1 1 (metamathematics) ( ) ( ) ( ) a b = c d = e f a b = c d = e f = pa + qc pb + qd = pa + qc + re pb + qd + rf a b = c d = e f = k ( 0) a = bk c = dk
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1 7 (Euclid (Eukleides : EÎkleÐdhc) : 300 ) (StoiqeÐwsic) 3 ( ) 19 ( ) (Nicolas Bourbaki) (Éléments de Mathématique) 1 (Théorie des ensembles : 1966 ) ( 1 p.1) (qui dit mathématique dit démonstration : ) (Julius Wilhelm Richard Dedekind : ) (Was sind und was sollen die Zahlen? : 1887 ) ( 2 p.41) 1 1
2 1 1 (metamathematics) ( ) ( ) ( ) a b = c d = e f a b = c d = e f = pa + qc pb + qd = pa + qc + re pb + qd + rf a b = c d = e f = k ( 0) a = bk c = dk e = fk pa + qc pb + qd pa + qc + re pb + qd + rf = p(bk) + q(dk) pb + qd = p(bk) + q(dk) + r(fk) = pb + qd + rf k a b = c d = e f pa + qc = pb + qd = pa + qc + re pb + qd + rf (pb + qd)k pb + qd = = k (pb + qd + rf)k pb + qd + rf = k 2
3 (1) (Thales (Jal hc) : 640? 548?) ( 3 pp.9899) (Pythagoras (Pujagìrac) : ) 6 5 (Parmenid es (ParmenÐdhc) : ) (Zenon (Zănwn) : 490? 429?) (Aristotel es (>Aristotèlhc) : ) ( 3 p.191) 5 (Árpád Szabó : ) ( ) (Analutika Protera) ( 4 pp ) 3
4 ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) (Analutika Ustera) ( 4 p.613 p.633) ( 4 p.617) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ( ) ( ) ) ( ) ( 6 ) 4
5 23 ( 5 pp.12) ( ) 9 ( 5 ) ( ) ( 5 p.2) ( ) (9 )
6 ( 5 pp.56) 1 5 A ABG AB AG BD GE AB AG ABG AGB GBD BGE B G BD Z AE Z H AZ AH ZG HB D E AZ AH AB AG 2 ZA AG 2 HA AB ZAH ZG HB AZG AHB AGZ ABH AZG AHB AZ AH AB AG BZ GH ZG HB 2 BZ ZG 2 GH HB BZG GHB BG BZG GHB ZBG HGB BGZ GBH ABH AGZ GBH BGZ ABG AGB ABG ZBG HGB (íper êdei deĩxai : quod erat demonstrandum) Q.E.D. 3 (íper êdei poi hsai) 6
7 ( ) P P 2 ( 5 pp ) 2 ( 7 ) ( 5 p.149) A J 7 1 Z G H 2 AB Γ B D E AB Γ AB Γ ( ) AB GD E GD BZ ZA AZ DH HG HG ZJ JA a b P a b e ( 1) a pb = c (< b) b qc = d (< c) c rd = 1 7
8 E GD GD BZ E BZ E BA AZ AZ DH E DH E DG GH GH ZJ E ZJ ZA E AJ AB GD AB GD a b e a = ek b = el c = a pb = ek p(el) = e(k pl) e a B pb Z c A b c e d = b qc e c d e 1 = c rd e e ( 1) 1 ( ) a b a b 1 ( 5 pp ) A B 2 A B 2 D B G A B A B E A G D 2 2 G A D B G A E D B E G A E E A G E B D E A B A B A B A B 2 2 a b e 8
9 (2) 17 (René Descartes : ) ( ) (Discours de la méthode pour bien conduire sa raison, & chercher la verité dans les sciences : 1637 ) (Regulae ad directionem ingenii : 1628 ) 1 ( 7 p.11 p.23) 1 (ingenium) 4 (Methodus) (Arithmetica) (Geometria) (experientia) (deductio) 2 (intuitus) (inductio) 2 ( 2 7 p.17) ( 4 7 p.24) ( 7 p.31 p.33 p.39 p.44) 5 (dispositio) 9
10 ( 2 6 p.26) ( 2 6 p.26) 10
11 (Blaise Pascal : ) (De l'esprit géométrique : ) ( 8 pp ) 1 ( 8 p.118) 2 ( 8 pp ) 11
12 ( 8 p.145) ( ) 12
13 (3) 19 (Georg Cantor : ) 20 (David Hilbert : ) 1 (Grundlagen der Geometrie : 1899 ) VI ( 9 p.203) ( 1 ) ( 0 ) 3 ( IX 9 p.244) 13
14 & (x) (Ex) ( ) ( ) ( ) ( ) ( ) ( ) S S T T S S T ( ) ( 1971 ) ( IX 9 p.253) 1931 (Kurt Gödel : ) 14
15 (4) ( Ÿ5 2 pp.8182) 66 S s S s s S 1 s ϕ(s) S ϕ S S S S S s S ( ) 1 a b S a b ϕ ( ) S (Bernard Bolzano : ) (Paradoxien des Unendlichen : 1851 ) 13 ( 10 pp.1516) A A A A A B B A 3 C ( ) 8 15
16 ( 11 pp.36 p.11 p.15) ( ) A K ? (1) (2) C C 1 N. ( ) 1 ( ) 1968 ( 43) 2 R. ( ) ( ) 1961 ( 36) 3 ( ) 1990 ( 2) 4 ( ) ( ) ( 1 ) 1971 ( 46) 5 ( ) 1971 ( 46) 6 R. ( ) ( 1 ) 1973 ( 48) 7 R. ( ) ( 4 ) 1973 ( 48) 8 B. ( ) ( 1 ) 1959 ( 34) 9 D. ( ) ( 7 ) 1970 ( 45) 10 B. ( ) 1978 ( 53) 11 ( 1345) 1997 ( 9) 16
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