280 François charette
precise formulation of the idea opposing the apodictic rationality of Greek
mathematical practice to the more intuitive one of the Indians. He con-
tended that whereas the Greeks were trying to recognize that which is
given ( das Gegebene ) and has a form ( das Gestaltete ), the Indians were
creating forms ( Gestaltungen ) through active research, satisfying them-
selves to know that something exists, without concern for knowing how it
is so. 11 Both styles were one-sided, but necessary. Th e rapid development of
modern mathematics, according to Arneth, owed much to the mingling of
these two contrasting styles of mathematical practice.
Let us turn our attention to Hankel, Cantor and Zeuthen’s writings.
Hankel’s original views on mathematical demonstration contrast with the
coarse dogmatism of Biot, on the one side, and the more sophisticated
conservatism of Cantor, on the other. 12 Hankel devoted thirteen pages to
the Greek concepts and practice of analysis and synthesis, presenting a
competent and inspiring survey of the topic. 13 For him, the painstaking
care associated with analysis and synthesis and the ‘dry dogmatic syllogism’
so peculiar to Greek mathematicians was not a ‘useless burden’ to them; in
fact, he says, ‘for their mental strength, this form, so annoying to us, was
the appropriate one’. 14
Hankel’s account of Indian mathematics is still permeated with the
German romantic fascination for India and its philosophy. Like Playfair, he
noted the occasional and partial use of certain forms of demonstration in
Indian mathematical texts: ‘Th ere is also little to fi nd among the Indians of...
a practice of proof. Only here and there does a commentator add some
remarks to the rules and theorems, which can pave the way to their deriva-
tion.’^15 Indian geometry, radically diff erent from that of the Greeks, was also
characterized by the absence of demonstration in the traditional (Greek)
sense; there is simply a reference to a fi gure accompanied by the exclamation
‘Look!’ Th is kind of ‘illustrative demonstration’ ( Anschauungsbeweis ), as we
have seen in our previous quotation of Günther, strongly fascinated histori-
ans of mathematics. Cantor saw this as a typically Indian mode of thought:
‘Th is form of demonstration, which does not appear in Brahmagupta, must
certainly be considered as (typically) Indian. Combined with the algebraic11 Arneth 1852 : 141.
12 Cantor’s views are analysed further below.
13 Hankel 1874 : 137–50.
14 Hankel 1874 : 208.
15 ‘Von solcher Entwickelung und Beweisführung ist nun auch bei den Indern nicht eben
so viel zu fi nden. Nur hie und da fügt ein Commentator zu den Regeln und Sätzen einige
Bemerkungen, welche den Weg zu deren Ableitung geben können.’ Hankel 1874 : 182–3.