18032426 Στυλιανός Νεγρεπόντης Βασιλική Φαρμάκη Η Παράλογη Αποτελεσματικότητα Των Μαθηματικών Στις Άλλες

Embed Size (px)

Citation preview

. , . 1. [ ?] , , . , , , , . 17 , . , , 718, . , , Heisenberg , -, . , , ( , ,), ( ),

, (, ) , . , : , , , , , , ? E. Wigner, 1963 o , [W], 1960, : (unreasonable effectiveness) , . , .. Hamming [H] 1980, , Wigner . 2001 Halpern, Harper, Immerman, Kolaitis, Vardi Vianu, [HHIKVV], Wigner, (unusual effectiveness) . Wigner, . , , , , . , ( , ), .2.[ ] , , 1800 .. . , 600 .., ,

,

, , , , ,

.

, , , , .

,

,

.

,

.

: 1 (5, 5, 8) (4.95, 4.95, 7). ? . , 1-1 , , .2 , , , , 12, , =7, 3.5 (, , 2=(4.95)2 (3.5)2 = 24.5025 12.25 =12.2525>12.25= (3.5)2), /2=3.5 3.5= 12.25, 12! , .

, (. , 403,14-404,14), :3 , (5,5,8), (5,5,6), ? , , (5,5,8) . .4 (3,4,5) (32+42=52).5, < < ( 2+2=2), (, , 2) (, , 2), ...

(, (2-1)/2, (2+1)/2)) () ,

((2+1)/2, (2+1)/2, 2)) ((2+1)/2, (2+1)/2, 2-1)) ,(.. (13,13,10) (13,13,24)),

(61,61,22) (61,61,120)).

.

, .3. [ .35, 37 ] 35 37 :6 , (, ), , . , . , .37, , , . , . :

[.35] , , [.37] . , . (396,10-397,6) . , , , . (403,5-14) . , , .35, 37, , . : . . : , , , , , , , , . ` : , , , . , , , , , , , . , , , , . , .4. [ ] 819d-e , , , , , , . , , , : 7 , , , () ?

, =, =.

, . ?

( ), ( ). , , .

, . , , , , , .

. , .8 , , , >, /=/(-), , 2=+2. ( 11 ):

. /2. . >>/2, =+(/2). , (/2,,), . ( ). , .

9 10 () . ( . i o ,

). . , : =, (,)=.

>.

. 1, , 1, , =1+1, 10, , 11>2>

, ( ) .

, , ,. .

, , .

, . 11 , =, =, , , , .12 >.

=+(-), -. 1 1=-. ,1 . , /1= 1/-1 (-1)=12, 2-(-)=(-)2 2+2-=2+2-2, 2+=2, !13 , , 1. 14 - .

, , .5.[ ] . (1+51/2)/2, 21/2. , . , , . , : , , , , , , , , , , 2. .

? , , , , , , , . , , . , , , ! , , .

, ( ) ( , 2). , . , , , , , .

, , , , . , , . , ( ) , . Cantor, , 1-1 . , . 6. [ ] , , , , . , , , , , . , . , , , . , . . ? , .. , 100 . , . , , , . ?

, , . ( ) ( ). , . , DNA . , , , , DNA. DNA, . DNA .

, D.

. , , , . .

: ? , , , . 11 , . 15

- ( von Fritz, [F]): , , , .

, , .

, ( ) -.7. [Fractals ] , , . , , , , . , , , , , . fractals, , - , . , fractal, 1915 Sierpinski:16 ( 0):

0

( 1), ():

1

( 2), 9 :2

17 k (.. 3 4 ): 3

3

4

, , , () Sierpinski, . fractal, . fractal, , , .

, , , (game of chaos), Internet

(.. http://mathworld.wolfram.com/SierpinskiSieve.html,

http://ejad.best.vwh.net/java/fractals/sierpinski.shtml,

http://math.bu.edu/DYSYS/chaos-game/node1.html,http://www.efg2.com/Lab/FractalsAndChaos/SierpinskiTriangle.htm,http://school.discovery.com/lessonplans/activities/chaosgame/,http://www.cut-the knot.org/Curriculum/Geometry/SierpinskiChaosGame.shtml,http://www.cevis.uni-bremen.de/fractals/nsfpe/Chaos_Game/Chaos1.html), , , - .18

() , , C, :

.

( ), , , 1.(1) 1 2, 1 .

(2) 3 4, 1 .

(3) 5 6, 1 C.

, ( ), 2, 1 ( ). ( ) , 1, 2, 3, , ,

? , , , , . 19 , 30 , (1,,30):

, , . , . ? .

19 100 (1,,30,,100):

20 400 (1,,30,,100,,400):

21 30 000 (1,,30,,100,,400,,30 000)

, , , fractal Sierpinski:

, , , .8. [ ] , , , , - . Wigner, : , ?

.22

p q :

p q p q

p pp p p p

() p .

23 'p q:pqp q

p q () p (), q.24, , p p q p pqp p q

() q.

( -, ) , p p p, q ,

, . , !,

1 0 1 0,

, ...

, , . , . , ? , , . ? , , , , (. [] ). , , . -, , . - . ( , , , , ) , , , ( ). Solovay, ( [S]) , , Russell, . , Wigner, , , , , , , .

9. [ Bertrand Russell ] , , . . ? , - , . . . , , . [R], Wigner [W], Bertrand Russell ( ) , , , , , , , , , , . , , , , . , .,

,

, [F] K. von Fritz, The Discovery of Incommensurability by Hippasos of Maetapontium, Annals of Mathematics 46 (1945), 242-264.[] R. W. Hamming, The Unreasonable Effectiveness of Mathematics, American Mathematical Monthly 87 (1980), 81-90.[IKVV] J. Y. Halpern, R. Harper, N. Immerman, P. G. Kolaitis, M. Vardi, and V. Vianu, On the unusual effectiveness of logic in computer science, The Bulletin of Symbolic Logic 7(2), (2001), 213-236.[N] . , , (1979), 161-173.[R] B. Russell, Philosophical Essays: The Study of Mathematics, 1910.[S] R. Solovay, On the cardinality of 12 sets of reals, Foundations of Mathematics (Symposium commemorating Kurt Godel, Columbus, Ohio, 1966) pp.58-73, Springer, New York, 1969.[W] E. P. Wigner, The Unreasonable Effectiveness of Mathematics in Natural Sciences. Comm.Pure and Applied Math.13 (1960), 1-14.

, , 157 84

[email protected], [email protected] PAGE 10