308 g e o f f r e y L l o y d
that he promoted in part to create a gap between demonstrative reasoning
and the merely plausible arguments of orators and others. Whether or how
far Aristotle was infl uenced by already existing mathematical practice is a
question we are in no position to answer defi nitively. But certainly his was
the fi rst explicit defi nition of such a style of demonstration, and equally
clearly soon aft erwards Euclid’s Elements exemplifi ed that style in a more
comprehensive manner than any previously attempted.
From this it would appear that it was the particular combination of
cross-disciplinary and interdisciplinary rivalries in Greece that provided
an important stimulus to the developments we have been discussing.
Elsewhere in other mathematical traditions there was certainly competi-
tion between rival practitioners. It is for the comparativist to explore how
far the rivalries that undoubtedly existed in those traditions conformed to
or departed from the patterns we have found in Greece.
Th en on the second question I posed of the consequences of the proposal
by certain Greeks themselves of a hierarchy in which axiomatic–deductive
demonstration provided the ideal, we must be even-handed. On the one
hand we can say that with the development of axiomatics there was a gain
in explicitness and clarity on the issue of what assumptions needed to be
made for conclusions that could claim certainty. On the other there was evi-
dently also a loss, in that the demand for incontrovertibility could detract
attention from heuristics, from the business of expanding the subject and
obtaining new knowledge. Th is is particularly evident when Archimedes
remarks that conclusions obtained by the use of his Method had thereaft er
to be proved rigorously using the standard procedures of the method of
exhaustion. If we can recognize – with one Greek point of view – that there
was good sense in the search for axioms insofar as that identifi ed and made
explicit the foundations on which the deductive structure was based, we
should also be conscious – with another Greek opinion indeed – of a pos-
sible confl ict between that demand for incontrovertibility and the need to
g e t o n w i t h t h e b u s i n e s s o f d i s c o v e r y.Bibliography
Barker , A. D. ( 1989 ) Greek Musical Writings , vol. ii. Cambridge.
( 2000 ) Scientifi c Method in Ptolemy’s Harmonics. Cambridge.
Brownson , C. D. ( 1981 ) ‘ Euclid’s optics and its compatibility with linear
perspective ’, Archive for History of Exact Sciences 24 : 165 –94.