自然な図形と不自然な図形： 幾何図形の二つの「意味」
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- しゅんすけ いいはた
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1 ( ) mail: [email protected]
2 1 1 ( ) ( )
3 Shin Manders Avigad, Mumma, Mueller (Macbeth ) Fowler, Netz [2014]
4
5 Theorem c a, b a 2 + b 2 = c 2
6 Proof. c 4 b a 180 b c a c (a + b) 2 = c ab c b a c b a a 2 + b 2 = c 2
7 1 1 Theorem ( )
8 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
9 D G E A B
10 co-exact exact
11 (Shin[1994]) (Shin[2012]) epiphany (Catton and Montelle[2012]) (Azzouni[2013]) (Macbeth)
12 ( )
13 ( )
14 pop up (Manders[1995]) Macbeth[ ]
15
16 seeing - as Coliva[2012]
17 Grice non natural meaning (Macbeth[ ])
18 (natural meaning) (non natural meaning)
19 instance picture (icon) (Peirce[ ] (Macbeth[ ] ))
20 (contradictory content) Macbeth[ ]
21 I (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" (( ) )
22 II 0 = 1 {Pa, Pa} exact co-exact 3 co-exact
23 3 5 Theorem 2 2
24 A E D G B Z H
25 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
26 3 13 Theorem 1 2 1
27
28 K L A G 3 13 ( )
29 1. AG AG AKGL 3-2 AG AKGL
30 3 2 Theorem 2 2 2
31 D G A Z B E
32 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
33 1 27 Theorem
34 A E B H G Z D
35 Proof. AB,GD B,D H HEZ AEZ EZH AB,GD B,D A,G
36 I 3 5 ( ) 3 13 ( )
37 I (1) (2) Coliva[2012] (1) (2) taking-as (p.131) (1) (2)
38 II (1) 1 (2) 2 ( )
39 III (Macbeth[ ]) Catton and Montelle[2012] (p.54)
40 I
41 II cf.rabouin[2015] diagram control
42 I ( )
43 II Johansen diagram figure (Johansen[2014])
44 I Jeremy Avigad, Edward Dean, John Mumma, "A formal system for Euclid s Elements", Review of Symbolic Logic, 2, 2009, pp Jody Azzouni, "That We See That Some Diagrammatic Proofs Are Perfectly Rigorous", Philosophia Mathematica, 21-3, 2013, pp Philip Catton, Clemency Montelle, "To Diagram, to Demonstrate: To Do, To See, and To Judge in Greek Geometry", Philosophia Mathematica, 20-1, 2012, pp Annalisa Coliva, "Human diagrammatic reasoning and seeing -as", Syhthese, 186, 2012, pp 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 Danielle Macbeth, "Seeing How It Goes: Paper-and-Pencil Reasoning in Mathematical Practice",Philosophia Mathematica, 3-20, 2012, pp
45 II Danielle Macbeth, "Diagrammatic reasoning in Frege s Begriffsschrift", Synthese, 186, 2012, pp Danielle Macbeth, Realizing Reason: A Narrative of Truth and Knowing, Oxford University Press, Kenneth Manders, "Diagram-Based Geometric Practice", Paolo Mancosu, ed, The Philosophy of Mathematical Practice, Oxford University Press, 2008, pp Kenneth Manders, "The Euclidean diagram(1995)", Paolo Mancosu, ed, The Philosophy of Mathematical Practice, Oxford University Press, 2008, pp Charles Sanders Peirce, Collected Papers of Charles Sanders Peirce, vol.2., Harvard University Press, 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, pp Sun-Joo Shin, The Logical Status of Diagrams, Cambridge University Press, 1994.
46 III Sun-Joo Shin, "The forgotten individual: diagrammatic reasoning in mathematics", Synthese, 186, 2012, pp Bernard Vitrac, " Les démonstrations par l absurde dans les Éléments d Euclide : inventaire, formulation, usages" pp K02002
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