Some elements in the history of Arab mathematics

Some elements in the history of Arab mathematics From arithmetic to algebra Part 2 Palerme, 25 -26 novembre 2003

Summary Introduction The roots of algebra : arithmetic or geometry The roots of al-Khwarizmi's algebra The roots of Arabic algebra Conclusion Palerme, 25 -26 novembre 2003 2

Luis Radford 1996 The roots of algebra : arithmetic or geometry ? • • • He discusses some hypotheses on the origins of Diophantus’s algebraic ideas. He suggests that the historical conceptual structure of the concept of unknown and the concept of variable are quite different. He raises some questions about teaching algebra in high schools. Palerme, 25 -26 novembre 2003 3

Luis Radford 1996 • Classical interpretation : Babylonian math know the general formula for solving quadratic equations without being able to express it as such, because they lacked the symbols to do so. • Interpretation of Hoyrup (1990) : Babylonian algebra cannot have been arithmetical. Instead it appears to have been “naïve” , non deductive geometry, consisting of “cut-and-paste geometry”. • Interpretation of Radford : The procedure of solving some types of problems consists of an arithmetical method of false position, based on an idea of proportionality. Palerme, 25 -26 novembre 2003 4

Luis Radford 1996 • • Unknowns and variables While the unknown does not vary, a variable designate a quantity whose value can change in Diophantus' Arithmetica "the kew concept of unknown (the arithme) is not represented geometrically. Nichomacus uses empirical set values, that is a concrete-arithmetical treatment of numbers, Diophantus deals with abstract set values. "The propositions in his theory of polygonal numbers are supported by a deductive organization. He is concerned about variables, not through the concept of function but through the concept of formula Palerme, 25 -26 novembre 2003 5

Luis Radford 1996 Implications for the teaching of Algebra 1. Introducing cut-and-paste algebra to facilitate acquisition of basic algebraic concepts. 2. Introducing certain elements of the "false position" method prior to introducing the concept of the unknown in solutions of word problems. 3. Use proportional thinking as a useful link to algebraic thinking. 4. Introduce an appropriate distinction between the concepts of unknown and variables, using historical ideas. 5. Try to develop the concept of abstract set value in a deductive context as a prerequisite to a deep learning of the concept of variable. Palerme, 25 -26 novembre 2003 6

The roots of al-Khwarizmi's algebra • • Indian origin Greek origin Babylonian tradition Our conclusion : Epistemological arguments Palerme, 25 -26 novembre 2003 7

The roots of al-Khwarizmi's algebra Yes : Indian origin • He has written a treatise on Hindu-Arabic numerals and an astronomical table Sindhind zij No : it is not from Indian origin (Rodet, 1878) • Al-Khawarizmi does not know negative numbers. Hindus use them abondantly as they use « 0 » . • Al-Khwarizmi "al-jabr" means "completion", it is the process of removing negative terms from an equation. • There are six canonical forms for Arabic equations, while Indian had only one canonical equation. Palerme, 25 -26 novembre 2003 8

The roots of al-Khwarizmi's algebra Al-Khwarizmi "al-jabr" means "completion", it is the process of removing negative terms from an equation 50 x 2 + 300 - 6 x = 10 x - 100 - x 2 Arab method : Complete each side by removing negative terms 50 x 2 + 100 + x 2 + 100 = 10 x + 300 + 6 x Indian method : Sustract from right side the unknown and from the left side the number even if it is negative, so all unknowns are on the left and numbers on the right. 50 x 2 + 300 - 6 x - 300 - 10 x - (-x 2) = 10 x -100 - x 2 - 300 - 10 x - (-x 2) Palerme, 25 -26 novembre 2003 9

The roots of al-Khwarizmi's algebra There are six canonical forms for Arabic equations, while Indian had only one canonical equation Arab canonical equation : ax 2 = bx ax 2 = c bx = c ax 2 + bx = c ax 2 + c = bx x 2 = bx + c with a, b and c strictly positive numbers Indian canonical equation : ax 2 ± bx = ±c with a, b and c positive numbers or nil Palerme, 25 -26 novembre 2003 10

The roots of al-Khwarizmi's algebra Yes : Greek origin (Rodet, 1878) : "he is purely and simply a disciple of the Greek School" (a) He never uses negative numbers. (b) He gives cut-and-paste geometrical proofs for solutions of quadratic equations. (c) No : (Gandz, 1931) : "Euclid's Elements" in their spirit and letter are entirely unknown to al. Khwarizmi who has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind. « (d) May be : Rashed (1997) suggested an intermediary position : al-Khwarizmi's : "treatment was very probably inspired by recent knowledge of Euclid's Elements". Palerme, 25 -26 novembre 2003 11

The roots of al-Khwarizmi's algebra Babylonian tradition : Hoyrup • Babylonian algebra "did not deal with known and unknown numbers represented by words or symbols. Strictly speaking it did not deal with numbers at all, but with mesurable line segments. . . • The operations used to define and solve these problems were not arithmetical but concrete and geometrical. . . • In all cases, the geometry involved can be characterized as "naïve". . . This naïve geometry is fairly similar to the proofs given by al-Khwarizmi. . Palerme, 25 -26 novembre 2003 12

The roots of al-Khwarizmi's algebra Epistemological arguments 1 When we look at the structure of al-Jabr wa almuqabala, we notice specific features who are common to Hawa'i arithmetic and absent from theoretical arithmetic and from Indian arithmetic : 1. He starts by a very short description of numbers. 2. Then he introduces all tools of algebra. 3. This textbook is a completely rhetorical algebra. 4. He uses unit fractions and expresses all fractions in function of them. 5. He works in numerical settings : positive coefficients. Palerme, 25 -26 novembre 2003 13

6. 7. 8. 9. The roots of al-Khwarizmi's algebra Epistemological arguments 2 He deals with "shay“ as if it was a known number. But he stays in the realm of geometric conception of numbers (lines, surfaces and solids). He uses cut-and-paste geometrical arguments. He gives applications of algebra techniques to four kinds of problems : (1) Numerical ones (for example : division of ten in two parts). (2) Commercial problems. (3) Mensurations of geometrical figures. (4) Inheritance problems (this section is about the half of the book). Palerme, 25 -26 novembre 2003 14

The roots of al-Khwarizmi's algebra Our conclusion It is clear from the contents of al-Khwarizmi's algebra that it is in fact part of the usual arithmetic textbooks. The author intention is to help people to solve their practical problems. He had collected techniques people used to transmit orally without it being recorded in any book. Palerme, 25 -26 novembre 2003 15

The roots of Arabic algebra 1. 2. 3. 4. 5. 6. A numerical origin Babylonian geometrical influences Euclidian geometrical influences A geometric theory of equations An arithmetic of polynomial expressions Algebra becomes a section of textbooks of Indian arithmetic Palerme, 25 -26 novembre 2003 16

The roots of Arabic algebra The epistemical status of algebra did not stabilise for centuries. 1. The philosopher al-Farabi (d. 950), for example, considered arithmetic as a science having a theoretical part and a practical one ; but for him algebra was not a science but one technique common to arithmetic and geometry. 2. Ibn Sina (Avicenna 980 -1038) subdivised arithmetic into Indian calculus and the art of algebra", that is into two different subjects. Palerme, 25 -26 novembre 2003 17

The roots of Arabic algebra A business school origin No : Abu l-Wafa (d. 998) • For al-Khwarizmi, algebra is a section of business arithmetic • However Abu l-Wafa who is the author of the most read business textbook says : All transactions are solved by the use of one unique Euclidian proposition, the one requiring finding an unknown placed in a proportion. Palerme, 25 -26 novembre 2003 18

The roots of Arabic algebra A business school origin Yes : al-Karaji (d. 1029) : al-Kafi fi al-hisab • He considers the first one - when the unknown is in a proportion • He includes a section of algebra in his business arithmetic textbook. • He completes by many solved numerical and practical problems. Al-Karaji is also the author of two important books of algebra al-Fakhri and al-Badii . Palerme, 25 -26 novembre 2003 19

The roots of Arabic algebra Babylonian geometrical influences • • • al-Khwarizmi's approach is a concrete-arithmetical treatment of numbers, and does not deal with abstract set values. references to Euclid are non-existent and his direct influence never acknowledged. However, al-Khwarizmi's innovation with respect to Babylonians consists in introducing the unknown into account through the problem solving-procedure, being of the object of calculation. Palerme, 25 -26 novembre 2003 20

The roots of Arabic algebra Euclidian geometrical influences : Thabit ibn Qurra (826 – 900) (a) al-Khwarizmi’s pragmatic proofs which are not based on Euclid's Elements did not please him. (b) He then proofs them referring directly to Euclid, (c) He has a more general and more intellectual character than that of al-Khwarizmi. In fact he deals with abstract set values and he never gives a numerical example. Palerme, 25 -26 novembre 2003 21

The roots of Arabic algebra Euclidian geometrical influences : Abu-Kamil (850 – 930) (a) al-Khwarizmi’s pragmatic proofs are said « visual » . (b) He introduces proofs referring directly to Euclid. (c) But he has a concrete-arithmetical treatment of numbers, and does not deal with abstract set values. Generic examples (implicite induction). (d) He enriches the toolbox by including in it numerous algebraic identities proved geometrically and never gives a numerical example. Palerme, 25 -26 novembre 2003 22

The roots of Arabic algebra Euclidian geometrical influences : Al-Karaji (d. 1029), Omar al-Khayyam (1048 -1131) and as-Samaw'al (1130 -1174) (a) They take up the geometric proofs of al-Khwarizmi and those of Abu Kamil and extend them systematically. (b) They complete the algebraic toolbox by placing in it all the arithmetic propositions on whole numbers, on fractions and on quadratic irrationals, adding algebraic identities, all proved geometrically using Books II and VII to X of the Elements. Palerme, 25 -26 novembre 2003 23

The roots of Arabic algebra A geometric theory of equations Omar al-Khayyam (1048 -1131) , 1. He classify all third degree equations and solve them. 2. For each of the types, he finds a construction of a positive root by the intersection of two conics 3. He works entirely within a Euclidean framework Palerme, 25 -26 novembre 2003 24

The roots of Arabic algebra An arithmetic of polynomial expressions 1. The major obstacle encountered in legitimizing the algebraic reasoning concerns the nature of the product of numbers. 2. Al-Karaji gets around this difficulty by creating the field of "known numbers" in parallel with the field of "unknown numbers". 3. “Operating in the field of knowns keeps them in this field no matter what the operation” 4. 3. It is thus no longer a matter of reasoning on geometric figures but directly on numbers Palerme, 25 -26 novembre 2003 25

The roots of Arabic algebra An arithmetic of polynomial expressions Al-Karaji (d. 1029) and as-Samaw'al (1130 -1174) 1. Representing polynomials by tables Palerme, 25 -26 novembre 2003 26

The roots of Arabic algebra Algebra becomes a section of textbooks of Indian arithmetic The legacy of al-Karaji is found in later arithmetic textbooks written in North Africa (beginning in the XIIth century) : They combine three heritages : (1) Indian arithmetic. (2) algebra as an autonomous science, with no geometrical proofs. (3) Business textbooks heritage. how to find an unknown from known quantities or numbers : - the unknown placed in a proportion - the double false position method - algebra Palerme, 25 -26 novembre 2003 27

Conclusion We have been analysing the roots of Arabic algebra, using Luis Radford and Hoyrup's schemes. Two distinct phases have been pinpointed : (1) The birdh of algebra as a new art or science. (2) After al-Khwarizmi, Arabic algebra has moved from cut-and-paste techniques to arithmetic of polynomials by gaining credibility through Euclidian justifications (3) After al-Karaji it became autonomous as a science with its own way of reasoning and its proper proofs. (4) This autonomy is illustrated by business textbooks of North Africa combining Indian arithmetic with algebra. Palerme, 25 -26 novembre 2003 28