The History of Mathematical Proof in Ancient Traditions

(Elle) #1

Th e pluralism of Greek ‘mathematics’ 295


about that) related to, or used, one or other branch of mathematics, namely
Hippocrates of Chios. He was responsible not just for important particular
geometrical studies, on the quadrature of lunules, but also, maybe, for a fi rst
attempt at systematizing geometrical knowledge, though whether he can
be credited with a book entitled (like Euclid’s) Elements is more doubtful.
Furthermore in his other investigations, such as his account of comets,
reported by Aristotle in the Meteorology , he used geometrical arguments to
explain the comet’s tail as a refl ection.
Yet most of those to whom both ancient and modern histories of pre-
Euclidean Greek mathematics devote most attention were far from just
‘mathematicians’ in either the Greek or the English sense. Philolaus,
Archytas, Democritus and Eudoxus all made notable contributions to one
or other branches of mathēmatikē , but all also had developed interests in
one or more of the studies we should call epistemology, physics, cosmol-
ogy and ethics. A similar diversity of interests is also present in what we are
told of the work of such more shadowy fi gures as Th ales or Pythagoras. Th e
evidence for Th ales’ geometrical theorems is doubtful, but Aristotle (who
underlines the limitations of his own knowledge about Th ales) treats him as
interested in what he, Aristotle, termed the material cause of things, as well
as in soul or life. Pythagoras’ own involvement in geometry and in harmon-
ics has again been contested, 2 and the more reliably attested of his interests
relate to the organization of entities in opposite pairs, and, again, to soul.
Th ese remarks have a bearing on the controversy on the question of
whether deductive argument, in Greece, originated in ‘philosophy’ and
was then exported to ‘mathematics’, 3 or whether within mathematics it was
an original development internal to that discipline. 4 Clearly when neither
‘philosophy’ nor ‘mathematics’ were well-defi ned disciplines, it is hard to
resolve that issue in the terms in which it was originally posed, although, to
be sure, the question remains as to whether the Eleatic use of reductio argu-
ments did or did not infl uence the deployment of arguments of a similar
type by such fi gures as Eudoxus.
If we consider the evidence for the investigation of what Knorr, in other
studies,^5 called the three ‘traditional’ mathematical problems, of squaring
the circle, the duplication of the cube and the trisection of an angle, those
who fi gure in our sources exhibit very varied profi les. Among the ten or so
individuals who are said to have tackled the problem of squaring the circle


(^2) Burkert 1972.
(^3) Szabó 1978.
(^4) Knorr 1981.
5 Knorr 1986.

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