( ) mail: hiroyuki.inaoka@gmail.com 48 2015.11.22
1 1 ( ) ( ) 3 5 3 13 3 2 1 27
Shin Manders Avigad, Mumma, Mueller (Macbeth ) Fowler, Netz [2014]
Theorem c a, b a 2 + b 2 = c 2
Proof. c 4 b a 180 b c a c (a + b) 2 = c 2 + 4 1 2 ab c b a c b a a 2 + b 2 = c 2
1 1 Theorem ( )
Proof. AB AB A AB BGD [I 3] B BA AGE [I 3] G A B GA GB [I 3] A GDB AG AB [I. 15,16] B GAE BG BA [I. 15,16] GA AB GA GB AB [I. 1] GA GB 3 GA, AB, BG ABG [I. 20] AB
D G E A B
co-exact exact
(Shin[1994]) (Shin[2012]) epiphany (Catton and Montelle[2012]) (Azzouni[2013]) (Macbeth)
( ) 1. 2. 3. 4. 5.
( ) 1. 2. 1 3. 4. 5. 2 2 2 2
pop up (Manders[1995]) Macbeth[2014 87-9]
seeing - as Coliva[2012]
Grice non natural meaning (Macbeth[2014 81-])
(natural meaning) (non natural meaning)
instance picture (icon) (Peirce[1932 159] (Macbeth[2014 83] ))
(contradictory content) Macbeth[2012 72]
I 465 110 (Vitrac[2012]) 1 ( ): 6, 7, 14, 19, 25, 27, 39, 40 3 ( ): 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 18, 19, 23, 27 4 ( ): 4 5 ( ): 9, 10, 18 6 ( ): 7, 26 "Eι (γαρ) δυατoν" (( ) )
II 0 = 1 {Pa, Pa} exact co-exact 3 co-exact
3 5 Theorem 2 2
A E D G B Z H
Proof. 2 ABG,GDH B,G 2 E EG EZH E ABG EG EZ E GDH EG EH EG EZ EZ EH E 2 ABG,GDH
3 13 Theorem 1 2 1
K L A G 3 13 ( )
1. AG AG AKGL 3-2 AG AKGL
3 2 Theorem 2 2 2
D G A Z B E
Proof. ABG D DA,DB DZE DA DB DAE DBE DAE 1 AEB DEB DAE DAE DBE DEB DBE DB DE DB DZ DZ DE
1 27 Theorem 2 2 2 2
A E B H G Z D
Proof. AB,GD B,D H HEZ AEZ EZH AB,GD B,D A,G
I 3 5 ( ) 3 13 ( ) 3 2 1 27
I (1) (2) Coliva[2012] (1) (2) taking-as (p.131) (1) (2)
II (1) 1 (2) 2 ( )
III (Macbeth[2014 83-4]) Catton and Montelle[2012] (p.54)
I
II 1 14 27 cf.rabouin[2015] diagram control
I ( )
II Johansen diagram figure (Johansen[2014])
I Jeremy Avigad, Edward Dean, John Mumma, "A formal system for Euclid s Elements", Review of Symbolic Logic, 2, 2009, pp.700-68. Jody Azzouni, "That We See That Some Diagrammatic Proofs Are Perfectly Rigorous", Philosophia Mathematica, 21-3, 2013, pp.323-38. Philip Catton, Clemency Montelle, "To Diagram, to Demonstrate: To Do, To See, and To Judge in Greek Geometry", Philosophia Mathematica, 20-1, 2012, pp.25-57. Annalisa Coliva, "Human diagrammatic reasoning and seeing -as", Syhthese, 186, 2012, pp.121-48. Mikkel Willum Johansen, "What s in a Diagram?: On the Classification of Symbols, Figures and Diagrams", Model-Based Reasoning in Science and Technology: Theoretical and Cognitive Issues, Studies in Applied Philosophy, Epistemology and Rational Ethics 8, Springer, 2014, pp.89-108. Danielle Macbeth, "Seeing How It Goes: Paper-and-Pencil Reasoning in Mathematical Practice",Philosophia Mathematica, 3-20, 2012, pp.58-85.
II Danielle Macbeth, "Diagrammatic reasoning in Frege s Begriffsschrift", Synthese, 186, 2012, pp.289-314. Danielle Macbeth, Realizing Reason: A Narrative of Truth and Knowing, Oxford University Press, 2014. Kenneth Manders, "Diagram-Based Geometric Practice", Paolo Mancosu, ed, The Philosophy of Mathematical Practice, Oxford University Press, 2008, pp.65-79. Kenneth Manders, "The Euclidean diagram(1995)", Paolo Mancosu, ed, The Philosophy of Mathematical Practice, Oxford University Press, 2008, pp.80-133. Charles Sanders Peirce, Collected Papers of Charles Sanders Peirce, vol.2., Harvard University Press, 1932. David Rabouin, "Proclus Conception of Geometric Space and Its Actuality", Vincenzo De Risi, ed., Mathematizing Space The Objects of Geometry from Antiquity to the Early Modern Age,Springer, 2015, pp105-42. Sun-Joo Shin, The Logical Status of Diagrams, Cambridge University Press, 1994.
III Sun-Joo Shin, "The forgotten individual: diagrammatic reasoning in mathematics", Synthese, 186, 2012, pp.149-68. Bernard Vitrac, " Les démonstrations par l absurde dans les Éléments d Euclide : inventaire, formulation, usages" 2012. https://hal.archives-ouvertes.fr/hal-00496748v2 2015 11 20 47 1 pp.67-82 2014. 1 2008. 15K02002