Dipl Dimitris Giannopoulos

M ,

&

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:

.

2007

30-4-2007 , : 1) . ..

( )

2) . .. 3) . ..

mpeira mn poie tn ana mn poreesqai kat tcnhn,

peira d kat tchn.

2006

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, 2007

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.

............................................................................................................... 1

1 ................................ 4

1.1 ........................................................................ 4

1.2 ............................................... 9

2 ................................... 13

2.1 - ......................................... 13

2.2 - ................................................................. 17

2.3 .......................................................... 22

2.4 ................................. 27

3 ............................................ 33

3.1 ........................................................................ 33

3.2 ..................................................... 48

3.2.1 .................................................................... 48

3.2.2 ..................................................................... 54

3.3 ........................................... 72

3.4 .............................. 87

3.5 ........................ 97

3.5.1 ........................................................................ 97

3.5.2 ....................................................................... 99

3.5.3 .................................................................................. 101

3.5.4 ................................ 103

3.5.5 .............................................................. 105

4 ... 109

4.1 ................................................... 110

4.2 ........................................ 112

4.3 ............................... 119

4.4 .............................. 127

5 ...................................................................................... 130

6 ........................................................................................... 133

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3.11 ()

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(, 2000).

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[Zeichen, sign], [Sinn, sense],

[Bedeutung, reference],

10

--. F. de

Saussure (1916), (2001),

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(, 2000).

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(2000),

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(, 1993). Wittgenstein (1978) ,

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, (, ), (

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Dummett (1977),

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11

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Cajori, 1919),

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10

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1997) , ,

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Vieta

(Spangolo, 2006, Eves, 1981).

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, (, 1991)

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14

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-

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.

(, 1991,

. 38). Frege

(, 1991, . 67). ,

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,

. Girolano Cardano

1545, Ars Magna

, m

,

.

(, 1985, Cajori, 1928).

19 ,

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15

(, 985b). Descartes

1701, ,

,

(, 1991, . 35), Wronski 1803

(Davis & Hersh, 1981, . 75), Lagrange

1800 (, 1999), Jacobi 1841,

man muss immer generaliesieren, (Davis &

Hersh, 1981). ,

, Dedekind

, ,

Cantor

(, 2005),

Felix Klein

19 (Davis & Hersh, 1981).

Leibniz,

(caracteristica universalis)

,

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Leibniz (, 1985).

Bell (1992, . 1, . 195), Leibniz, ,

De arte combinatorya,

.

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(, )

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. Frege Leibniz ,

16

. , calculus

ratiocinator,

(, 1991, . 68).

Leibniz,

,

(Bell, 1992).

Leibniz,

.

Boole 1848

(, 2005) De

Morgan, Jevons, Pierce Schroeder

(Boyer, & Merzbach, 1997). Russell ,

19 ,

Boole

(Boyer & Merzbach, 1997).

17

2.2 -

18 ,

,

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Euler,

D Alembert Lagrange. ,

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19 .

Cauchy, Gauss, Dirichlet, Abel Bolzano

(, 2005).

Peacock

1830,

. Boole ,

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... (Boyer & Merzbach, 1997, .

648). ,

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19 Frege Russell

(, 2005).

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18

(1991, . 64) ,

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Tarski. ,

() , ()

., .

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(, 1985).

Peano Frege 19 ,

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20 .

Schroeder

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1993, . 75).

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, (1991)

211 [ Jourdain, (1912, .238)]

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Russell (1908) :

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Russell

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[Sinn, sense] [Bedeutung,

reference]. .

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20

,

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,

(, 1985).

- Wittgenstein (1978) ,

() .

(

) (.3.32, 3.321).

(.3.325).

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20

.

Frege Russell

.

(1993), Frege,

, , Russell,

(. 32)

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Brouwer, -,

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(Davis &

Hersh, 1981, , 1993). ,

21

,

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(,

1985, , 1993)

, Hilbert

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(, 1985).

Gdel 1931 , Hilbert

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(, 1993). ,

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Bourbaki. ,

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Gdel 1930,

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(, 1985).

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22

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20 , ,

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1991). ,

(, 1993).

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www.math.uoa.gr/me,

Latex:

23

%Here begins the document!

\begin{document}

$\NN,\ZZ,\QQ,\RR,\CC,\TT,\DD$

$\grA,\grB,\grG,\grD,\grE,\grZ,\grH,\grU,\grI,\grK,\grL,\grM,\grN,\grJ,\grO,\grP,\grR,\grS,\grT,\grY,\grF,\grX,\grC,\grV$

$A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z$

$\gra,\grb,\grg,\grd,\gre,\grz,\grh,\gru,\gri,\grk,\grl,\grm,\grn,\grj,\gro,\grp,\grr,\grs,\grt,\gry,\grf,\grx,\grc,\grv$

$a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z$

Two ways to turn from English to Greek

\gt

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(

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24

, (, 1991).

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Peano,

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Boole

Schroeder , Frege (,

1991).

Frege,

(),

,

.

.. (a+b)c=ac+bc.

(, 1991, . 188). , Russell

Frege,

. Russell, (1985, . 232),

,

.

(1985, . 232)

,

.

ermelo-Fraenkel yx ( )yzxzz .

25

Principia Mathematica Russell Whitehead,

.

,

,

, Quine (1967) (1993,

. 81). , ,

(, 1993).

Principia Mathematica Russell Whitehead.

,

. ,

, , (Davis &

Hersh, 1981). , Tomae

26

(game formalism) (1985, . 242),

:

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Tomae (, 1985, . 242),

a priori

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(1985) .

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(, 1985, . 243).

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(Davis & Hersh, 1981, , 1985).

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(Davis & Hersh, 1981 . 135).

27

2.4

.

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,

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Hermann Grassman, 1844

Die Lineale Ausdehnungslehrer.

. ,

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(Davis & Hersh, 1981).

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. 16 ,

.

Nesselmann

,

(Eves,

1981, Spagnolo, 2006). Vieta Decartes

, ,

, ,

(Eves, 1981, Cajori, 1919).

28

, ,

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() , () ()

(Boyer & Merzbach, 1997)

. , 10841169

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, MLXXXIV

MCLXIX , ,

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, Van der Waerden (2003, . 44) :

.

.

Newton Leibniz. , 0/0

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Euler,

,

. Euler

.

Leibniz,

18 (Boyer, 1959, . 242).

Leibniz,

, Newton

,

,

(Bell, 1992).

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29

,

Leibniz, (Boyer & Merzbach, 1997).

,

,

.

Bombeli 16

x2=1

. ,

(, 1985,

. 241)

, , (pattern)

. Oresme ,

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,

. 6443 = 864 2/1 = 84 211 =

: (Cajori, 1919, . 127).

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, ,

2x .


Boyer & Merzbach (1997),

Newton, ,

.

Cardano Pascal,

Descartes,

211p

1p 21 4 4

30

. Newton,

, Wallis.

,

x2+y2=1 x2-y2=1

18 , Vincenzo Riccati

. Lambert sinh(x)

cosh(x) tanh(x)


,

,

. 18 ,

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17 ,

(, 1999, . 33).

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, ,

(, 1999). , 19 ,

(, 1991). Boyer & Merzbach (1997),

Hamilton 1834,

,

,

.

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(Davis & Hersh, 1981), Spinoza more

geometrico, , Ethica (Ordina

31

Geometrica Demonstrata), (Bell,

1992).

,

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,

.

,

.

dm=K(dp/p), m , p

. (Boyer & Merzbach, 1997)

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Fregge (, 1991, . 67)

...

Russell (Wittgenstein, 1978, . 35)

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32

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3.1

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(Struik, 1966).

, Wedda,

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, - sticks (Menniger, 1969).

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33

, .

,

(Struik, 1966).

,

, Damara,

(Menniger, 1969). ,

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, (, 1999, Menniger, 1969).

, Menniger (1969)

, 4 ,

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( 1). ,

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1. ,

(Menniger, 1969, . 17).

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(Struik, 1966).

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34

, .

,

(ordering and grouping),

. ,

(Menniger, 1969). ,

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Menniger, (1969), (4, 412-413),

atr pn psaj pempssetai d dhtai, lxetai n mssVsi, nomej j pesi mlwn.

,

. , Dene-Dinjie

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20

, 100 (100=5x20) (Menniger, 1969),

(Hughes, 1996).

18 (Menniger, 1969).

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20. , 18 ,

- achtzehn, - - duo-de-

viginti, (Menniger, 1969).

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20 60 .. 12 (-

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35

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, (abstract concept)

(Menniger, 1969).

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(Hughes, 1996, Struik, 1966).

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(Hughes, 1996).

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36

, ( 2),

(Struik, 1966, Van der Wearden, 2003).

2. (Menniger, 1969, . 42)

. ,

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,

( 1).

8 VIII (5+3)

9 IX (10-1)

19 XIX (10+9)

1. 8, 9 19

,

, .

,

1/ (Van der Wearden, 2003, Struik,

1966).

37

, .

, 1/12 (Van der Wearden, 2003).

,

,

. , Cajori (1928),

, .

, ( 3),

, 800 .. ,

(Cajori, 1928).

3. , ,

(Cajori, 1928, . 12)

, .

,

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38

,

, (Van der

Wearden, 2003).

. -

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-,

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15 ,

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-

, 400 (Eves, 1981).

,

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, (Van

der Wearden, 2003). 1 60

60 ( 4). Van der Wearden (2003,

.33), , .

4. (Van der

Wearden, 2003, . 33)

39

,

. ,

1, , 10. 35

. ,

, 60 1,

, 10 10 x 60.

. , ,

1, 60 , 602 1/60 1/602 .

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. 60

. ,

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.

,

(Van der Wearden, 2003,

Bunt, 1981). ,

60, .

, ,

, . ,

, 1x602+0x60+4=3604

1x60+4=64 (Van der

Wearden, 2003).

, ,

. , ,

,

( 5). , ,

, , , ,

, . ,

(Van der Wearden, 2003, Heath, 2001).

40

5. , (Va der

, 24

=1 =10 =100

n

Wearden, 2003, . 42).

,

( 2).

,

999,

9

. ,

.

1305 , 3.000,

4.352. Cajori (1928)

, , .

, ,

=2 =20 =200 =3 =30 =300 =4 =40 =400 =5 =50 =500 =6 =60 =600 =7 =70 =700 =8 =80 =800 =9 =90 =900

2.

,

(Heath, 2001).

41

=10.000,

,

M 250043 (Van der Wearden, 2003).

,

(Heath, 2001).

, ,

. ,

, (), () (). ,

,

, 10,

. , 3 4 7,

, 30 40 70 (Hughes, 1996).

, Van der Wearden (2003, . 43)

,

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, Boyer & Merzbach (1997),

,

. 3520 .

. ,

221 .., ,

(Lam Lay-Yong, 1986, Cajori,1928).

( 3),

,

, ,

, , :

42

1 2 3 4 5 6 7 8 9

3. (Lay-Yong, 1986).

Shang

,

, ,

, .

94571 . ,

,

(Lay-Yong, 1986).

,

. ,

.

.. ,

.

, ,

Kharosti (Boyer & Merzbach, 1997, . 237).

6. Kharosti (Menniger, 1 9, . 50).

Kharosti

Brahm

, 10

96

i ( 7),

, ,

43

,

(Van der Wearden, 2003,

Menniger, 1969).

7. Brahmi (Menniger, 1969, . 395).

Gvalior

,

Brahmi,

(Menniger, 1969).

, Freudenthal, Van der Wearden (2003)

,

. 0

Menniger, (1969) Cajori (1928),

o-

Brahmi

( 7). ,

. ,

,

(Van der Wearden, 2003).

,

,

,

(Van der Wearden, 2003, Boyer &

Merzbach, 1997, Lay-Yong, 1986, Struik, 1966).

Gvalior, () (gubar),

()

. Brahmi,

44

8,

(Menniger, 1969)

8.

, Menniger, (1969, .418)

, Menniger, (1969,

. 4

-

06) (The Indian

numerals in Arab hands). ,

,

(Cajori, 1928,

Struik, 1966, Lay Yong, 1986). ,

Severus Sebokht ( )

622 ... ,

. ,

. ,

, 595

45

346 (Boyer & Merzbach, 1997, Struik, 1966)

,

(),

(Menniger, 1969).

,

. , 8

9 ,

. ,

, (Menniger, 1969, Van der

Wearden, 2003). -

,

Gerbert ( ) 10

. , 12 , ,

Al Khwarizmi. ,

, Algorithmi de numero Indorum,

Hisab aljabr walmuqabala. ,

(Boyer & Merzbach, 1997, Van der

Wearden, 2003, Menniger, 1969, Struik,1966).

, Leonardo di Pisa Liber abaci,

1202

, .

, , 0,

- (Boyer & Merzbach, 1997).

, o ,

, ,

Leonardo di Pisa. , ,

- (Boyer & Merzbach, 1997, Vogel, 1996).

46

-

, , ,

(Van der Wearden, 2003).

IImIIIcXV

2315 (Struik, 1966). ,

- .

400 ,

, . ,

, ,

, Adam Riese ( 9) .

9. Adam Riese (Menniger, 1969 .

437 & 35 )

15

,

2

-

. - ,

.

Van der Wearden, (2003, . 55)

,

Menniger, (1969, . 55),

, , .

47

3.2

, .

,

, .

3.2.1

.

,

, .

,

, .

,

3, 4 7 (Struik,

1966).

,

. , ,

, , ,

(, 2003).

,

(, 1975, Van der Waerden, 2003).

,

- , .

,

17 (, 2003).

,

,

48

.

, ,

, . ( 10)

1342, x2+10x=39

, .

10.

(Hairer et al., 2000)

16 .. ,

, , . x2, x3, 6x

3 .

, Cardano Ars Magna

, x3 + 6x =20,

20 (Boyer & Merzbach,

1997).

,

(, 2005),

.

,

, , , , .

. ,

- .

,

.

.

49

, 1

Estw doqesa eqea peperasmnh AB. De d p tj AB eqeaj trgwnon spleuron sust- sasqai. KntrJ mn t A diastmati d t AB kkloj gegrfqw BGD, ka plin kntrJ mn t B diastmati d t BA kkloj gegrfqw AGE, ka p to G sh- meou, kaq' tmnousin ll- louj o kkloi, p t A, B shmea pezecqwsan eqeai a GA, GB. Ka pe t A shmeon kntron st to GDB kklou, sh stn AG t AB plin, pe t B shmeon kntron st to GAE kklou, sh stn BG t BA. decqh d ka GA t AB sh katra ra tn GA, GB t AB stn sh. t d t at sa ka llloij stn sa ka GA ra t GB stn sh a trej ra a GA, AB, BG sai lllaij esn.

, ,

,

,

. 1342 ( 11),

,

2 .

11. (Hairer et al.,

2000).

Leonardo di Pisa Regiomontanus

c f, Stevin 1634

, , , , . Leibniz 1676

,

1B, 2B, 3B, 1D, 2D, 3D 1b, 2b, 3b

50

. Leibniz,

. 12,

Leibniz .

12. Leibniz (Hairer et al., 2000).

Euler, ,

x, x, x y, y, y,

,

a, b, c, A, B, C,

, . , r, R

s ,

, . 4rRs=abc

,

Euler, (Boyer

& Merzbach, 1997).

,

, , , .,

,

. , ,

.

(), .

51

150 .. , ,

, .

, 5 .. , , ,

(Cajori, 1928).

==

, , , , , . .

Maurolykus 1555 , , ,

. 1634 Herigone Cursus mathematicus

:

< , , , , _| , = , , , , , , 5< 6

(identity).

, -

is simil. (Cajori, 1928).

( 4),

Cantor 1801 De la Correlationdes figures de

geometrie Cajori, (1928) :

A, B, C,

, ,

BCD

CDAB

F C D AB

F

AB CD

= | =

AB BC

AB BC

ABC ABC

ABC BC A

ABC ABC

4. Cantor 1801 (Cajori, 1928, . 422).

ABCD

ABC

CD

BCD

53

3.2.2

, ,

18 .

,

, .

,

. 18 ,

,

. ,

,

.

, ,

, ,

.

,

.

: Ka t farmzonta p' llhla sa llloij

stn. Frege, ,

Leibniz: Leibniz ,

.

.

Leibniz . [dasselbe]

, [gleich]

(, 1993, . 52).

, Russell :

x=y.=():!x. !yDf. , x y

x,

y (, 1993).

15 ,

54

, . ,

(it

gives) (Cajori, 1928).

,

aequales, aequantur, eguale, esgale, faciunt, ghelijck, gleich

(Cajori, 1928). ,

. , (

) ,

.

pha phala,

Al-Qalasadi

(Cajori, 1928). Regiomontanus

Pacioli ( 13), ()

, Cardano

(Cajori, 1928).

13. , ,

Summa Pacioli (Cajori,1928, . 110).

= Robert Record 1557

The Whetstone of Witte, ( 14),

, (Cajori, 1928).

:

55

(Eves, 1981, , .

150).

, ,

,

(Cajori, 1928).

,

, [ , || , | ,

2|2 Herigone (1634), . Herigone

3|2 2|3 .

|| Wilchelm

Holzmann, Xylander 1571

. , Cantor

, Xylander

(Cajori, 1928).

14. Robert Record

(: http://members.aol.com/jeff94100/witte.jpg)

= ,

,

56

ae aequales,

(Cajori, 1928). ,

Descartes (1637) ,

, Huygens 1646, . Rolle

1690 Jac. Bernoulli 1713 (Cajori, 1928).

=, .

Boyer & Merzbach (1997),

,

.

1557, 1618

Artis analyticae praxis

T. Harriot, Clavis mathematicae W. Oughtred Trigonometria R.

Norwood (Cajori, 1928). ,

, , Newton

Leibniz ,

(Cajori, 1923a, Cajori,

1928). 18 ,

. Jac. Bernoulli,

Ars Conjectandi, 1713 ( 15),

(Cajori, 1928).

15. Jac. Bernoulli (Hairer et al., 2000).

57

,

=

. Record, ,

(Eves, 1981).

> <

T. Harriot Artis analyticae praxis 1631, (Cajori, 1928,

Boyer & Merzbach, 1997), Oughtred,

, . O Isaac Barrow (1630-1677),

Lectiones Opticae & Geometricae (London 1674),

: A major est quam B A

minor est quam B (Cajori, 1928).

,

. ,

.

, ,

( 16),

.

16. (Cajori, 1928)

, ,

,

. leyij parxin,

, .

, ,

.

.

74-1/14 (Heath, 2001).

, Arithmetic Bakhshali 8 10

.. , + .

12 7+ 12-7

yu, yuta,

(Cajori, 1928). , ,

58

. 3 4

7 3 4 . , 15 ,

, , ,

(Cajori, 1928).

1

, + -,

, Mercantile Arithmetic

Behende und hpsche Rechenung auf allen Kauffmanschafft Johannes Widmann,

1489. ,

,

. Widmann

( 17), "4 centner + 5 pfund" "5 centner - 17 pfund,"

"Was - ist / das ist minus ... und das + das ist mer."

(Cajori, 1928).

17. Widmann, ,

Augsburg 1526. (: http://members.aol.com/jeff94100/widman.jpg)

59

+,

et.

15 , 1486,

et . Cajori,

et ,

+ ,

t ( ) ,

1417. ,

Widman ,

, (Cajori, 1928).

Widman,

+

.

Henricus Grammateus Rechnenbuch 1518 Rudolf

Coss 1525, + ,

(Cajori, 1928). , +

Robert Recorde The Whetstone of Witte, 1557 (Boyer

& Merzbach, 1997). , Paciolli

Chuquet, 1484, ,

piu minus, p p~

m (Cajori, 1928, Boyer & Merzbach, 1997).

Regiomontanus 1460,

m~

9i ( 18). ,

(florescentform) (Cajori, 1928). m~

18. Regiomontanus (Cajori, 1928).

60

16

+ ,

17 (Cajori, 1928).

p m,

+ Rudolf (1525)

Stiefel (1544) (Boyer & Merzbach, 1997), Vieta,

,

| ( 19)

(Cajori, 1919, Cajori, 1928).

19. Vieta (Hairer et al., 2000, . 7)

+ ,

,

, .

? , . . ,

, . , ,

, .

, Herigone ~, Descartes Geometrie

. 18 ,

1---R---11 11--1 (Cajori, 1928).

61

+ , ,

,

.

,

. ,

, +a a a

. Wolfgang Bolyai 1832

+ (Cajori, 1928).

Albert Girard 1626,

ou, ou+

, .

, 1631 Oughtred Cavis mathematicae,

, Wallis,

Jones . , Schooten ,

Newton , , . , Euler, 1755 (Cajori, 1928).

m

,

.

, (-DU),

. , Rhind,

. ,

,

. 8, 9 10 ,

Bakhshali,

(indicated by placing numbers side-by-side). al-

Qalasadi 15 ,

(Cajori, 1928).

Bahaskara ,

bha,

, . Vieta,

, A in B Stifel, 1545,

62

Deutsche Arithmetica (M)

D (D) .

Stifel Stevin, 3xyz2 31 sec1M ter2 , sec ter , . 1553, Stifel Coss

Rudolff,

.

,

Herigone,

, * Rahn Banker

Teutsche Algebra Rahn,

17 (Cajori, 1928).

, ( Cajori (1928)), William Oughtred Clavis

Mathematicae - - 1631

1647. Oughtred .

John Wallis Seth Ward.

Oughtred ,

150 ,

. ,

. ("Appendix to the Logarithmes")

Descriptio John Napier, 1618,

.

Oughtred. ,

, , ,

(Cajori, 1928, . 251):

315172 295448

395093 174715

63

,

,

, (Cajori, 1928).

(.) Leibniz,

Oughtred,

x . John Bernoulli

: "I do not like (the cross) as a symbol for multiplication, as it is easily

confounded with x; .... often I simply relate two quantities by an interposed dot and

indicate multiplication by ZCLM.". Leibniz, Harriot Artis analyticae praxis 1631

aaa-3bba=+2ccc (Cajori, 1928). Cajori (1928) , Harriot,

Gibson 1655 Syntaxis mathematica,

,

. Leibniz

De arte combinatoria 1666, C

.

18 , Cristian Wolf,

Euler (Cajori, 1928).

, ,

.

*, , .

Rene Descartes, Gullberg (Miller, 2004). Cajori (1928),

Gauss, 1812.

, ,

,

. ,

64

,

, . ,

% ,

o

M , % o

M , (60x2+2520)/(x4+900-60x2) (Heath,

2001). ,

, .

,

(Boyer & Merzbach, 1997), Bakhshali

bh bhga

(part)(Cajori, 1928). al Hassar 12 ,

Leonardo di Pisa

, Liber abbaci, (Cajori, 1928).

16 /

, 1845, De Morgan (Boyer &

Merzbach, 1997).

,

Arithmetica integra Stifel 1544 Oughtred

Clavis mathematicae : 24:8 8)24 8)24(.

Stifel Stevin, ,

D (D) . Rahn Teusche Algebra,

, . Rahn ,

, Wallis

Barrow, ,

, (Cajori, 1928).

Leibniz, , De arte combinatoria

C . 1684, Acta eruditorium, : , x:y x

y yx . ,

65

(Cajori, 1928).

,

, : 36)116(3,

116:36=3, : 116)36(3 Bezout,

1833, :

14464 8

1808

Miller (2004),

. ,

1882, Complete Graded Arithmetic James B. Thomson,

,

( 20).

20.

, 1888

The Elements of Algebra G. A. Wentworth

( 21).

.

21.

, .

,

66

, . ,

. 16o ,

,

.. Radix, quadrata

R ra, radix , latus (Cajori, 1928).

,

ka, karana

(Eves, 1981).

l

, 1220

Leonardo di Pisa Practica geometriae,

(Cajori, 1928).

, Ars Magna Cardano. Chuquet

2 ,

(Boyer & Merzbach, 1997, Cajori, 1928),

Bombeli Rq Rc

(Eves, 1981). ,

Q cu C . 1525 Die Coss Christoff Rudolff

,

(Cajori, 1928). ,

r,

radix. Institutiones calculi

differentialis (1775) Euler, Cajori (1928), ,

, ,

. 1637, Descartes La Geometrie,

, ,

.

Albert Girard 1629 Invention nouvelle,

3).20+392 3 39220+ (Cajori, 1928). Stevin,

67

(Boyer &

Merzbach, 1997) e (Cajori, 1928). Cajori (1928), Girard

John Wallis,

320

3 20 Michel Rolle (1652-1719), 1690

Trait d' Algbre .

. ,

, Newton

(Cajori, 1919).

,

.

,

.

, , .,

.

, , ,

.

u v,

Cajori (1928) Eves (1981), Pacioli Summa,

1494 1523 : R7pR14, 7 14+ , R (radix universalis niversalis)

.

,

.

Cajori (1928), Le Triparty en la Science des Nombres

Chuquet 1484. Chuquet

14 180+ . , Alexander, Bombeli (Cajori, 1928).

68

Frans van Schooten,

Vieta , . ,

1646 ,

B(D2 + BD) Vieta

in . + Din Bquadratum D.

.

Newton, Analysi per Aequationes mumero terminorum Infinitas,

4 5 12 17 0y y y y + + = {[(y-4)y-5]y-12}y+17=0. , 1631 Hariot,

(Cajori, 1928).

, 16 17

Tartaglia Cardan, Ars magna,

Opera 1663, Stifel

. [ ]

Algebra Bombelli, 1550 (Cajori, 1928). Bombelli,

7 14+ R[7pR14] (Eves, 1981). Cajori (1928), 1593 Zetetica

Vieta. 18 , ,

,

,

.

, ,

, ,

. 1202 Liber Abaci Leonardo di Pisa,

.

.

,

Cajori (1928), 1795, Charles Hutton

"Numeration" Mathematical and Philosophical Dictionary.

69

,

16 (Cajori, 1928).

Vieta .

141.421.356.24 ,

314.159.265.35

1.000.00 314.159.265.35

(Boyer & Merzbach, 1997). 1585 Stevin

La disme ( - The Tenth) (Arcavi et al 1991, Struik, 1966).

16 , ,

,

, 1616, Descriptio Napier


,

.

, ,

e, , i . Wallis , Barrow

Jones

. ,

Euler (Eves, 1981). Euler, Boyer & Merzbach

(1997),

. e,

,

i ,

. i,

Gauss, . , e i

Euler ,

0 1 ei+1=0.

Euler, f(x) x,

1734 1735 (Boyer & Merzbach,

1997).

70

13 ,

. ,

.

,

, -

, , ,

. ,

Vieta .

,

.

71

3.3

,

:

.

,

.

(Eves, 1981).

,

, ,

.

,

1700 ..,

(Van der Waerden, 2003).

,

, .

,

, .

, ,

,

xy x-y, (Van

der Waerden, 2003). Van der Waerden (2003),

,

, , ,

. ,

,

,

.

, ,

. (,

, 2):

72

'Ariqmj d t k mondwn sugkemenon plqoj, (, 1975)

,

, ,

. ,

, ,

.

, ,

, ., 1+3+5++(2n-1)=n2

, x2=(-x)

(Van der Waerden, 2003).

, (,

, .)

,

.

, Nesselmann

(Spangolo 2006, Eves, 1981, Heath, 2001).

, ,

17 .

, ,

,


,

,

, . Bashmakova

(1999a),

Vieta.

3 .. ,

. ,

73

, ,

, (Eves,1981).

, 189 .

,

,

.

,

(, 2003, Eves, 1981).

, al-

Khowarizmi (Van der Waerden, 2003). Heath (2001),

,

. , Eves (1981)

, , (2003)

Bashmakova (1999a),

,

, , .

. , .

= ( )( )( ) , .

,

(Bashmakova 1999a). Heath (2001),

,

.

,

,

(,

74

2003). , ,

,

, .

,

(Heath, 2001, Van der

Waerden, 2003). ,

(Bashmakova 1999a, Van der

Waerden, 2003). ,

, : , x2, , x3,

x4, x5 .

,

, , 1/x=x-1,

, x-2=1/x2 .

o

M ,

Bashmakova (1999a), .

. ,

2x5 2/x5 .

,

. , ,

,

.

x3-2x2+10x-1=5 :

o

M o

M

,

.

, (2003, . 110),

,

75

. ,

() ,

() .

, al-Khowarizmi,

al-jabr al-muqabala, al-jabr

algebra - (, 2003).

,

. Bashmakova (1999a),

()=.

, VI14, 90-15x2

54 15x2-36.

-15x2 . ,

(intermediate

computations), Bombeli.

, . Nesselmann

(Klein, 1968, Eves, 1981).

, Hultsch,

(2003, . 108-109),

, .

,

. ,

,

(determinate and indeterminate) .

Descartes,

, (Bashmakova, 1999a).

Vieta, Descartes, Fermat

17 , J.

Klein:

Vieta .

76

Vieta (,

2003, . 110). , Bashmakova (1999a),

Poincar,

20

(algebraic curves), .

,

. ,

. ,

. , Bhaskara x2-

45x=250 50 -5 (Struik, 1966). , Brahmagupta 7

ya yavattavat

ru rupa

. ,

, ka

kalaca , ni nilaca , .

yu pha (Eves, 1981,

Cajori, 1928). , pha 12 yu 1175

1217

15 =+

(Cajori, 1919). Bakhshali

sunya . sunya

(empty) ,

. ,

,

(Cajori, 1919).

.

al Khowarizmi,

825. ,

, ,

.

, , ,

(Struik, 1966, Eves, 1981, Cajori, 1928). ,

77

,

,

Horner.

, (Struik,

1966).

,

(Heath, 2001).

,

,

, .

,

,

, . ,

888 ..

. ,

,

,

=4, =5 =6 (Vogel, 1996).

, ,

,

. ,

, Leonardo di Pisa. 1202,

Leonardo Liber Abaci,

, al Khowarizmi,

. T Liber Abaci, ,

-

(Struik, 1966). ,

, ,

,

.

78

, . radices radix

x, x2 census. ,

radices 1221 x

2112 ,

factus ex...

,

. ,

Leonardo di Pisa (Eves, 1981,

Cajori, 1928, , 1996).

, 14 15 ,

.

,

. , Pacioli Summa de Arihmetica 1494

co x cosa

, ce x2 censo, cu x3 cuba, cece

x4 censocenso Stifel,

,

coss, zensus, cubus zenzizensus (Cajori, 1919, Boyer & Merzbach, 1997).

Regiomontanus ,

, .

,

(Boyer & Merzbach,

1997, , 1996). 1484,

Triparty Chuquet. ,

p~ , m~ 2

, 3 . res, chose,

cosa, coss premier .

, , Chuquet

.

5x, 6x2 10x3 .5.1, .6.2, .10.3. ,

79

9x0 .9.0 9x-2 .9.2.m .9.

seconds moins ( ). , Chuquet

.72.1 .8.3

.9.2.m, 72x / 8x3 = 9x-2.

Chuquet .4.1 .2.0, 4x=-2,

(Boyer &

Merzbach, 1997, Cajori, 1928).

m~

16

,

, ,

.

,

. , ,

: Coss Rudolf, Rechnung Apian

Arithmetica integra Stifel. ,

, ,

. ,

+ ,

. Stifel

, Harriot

Artis analyticae praxis 1631 (Boyer & Merzbach, 1997, Cajori, 1928).

, Ars magna Cardano Algebra

Bombeli ,

. ,

Ars magna ,

,

, Ferrari. Cardano

, Summa Pacioli

,

. p m,

. 1545,

80

10 40, 15-5+

15--5 , :

5p: m:15

5m: m:15

25m:m: 15 q~ d. est 40

Ars magna 1663,

:

5. p~ . . .15. m~

5. m~ . . m~ .15.

25.m.m. 15.quad. est. 40.

Cardano

.

,

(Boyer & Merzbach, 1997, Cajori, 1928).

Algebra 1560 Bombeli,

,

. ,

Bombeli

. Algebra

1579,

(Arcavi et al., 1991). Bombeli 1p.5Rm.4, 1 zenus

5 res 4 x2+5x-4, ,

,

(Boyer & Merzbach, 1997, Klein, 1968).

,

.

2x, 4x2 5x3 2 , 4 5 (Boyer & Merzbach, 1997,

Cajori, 1928). , Bombeli

,

3 1 2

81

.

,

Bombeli (Boyer & Merzbach, 1997).

,

Vieta 1591. Vieta

ars analytica

.

, nullum

non problema solvere. Vieta

- In artem analyticam isagoge, 1591

(Bashmakova 1999b). ,

, (Boyer &

Merzbach, 1997, Klein, 1968).

Bashmakova (1999b),

,

,

Vieta. , Regiomontanus Stifel , Leonardo

Cardano ,

Vieta . ,

, . Stifel ,

,

. Vieta,

,

.

,

. Vieta

,

2+CA+D=0, B, C D (Boyer

& Merzbach, 1997, Cajori, 1919). , Vieta

. ,

(homogeneous magnitudes) ,

82

. ,

, longitudo (),

latitudove (), planum ( ), solidum () .( 22).

: quadratum Ein C

3 solido B-cubusA , 3

, C ,

C . 23

cy3b-x x, y

b, c (Klein, 1968).

, Vieta

(cossist) + , =

= |-|. ,

in, applicare,

aequale. , quadratum, cubus

radix l

l. latus

l. 121 121 . ,

Van Schooten. ,

(Cajori,1928 , Bashmakova, 1999b).

22. Vieta 2+2=2 (Hairer et al., 2000).

In artem analyticam isagoge (Cajori, 1928) :

B in A quadratum plus D plano in A aequari Z solido BA2+D2A=Z3

Ad Logisticen speciosam notae priores,

1646

(Bashmakova, 1999b). (Cajori, 1928),

3+32+32+3 :

cubus, + quadrato in ter, + in quadratum ter, + cubo,

83

Vieta,

,

, . Vieta,

,

, .

,

, , ,

- ars analytice. 5

Isagoge, (Law of Zetetic-De Legibus

Zeteticis),

. , logistica

numerosa ,

logistica speciosa. , ,

, .

Vieta, x a, ,

,

(, 1996). Klein (1968)

. ,

- The modern concept of number is born (Klein,

1968, . 175).

, Vieta, ,

.

,

.

, .

,

,

(Klein, 1968, ,

1996). , Vieta

,

,

(Cajori, 1919).

84

. 16 , Stevin

,

Brgi , . , x4+3x2-7x

, Stevin

4 2 1 1 + 3 - 7

Brgi

iv ii i 1 + 3 - 7

Stevin 3/2

(Boyer & Merzbach, 1997, Klein, 1968).

17 ,

( Descartes),

,

Stifel Harriot. T 1634, Hrigone Cursus mathematicus

a, a2, a3, ., 1636 Hume L'Algbre de Vite d'vne

methode novelle, claire, et Facile

iii 3 (Cajori, 1928).

Hume, Boyer (1950)

3 Aiij iii. , Descartes

,

x3 xxx ,

, . Leibniz,

xm ,

Newton

(Cajori, 1928, Boyer, 1950).

Vieta, Descartes

. Geometry

Vieta,

85

. ( -

segments) (). ,

,

,

(Bashmakova, 1999b). Descartes,

(Klein,

1968). , Descartes

,

.

Newton Universal Arithmetic (Bashmakova, 1999b).

Vieta

Descartes Fermat,

.

. ,

, .

,

, -

(potential function)

, (were eventually recognized as forming a new analysis-

the calculus) (Kleiner, 1989, Boyer, 1959, . 98).

86

3.4

Boyer (1959),

,

,

. ,

, ,

,

( .., 1987).

, ,

,

,

, ,

(, 2005).

,

Bourbaki,

Riemann,

(, .., 1987).

800 ..,

,

,

, ,

, . 15

,

(, 2005).

, 16 ,

. ,

Vieta, Decartes

Fermat, ,

, .

87

, ,

(, 2005).

, 17 ,

,

, , -, ,

. ,

Stevin, , Kepler, Cavalieri, Torricelli, Fermat, Pascal Roberval,

Wallis, Gregory Barrow, ,

,

(, 2005). , ,

.

, ,

. ,

, .

, ,

y=xn.

, ,

. ,

, ,

.

Newton Leibniz 1675 (Maor, 2005).

Newton Leibniz,

,

. ,

.

, ,

.., (1987, . 210): ,

,

, ,

88

, ,

. ,

, Newton

Leibniz, . Leibniz,

:

(, 2005, . 59).

Newton, x, y,

(fluent), .

(fluxion) .

Methodus fluxionum, Newton,

,

. x t,

x ,

.

20 1665. , x ,

x(t)

x&

x&

dtdx . x y,

,

. ,

x& y& x&& y&&x& y&

xb

yy

-

xxaa - . ,

, x y,

. ,

Algebra

Wallis 1693, Newton, ,

Quadratura curvarum 1704 (Cajori, 1928, Boyer, 1959).

,

(moment) .

|x

|y

||x

||y

|x

|y

x&

89

y=xn =nxy& n-1 x& . , Newton,

(moment), (conatus) (indivisible), , e, a, o

(Boyer, 1959).

Newton, Principia, x, y, z,

p, q, r , X, Y, Z,

(Cajori, 1928). , Newton

, , , , , x, y, , ,

, , .

Newton ,

,

x&& y&& x& y&|x

|y

||x

||y

xaa

64 xdxaa64. (Cajori, 1928).

, .. (1987), Newton ,

. , ,

- quadratura

, ,

, .

Newton, ,

, .

, ,

x, , , x, x, x.

, Newton ,

, . ,

B. Taylor Methodus incrementorum 1715,

: = r adeoque p = r .

,

, ,

(Cajori, 1928, Boyer, 1959).

|x

||x

p& s& s&

Newton, Leibniz,

,

90

,

,

. (2005, . 68), C.

H. Edwards , Leibniz

, ,

Newton.

Leibniz,

26 1675,

y l dy

. 11 1675,

dx dy x y (Cajori, 1923).

dx dy,

Leibniz, Acta eruditotum 1684 d ad

dx, dx ad dy dx:dy, dydx . ,

, ddx:d 2y

dx3 (dx)3 3dx (Cajori, 1928).

Leibniz d : d, b ax , aeq. ba dx

b bax - (Cajori, 1923). Nova Methodus, Leibniz,

. x y

x, dx x dy

y (, 2005). , ,

calculus differentialis (Struik, 1966).

, Leibniz, Cavalieri

omn.pa=omn.yl=2

2y .

Mengoli Fabri

1659, omn. omnes linea,

.a, (Cajori, 1919, Rickey, 1992).

, .. (1987), omn.,

91

omnes (priores) magnitudines ad

omnes.

, 29 1675,

Leibniz : omn. omn.l, l. , Leibniz

, , :

Satis haec nova nobila, cum novum genus calculi inducant -

, , ,

d (Cajori, 1928, Rickey, 1992).

l

23. Leibniz

(Cajori, 1928, . B, . 243).

Leibniz ,

calculus summatorius- ,

. ,

92

Jacob Bernoulli, Acta eruditorum 1690.

Jacob, Johann Bernoulli, calculus integralis-

. Leibniz, , ,

,

(Cajori, 1928, Rickey, 1992).

Leibniz, 1695, Johann Bernoulli,

= n=-m .

, ,

Nieuwentijts ddx, d

md n

3x,

dddx d4x (Cajori, 1923).

18 , Leibniz

,

. Leibniz

, Euler,

,

(prominence) (Boyer, 1959, Rickey,

1992).

18 , Boyer (1959, . 243),

, Euler

Leibniz . , Leibniz,

Newton, ,

. ,

, ,

, . D Alembert ,

Euler, , ,

, Newton, prime

and ultimate ratios .

,

. D Alembert,

93

,

Leibniz Euler 18 .

,

,

,

(Boyer & Merzbach, 1997, Struik, 1966).

, Lagrange 1797, Theorie des functions

analytiques,

, .

Taylor,

fx, Lagrange

fx, fx, fx y, y, y, y (Cajori, 1923, Struik, 1966).

Arbogast, D

, .

a+bx+2.1

c x2+ a=Fa, b=DFa, c=DDFa .

D c

mDm

Dm

....3.2.1,

dx

xd , 22

.2.1

dxxd , x (Cajori, 1928).

, 1770,

Monge, Condorcet Legendre, Legendre,

1786

x (Cajori, 1928).

, ,

16 ,

. lim.

Simone Lhuilier 1786, : lim.q:Q Lim.xy

.

L Carnot 1797

Cauchy 1821. , Cauchy

. , x

94

, lim.(sin.x) 0, lim.((x1 ))

, lim.((sin.x1 )) . ,

lim. lim. 19 ,

,

. Limx=a,

Weierstrass

Hamilton { } (Cajori, 1923).

1lim

= == nn pLim .

=nlim

,

. G. H. Hardy 1908 )n1lim(

n

, :

Leathem Bromwich, : , , ,

, .

. , lim ,

nlim xlim a xlim =nlim

=xlim a =xlim

u' xu (Cajori, 1923, . 43).

19 ,

, ,

,

. ,

, .

, ,

, ,

17 18 Leibniz,

(Struik, 1966, . 182),

(2005, . 68) ,

,

95

.

,

, (2005, . 68) ,

Leibniz,

, ,

,

, Cajori (1928,

. B, . 181),

, Faust Goethe War es ein Gott, der diese

Zeichen schrieb?

96

3.5

,

. ,

Cantor

, . ,

, +1=, += ., ,

(Davis & Hersh, 1981).

, ,

. ,

, ,

Oresme (Boyer & Merzbach,

1997). ,

.

3.5.1

, 17

,

.

(1999), , ,

,

. ,

,

, ()()=(+) ()(+)=() (Bashmakova 1999a, , 1999). ,

97

Arithmetic Bakhshali 8 10 .. ,

+ . Cajori (1928),

, 12 7+, 12-7.

,

3 (Cajori, 1928). 4

1

, 12 ,

() ,

. Bombelli,

0-1069, -1069,

-1069 (Cajori, 1928).

Chuquet,

, .4.1 m~ .2.0, 4x = 2 (Boyer &

Merzbach, 1997). ,

, . ,

Stifel ,

numeri absurdi, (Boyer & Merzbach,

1997). , Cardano

numeri ficti, .

,


, (Vieta, Hariott .),

, Vieta

. ,

1629, Girard, Invention nouvelle en l algbre,


1657, Hudde ,

. ,

(Cajori, 1928).

98

,

. , +a

a a . ,

,

n p, . , as ar , , .

19 , + , 10

+10, .

-3,-2, -1, 0, +1, +2, +3, Continuum

Huntington 1917 (Cajori, 1928).

3.5.2

,

.

,

, 13 . ,

Chuquet 4+x2=3x,

. Chuquet : Tel

nombre est ineperible, (Boyer &

Merzbach, 1997, Cajori, 1928). ,

, , Cardano

Bombelli. , o Cardano

15-5+ 15--5 , 5p: m:15 5m: m:15.

Bombelli , , 4.p.R[0 4.~m ]

4-4+

. , ,

. di meno dm.

99

i= 1- . (+2i)(+i)=-2 : p.dm.2.via.p.dm.1 equal. m~ 2 (Boyer & Merzbach, 1997, Cajori, 1928).

,

. ,

, Cardano Tartaglia,

. ,

,


a- ,

, , Euler

a- , 1-

. Euler i

, 1777. , Euler

i . Euler

Jean

Bernoulli 1740 (Boyer & Merzbach, 1997, Cajori, 1928).

, , Hamilton, 1837,

(1, 2)=1+2 1- , 1903, Burkhardt (, )=+i (Cajori,

1928). i ,

Gauss Disquisitiones Arithmeticae

1801. , Gauss

,

.

, 1, -1 i, ,

,

directe, inverse, laterale Einheit (Boyer & Merzbach, 1997, Bell, 1992).

100

3.5.3

: () ,

() ,

. ,

,

. ,

. ,

,

-. (notion)

, 18 ,

.

,

.

: () ,

(Bombelli, Stifel

), () (Vieta, Descartes ), ()

, (Kepler, Galileo

) () (Descartes, Fermat

) (Kleiner, 1989).

, , ,

. ,

,

.

, ,

,

- (potential function). ,

, . ,

101

. Newton

Leibniz ,

(Kleiner, 1989, Boyer, 1959).

1692 Leibniz function -

.

1718 Johann Bernoulli.

, Leibniz :

x, x,

x, y, x (Cajori,

1928).

, Leibniz Johann Bernouli

, , n,

B, F, G, C , . x z b,

d, ., : x x z. Euler,

f(x) 1734 Commentarii Academiae Scientiarum Petropolitanae

(Cajori, 1928). , 1748, Euler Introductio in Analysin

Infinitorum,

(Kleiner, 1989).

x 1

x;y x 1 2

r.1 x

3.5.4

,

, .

,

.

, .

102

Regio

, ,

,

, .

,


.

montanus, .

, ,

Regiomontanus, Rheticus,

. Vieta,

,

. ,

2-2

)-(2

)(

A

BA

a

ba +=

+

,

1583 Thomas Geometria rotundi,

tangent secant

sin., tan., sec., sin.com, tan.com, sec.com.

(Cajori, 1928).

,

2Bb

Fink

. (Boyer & Merzbach, 1997). Thomas Fink

.

,

17 18 ,

. ,

s, S,

sin, sin., sen., sn , , cs, S.2., Cos., cos. cos,

t, T, tang., tang, tng, tg tan., (Cajori, 1928).

, ,

Introductio Euler.

103

, sin., sin, sn

, cos., cos, cs .,

(Boyer & Merzbach, 1997, Cajori, 1928):

sinx= 1-2

- -1--1 xx ee , cosx= 2

-1--1 xx ee + eix = cosx + 1- sinx .

, y2=1

x2-y2= 1, . 1757,

x2+

Riccati,

Sh.x Ch.x

. , Sc. Cc.

, ,

, Lambert

sinh(x) cosh(x) . ,

,

sinx, cosx eix

:

sinh(x)= 2- -xx ee , cosh(x)=

2

-xx ee + ex=sinh(x) + cosh(x).

,

3.5.5

,

, .

(Boyer & Merzbach, 1997, Cajori, 1928).

.

,

,

. ,

,

104

(, 1989).

,

1

20+21x+22y=0

30+31x+32y=0

,

10.32+11.32x-12.30-12.31x=0

x ,

x

:

a10.a21.a32+a11.a

.

Leibniz De L Hospital, 693. ,

Leibniz

, . ,

3 2 ,

:

10+11x+12y=0

. , Leibniz,

, - .

, Leibniz ,

:

10.22+11.22x-12.20-12.21x=0

:

10.21.32+11.22.30+12.20.31-10.22.31-11.20.32-12.21.30=0

22.a30+a12.a20.a31-a10.a22.a31-a11.a20.a32-a12.a21.a30=0

105

Leibniz :

10.12.32 10.22.31

11.22.30 = 11.20.32

12.20.31 12.21.30

.

, C. ,

2x2, 3x3 4x4,

Mac Laurin, 1729.

Leibniz, (, 1989, Cajori 1928).

,

Cramer,

Introduction a l Analyse des Lignes Courbes algebriques-

, 1750. ,

5

A+By+Cx+Dy2+Exy+x2=0, Cramer 5

5 .

,

. , 2x2 :

y= 12211221

--

ZZAZAZ z= 1221

1221

--

ZZAA

, Cramer

.

Bzout Euler

:

(, 1989).

1771, Vandermonde, ,

. ,

106

111 + 2

12 + 3

1 = 0

2 2 211 + 2 + = 0

2 3

a

b

a

b

.

, ,

2x2 :

=

a

b

- b

a

3x3

Vandermonde, ,

,

( , 1989,

Cajori, 1928).

, . Laplace

ax2+2bxy+cy2, (a, b, c)

b

a b c

a b c ab c ac

a b

, Laplace Lagrange,

,

,

, : 1a, 1b, 1c, 2a, 2b, 2c, .

, ,

. , 3x3

(1a 2b 3c), Binet (, 1989).

determinantem - ,

Gauss, 1801 . Gauss,

Disquisitiones Arithmeticae,

107

Cauchy Binet,

1815, determinant

bb-ac. , Lebesgue [g, i]

g i,

g, h i, k, (Cajori,

1928).

,

1812, . Cauchy

h kg i

, ,

. , Binet,

, resultante,

Laplace (, 1989).

Caley, 1841, :

'''

, a , '' a

a,

'''''' : , -, -+-+-,

. (Cajori, 1928).

,

.

, , ,

,

, .,

.

108

4

, ,

.

, ,

, .

,

, ,

.

,

,

(, 2001).

,

, ,

, .

,

,

,

, .

,

,

.

,

, ,

(, 2000).

109

4.1

,

,

, , ,

,

.

,

,

, ,

.

,

,

, .

,

,

. Bruner,

, ,

,

. : )

, , )

, ,

) ,

, (abstract)

(, 2000, , 1999).

,

. ,

,

. ,

,

.

(, , , )

110

,

(, 2000).

,

, ,

,

. ,

,

, ,

,

,

(, , ,

, ) ,

( , 1998, Hiebert, 1997, NCTM, 2004).

111

4.2

,

, ,

,

. , ,

, .

(symbolic relational

structures),

. ,

,

- - (Steinberg, 2000).

,

) - (sign/symbol) )

-, (object/reference, context) ) (concept)

Steinberg (2000),

.

,

(reciprocal)

. ,

,

(balance),

.

.

, ,

,

. Steinberg

(2000, . 141),

: (conveyance, signifier) ()

112

, (information, signified)

().

,

.

.

,

,

.

,

, .

,

(process) (concept).

, ,

(action) (entity),

.

.

,

(Gray & Tall,

1994, 2000, Hiebert & Lefevre, 1986 Gray & Tall, 1994).

, ,

- ,

, ,

. Gray & Tall, (2000),

,

, (interiorisation),

(encapsulation), (reification) ..

113

-

, Tall (2004),

,

. ,

(embodied), (proceptual) -

(formal-axiomatic) . ,

, .

. ,

.

,

,

,

.

Gray & Tall (1994, 2000),

procept,

(process) (concept)

. -

(elementary procept), :

.

- (procept), -

(elementary procepts) .

, -,

,

.

, ,

.

, ,

114

,

, .

,

(Tall, et al, 2001).

,

Gray & Tall (1994, . 1), : The ambiguity of

notation allows the successful thinker the flexibility in thought to move between the

process to carry out a mathematical task and the concept to be mentally manipulated

as part of a wider mental schema. ,

, ,

.

Sfard (1991),

,

(notion) , (operational) (structural).

. ,

(. 4).

, (deep

ontological gap) (. 4). , ,

, . , ,

,

. , ,

,

.

,

115

.

.

.

Sfard (1991)

: () (interiorisation) , ()

(condensation)

() (reification),

(operational) ,

, (structural) ,

.

Sfard, (1992)

.

,

.

,

.

,

,

. ,

. ,

, ,

,

. ,

,

,

.

,

. ,

, ,

.

, ,

116

(Sfard,

1995, 1992)

, ,

,

, ,

,

Vosniadou & Verschaffel, (2004), . ,

, ,

. ,

, ,

. ,

,

, .

, ,

,

(Vlassis,

2004).

,

Arcavi (1994, 2005),

, (symbol sense),

.

(behavior), ,

,

.

117

() ,

, ,

,

() ,

,

,

()

. ,

,

()

,

.

,

()

.

, Arcavi (1994) ,

.

,

,

.

118

4.3

,

, .

,

(it is subtle and difficult to

learn) (Davis & Hersh, 1981, Schoenfeld & Arcavi, 1988).

.

. ,

,

.

,

. ,

,

(Hughes, 1992,

1996).

,

. ,

.

,

(Schoenfeld & Arcavi, 1988).

-

,

. ,

. ,

119

(Tall, 1994).

, ,

. ,

,

, -

,

. , Vlassis

(2004), (structural signifier),

-

(operational signifier),

. ,

.

- Vlassis

(2004) :

() , , ,

, ,

,

() , ,

, ,

() , ,

.

, ,

, ,

. ,

-,

.

120

,

-

(Vlassis, 2004).

, Sfard (1991, 1992,

1994), ,

, .

,

,

,

. ,

,

,

, .

, ,

,

. ,

,

(Sfard,

1992).

,

, .

,

,

,

, .

, ,

,

121

,

(Sfard, 1994, 1995).

Sfard (1991), ,

,

( 5).

(Bourbaki, 1934)

(Skemp, 1971)

Property of a set or the class of all sets

of the same finite cardinality

0

1

5. (Sfard, 1991).

Sfard (1992),

, ,

, . ,

. : ()

() ,

. , Sfard (2000),

Ford (2003),

,

. ,

,

.

122

,

,

. ,

( ) .

,

, ,

. , ,

,

, Hilbert (Sfard, 1992).

-

,

, ( )

,

, .

, Tall et al. (2001),

( 6).

(process) (concept)

4

3+2

-3 3

3/4 ,

3+2x

v=s/t

y=f(x)

dy/dx

f(x)dx

6. (Tall et al., 2001).

123

, ,

,

, ,

(Tall et al., 2001).

, ,

.

,

, , . ,

, ,

, .

,

. ,

,

,

(proceptual thinking) (Tall et al.,2001,

Gray & Tall 1994).

Arcavi (1994, 2005),

,

,

, .

, ,

, .

,

-symbol sense

- Informal sense-making in formal

mathematics.

()

10% 10%;,

124

, ,

, ,

0,99.

() , () x2 y2=0

() (x-a)2+y2=1 0, 1, 2, 3, 4, 5, 6,7 8 ;, ,

,

.

() 3x+5=4x,

,

2412

++

xx =2

(4x+2)=2(2x+1),

,

, .

()

-1, , +1 , +1, +2 -2, -1, ,

,

,

.

() , y=b

, x=0 m=0 y=mx+b,

,

.

, , Arcavi (1994)

,

,

,

. , ,

,

,

125

. , Arcavi (2005), Sfard

(2000)

,

.

(serious tramp)

, .

, ,

Arcavi (2005), ,

,

,

,

.

126

4.4

.

,

,

(NCTM, 2004). (1989),

,

,

,

, ,

.

,

, - ,

, ,

,

, .

,

, ,

, . Sfard

(1995), ,

. ,

, ,

. ,

, ,

.

, ,

,

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