The History of Mathematical Proof in Ancient Traditions

286 François charette

Th us Egyptian theory is inductive, and Greek theory is deductive. Th ales,

when he obtained his geometrical knowledge in Egypt, must have off ered

diff erent kinds of demonstration than the Egyptians did (for example, the

theorem stating that the diameter divides the circle in two equal parts), for

the simple reason that he had a Greek mind! Cantor puts it as follows:

As a Greek he generalized, as a pupil of Egypt he grasped through the senses

what he then made comprehensible to the Greeks. It was an ethnic characteristic

[ Stammeseigentümlichkeit ] of the Greeks to get to the bottom of all things, and,

starting from practical needs, to reach speculative explanations. Nothing of the sort

with the Egyptians. 44

With the Egyptians, Cantor speculates, either the fi gure suffi ced for the

proof, or it was done through computation of the areas of both semicircles

according to the same, possibly uncomprehended, rule.

Th e problematic status of Islamic mathematics

Th e confrontation with Arabic mathematical writings and the bibliographi-

cal information about it forced historians of mathematics to adopt a diff erent

approach than with Indian mathematics. First it became increasingly obvious

that a large number of Greek mathematical works, including virtually all major

ones, had been not only translated into Arabic but also studied, commented

upon, adapted and transformed. Greek mathematics had been thoroughly

assimilated within Islamic culture. On the other hand, Indian infl uences

were obvious in several works, such as the arithmetic of al-Khwārizmī, or (it

was presumed) in the treatise on practical geometrical constructions by Abū

al-Wafāʾ. How was it possible, then, to treat Islamic mathematics within the

category ‘Oriental’? Which status did historians of mathematics grant to Arabic

mathematics? Another, related problem was of course the question of its origi-

nality. In this respect late nineteenth-century historians of mathematics proved

surprisingly severe, in spite of the excellent works of Franz Woepcke.

Th e question of originality

Although he presented an excellent summary of the available evidence

(mostly thanks to Woepcke’s works), Hankel claimed at the outset that the

Arabs had added little to what they had received from the Greeks and the

44 ‘Als Grieche hat er verallgemeinert, als Schüler Aegyptens sinnlich erfasst, was er dann den

Griechen wieder fassbarer gemacht hat. Es war eine griechische Stammeseigentümlichkeit,

den Dingen auf den Grund zu gehen, vom praktischen Bedürfnisse zu speculativen

Erörterungen zu gelangen. Nicht so den Aegyptern.’ Cantor 1894 : 140.