The History of Mathematical Proof in Ancient Traditions

288 François charette

Identifying the origin of al-Khwārizmī’s algebra presented more serious

diffi culties, since it diff ered from both the Greek and Indian algebraic

traditions. But for Cantor, Islamic algebra could under no circumstances

be autochthonous: it could only feature Greek and Indian elements, so he

assumed an amalgam of both traditions. In general, however, Cantor was

convinced that there existed two separate schools in Islamic mathematics,

bringing about a fundamental opposition between the disciples of Indian

methods and those, more numerous, who strictly adhered to the Greek

tradition.^51 Cantor’s entire section on Islamic mathematics shows precisely

his constant concern for associating every single mathematician or result

within one of the two groups.

How did the Arabs handle the Greek axiomatic–deductive methods?

Now we come to the more crucial question: were the ‘Arabs’ up to dealing

with Greek thought? How did ‘Oriental’ mathematicians come to terms

with the Euclidean axiomatic–deductive method? Hankel described the

nature of Euclid’s infl uence on Islamic mathematical practice with the

following words:

In the same way, one zealously occupied himself with the logical analysis of

his  (Euclid’s) method, his defi nitions and axioms, and one used his demonstra-

tions  for the exemplifi cation of the rules of formal logic in a similarly pedan-

tic  manner as what our German logicians still liked to do almost until our

century. 52

Th us the ‘Greek oversubtlety’, as Hankel says ( griechische Spitzfi ndigkeit ),

was a level too high for the Arabs. Its perversion he exemplifi ed with

[pseudo-]T. ūsī’s vain attempt to prove Euclid’s postulate of parallels. 53 T o

this observation Hankel adds the following statement concerning the status

and use of demonstrations in Arabic treatises:

Despite their even doctrinary acquaintance with the demonstrative method [of the

Greeks], the Arabs, most of the time, have refrained from providing the demonstra-

tions, and have dogmatically strung the theorems and rules together, exactly as the

51 Cantor 1894 : 718–19.

52 Hankel 1874 : 272.

53 Th e Arabic text of a recension of Euclid’s Elements wrongly attributed to Nas. īr al-Dīn al-T. ūsī

had been printed in Rome in 1586 and was available through Wallis’ analysis thereof in his

history of algebra, his interpretation being possible thanks to the collaboration of orientalist

Pococke. See Molland 1994 and Stedall 2001.