, H. and Chandrasekaran, B. (eds) (1995). Diagrammatic Reasoning, MIT Press, Cambridge Mass.
70. Goodman, N. (1955). Fact, Fiction and Forecast, Bobbs-Merrill, Indianapolis. 1965 Reprint.
71. Graham, R. (1887). Graphic and Analytic Statics in their Practical Application to the Treatment of Stresses in Roofs, Solid Girders, Lattice, Bowstring and Suspension Bridges, Braced Iron Arches and Piers, and other Frameworks, 2nd edn, Crosby Lockwood and Co., London.
72. Gray, J. (1998). The Riemann-Roch Theorem and Geometry 1854-1914, International Congress of Mathematicians Proceedings, Vol. III, pp. 811–822.
73. Grunbaum,¨ B. (1976). Lectures in lost mathematics. Unpublished mimeographed notes.
74. Hall, M. (1943). Projective planes, Transactions of the American Mathematical Society 54: 229 – 277.
75. Harris, J. (1992). Developments in Algebraic Geometry, Proceedings of the AMS Centennial Symposium, American Mathematical Society Publications, Providence.
76. Hartshorne, R. (1967). Foundations of Projective Geometry,Benjamin Press, Cambridge. Lecture Notes Harvard University.
77. Hartshorne, R. (2000). Geometry: Euclid and Beyond,Springer Verlag,Berlin.
78. Hennenberg, E. (1903). Die graphische Statik der starren Körper, Encyklopadie¨ der Mathematischen Wissenschaften, Vol. IV(1), pp. 345–435.
79. Hesse, M. (1961). Forces and Fields, Thomas Nelson and Sons, London.
80. Hilbert, D. (1899). Die Grundlagen der Geometrie, 1st ed. in Festschrift zur Feier der Enthullung¨ der Gauss-Weber-Denkmals in Gottingen ¨,Teubner, Leipzig.
81. Hughes, D. R. and Piper, F. C. (1973). Projective Planes, Vol.6 of Graduate Texts in Mathematics, Springer-Verlag, New York.
82. Humphries, P. (1993). Greater Unification equals Greater Understanding?, Analysis 53.3: 183 – 188.
83. Jackson, J. and Wirtz, H. (1983). Schaum’s Outline of Theory and Problems of Elementary Statics and Strength of Materials, McGraw-Hill, New York.
84. Jacobson, N. (1974). Basic Algebra, Freeman, San Francisco.
85. Jans, J. (1957). Review of (Artin, 1957), American Mathematical Monthly pp. 604 –605.
86. Kadison, L. and Kromann, M. (1996). Projective Geometry and Modern Algebra, Birkhäuser,¨ Boston.
87. Kim, J. (1994). Explanatory Knowledgeand Metaphysical Dependence, Philosophical Issues 5: 51 – 69. Truth and Rationality.
88. Kitcher, P. (1976). Explanation, Conjunction and Unification, Journal of Philosophy 73: 207–212.
89. Kitcher, P. (1982). Explanatory unification, Philosophy of Science 48: 507–531.
90. Kitcher, P. (1989). Explanatory Unification and the Causal Structure of the World, in P. Kitcher and W. Salmon (eds), Scientific Explanation, University of Minnesota
91. Press, Minneapolis. Klein, F. (1926/79). Development of mathematics in the 19th century, Vol. IXof Lie Groups: History, Frontiers and Applications, Math Sci Press, Brookline, Mass. With a preface and appendices by Robert Hermann, translated from the German by M. Ackerman.
92. Kuhn, T. (1977). Objectivity, Value Judgement and Theory Choice, in The Essential Tension, University of Chicago Press, Chicago.
93. Laugwitz, D. (1992). ‘Das letzte Ziel ist immer die Darstellung einer Funktion’: Grundlagen der Analysis bei Weierstraß 1886 historische Wurzeln und Parallelen, Historia Mathematica 19:341 – 355.
94. Laugwitz, D. (1999). Bernhard Riemann: 1826-1866 Turning Points in the Conception of Mathematics,Birkhauser,¨ Boston. A. Schnitzer (trans.).
95. Mancosu, P. (1999). Bolzano and Cournot on Mathematical Explanation, Revue d’Histoire des Sciences 52: 429 – 455.
96. Maxwell, J. C. (1864). On the Calculation of the Equilibrium and Stiffness of Frames, Philosophical Magazine 27: 294 – 299.
97. Morrison, M. (1997). Whewell and the Ultimate Problem of Philosophy, Studies in the History and Philosophyof Science 28: 315 – 324.
98. Morrison, M. (2000). Unifying Scientific Theories, Cambridge University Press, Cambridge.
99. Mumford, D. (1988). TheBigRed Book of Varieties and Schemes, Springer-Verlag, Berlin. Springer Lecture Notes in Mathematics: 1358.
100. Neuenschwander, E. (1980). Riemann und das ‘Weierstrass’schen Prinzip’ der analytischen Fortsetzung durch Potenzreihen, Jahresbericht der Deutschen Mathematiker-Vereinigung 82:1– 11.
101. Neuenschwander, E. (1981). Studies intheHistory of Complex Function Theory II: Interactions among the French School, Riemann, and Weierstrass, Bulletin of the American Mathematical Society 5: 87–105. New series.
102. Poincare,´ H. (1898). L’Oeuvre Mathmmatique de Weierstrass, Acta Mathematica 22: 1–18.
103. Poincare,´ H. (1903). Review of Hilbert’s Foundations of Geometry, Bulletin of the American Mathematical Society pp. 1 – 23. E. Huntington, trans., original, published 1902.
104. Pratt, I. and Lemon, O. (1997). Spatial Logic and the Complexity of Diagrammatic Reasoning, Machine Graphics and Vision 6(1): 89 – 108.
105. Railton, P. (1993). Essentially General Predicates, in P. French, T.Uehling and H. Wettstein (eds), Philosophy of Science, Vol. XVII of Midwest Studies in Philosophy, University of Minnesota Press, Minneapolis, pp. 166–176.
106. Rankine, T. (1869). On the Practical Application of Reciprocal Figures to the Calculation of Strains on Frameworks, Vol. XXV of Transactions of the Royal Society of Edinburgh.
107. Reisenfeld, R. (1981). Homogeneous Coordinates and Projective Planes in Computer Graphics, IEEE Computer Graphics and Applications 1(1): 50 – 55.
108. Remmert, R. (1991). Theory of Complex Functions, Springer Verlag, Berlin. R. Burkel (trans.).
109. Remmert, R. (1998). Classical Topics in Complex Function Theory, Springer Verlag, Berlin. L. Kay (trans.).
110. Rossi, P. (2000). Logic and the Art of Memory: the Quest for a Universal Language, University of Chicago Press, Chicago. S. Clucas (trans.).
111. Rota, G.-C. (1997). Indiscrete Thoughts,Birkhäuser,¨ Boston.
112. Salmon, W. (1989). Forty Years of Scientific Explanation, in P. Kitcher and W.Salmon (eds), Scientific Explanation,University of Minnesota Press, Minneapolis.
113. Scholz, E. (1989). Symmetrie Gruppe Dualitätt¨, Birkhauser, Basel.
114. Scholz, E. (1994). Graphical Statics, in I. Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Vol. II, Routledge, London, pp. 987–93.
115. Segre, C. (1904). On Some Tendencies in Geometrical Investigations, Bulletin of the American Mathematical Society pp. 412 – 468. J. Young (Trans.), Italian original published 1891 in Revista di Matematica.
116. Smith, K. E., Kahanpä¨a,¨ L., Kek¨al ainen,¨ P. and Traves, W. (2000). An invitation to algebraic geometry, Universitext, Springer-Verlag,NewYork.
117. Smith, K., Kahanpä¨a,¨ L., Kek¨al ainen,¨ P. and Traves, W. (2001). An Invitation to Algebraic Geometry, Springer, Berlin.
118. Stein, H. (1988). Logos, Logic and Logistiké:´ Some Philosophical Remarks on the Nineteenth-Century Transformation of Mathematics, in W. Aspray and P. Kitcher (eds), History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, pp. 238–259.
119. Steiner, M. (1978). Mathematical Explanation, Philosophical Studies 34: 135 –151.
120. Stillwell, J. (1993). Classical Topology and Combinatorial Group Theory, 2nd edn, Springer Verlag, Berlin.
121. Stolfi, J.(1991). Oriented Projective Geometry: A Framework for Geometric Computations, Academic Press, Boston.
122. Tappenden, J. (1995). Geometry and Generality in Frege’s Philosophy of Arithmetic, Synthése 102: 319 –361.
123. Tappenden, J. (2001a). “Geometrical Invariance and Mathematical Practice in Frege’s Foundations of Geometry”. In preparation.
124. Tappenden, J. (2001b). “Proof Style and Understanding in Mathematics II: Notation”. In preparation.
125. Tappenden, J. (2001c). A Reassessment of the Mathematical Roots of Frege’s Logicism I: The Riemannian Context of Frege’s Foundations. (Working title) manuscript in progress.
126. Timoshenko, S. (1953). History of the Strength of Materials, Dover Publications, New York. 1983 reprint.
127. Toulmin, S. (1961). Foresight and Understanding, Harper and Row, New York.
128. Truesdell, C. (1984). An Idiot’s Fugitive Essays on Science, Springer-Verlag, New York.
129. Tufte, E. (1983). The Visual Display of Quantitative Information, Graphics Press, Cheshire, Conn.
130. Tufte, E. (1997). Visual Explanations, Graphics Press, Cheshire, Conn.
131. van der Waerden, B. (1991). Algebra, Vol. II, Springer, New York. Translated by J. Schulenberger.
132. Weyl, H. (1913). Die Idee Der Riemannschen Flache¨, Teubner, Leipzig.
133. Weyl, H. (1932). Topologie und Abstrakte Algebra als zwei Wege mathematischen Verstandnisses,¨ Unterrichtsblatter¨ f¨ur Mathematik und Maturwissenschaften 38: 177 – 188. Gesammelte Abhandlungen vol. 3 pp. 348-358.
134. Weyl, H. (1955). The Concept of A Riemann Surface, Addison-Wesley, Reading, Mass. English translation of 3rd revised edition of (Weyl, 1913), G. Maclane, trans.
135. Weyl, H. (1995). Topology and Abstract Algebra as Two Roads of Mathematical Comprehension, American Mathematical Monthly pp. 453 – 460 and 646 – 651. Abe Shenitzer, trans., A translation of (Weyl, 1932).
136. Whiteley, W. (1985). The Projective Geometry of Rigid Frameworks, in C. Baker and L. Batten (eds), Finite Geometries, Marcel Dekker, New York.
137. Wilson, M. (1992). Frege: The Royal Road From Geometry, Noûs 26: 149 – 180. Page references to the reprinting in (Demopoulos, 1995).
138. Yates, F. (1966). The Art of Memory, University of Chicago Press, Chicago.
139. Zariski, O. (1978). Collected Papers,Vol. III, MIT Press, Cambridge, Mass. Edited by M. Artin and B. Mazur.
140. Ziwet, A. (1904). Theoretical Mechanics, Norwood Press, Norwood, Mass.