1 The Birth of Science - MSRI

156 6. The Hellenistic Scientific Method
Typical definitions fit for creating new scientific terms are the following,
made by Archimedes:
. . . we suppose the following: if an ellipse, its major axis staying still,
rotates so as to get back to its initial position, the figure enveloped by
the ellipse will be called a lengthened ball-shape [ ¦
]. If an ellipse, its minor axis staying still, rotates so as to get
back to its initial position, the figure enveloped by the ellipse will be
called a flattened ball-shape [ ¦]. For either ballshape.
. . 28
What we have translated as “lengthened”, “ball-shape” and “flattened”
are everyday Greek words, which acquire a new and precise meaning after
this point in the text, becoming abbreviations or tags for certain long expressions
made up of other terms already known. Note the contrast with
the essentialist mode of definition: Archimedes is not at all troubled by
the fact that his ball-shape may look nothing like a ball, but rather like a
needle or a lentil. 28a We will call definitions of this type nominalist; they
are common in the writings of Hellenistic scientists. 29 There is clearly a
close connection between a nominalist notion of definition and linguis- page 201
tic conventionalism, which, as already seen, arose at that time. 30 The use
of nominalist definitions in mathematics was accompanied in Hellenistic
times by a new concept of mathematical entities. For example, we know
from Proclus that Apollonius of Perga described the origin of fundamental
geometric concepts from everyday experience, saying for instance that the
notion of a line arises from considering things such as roads about which
one can say “Measure its length” without fear of misunderstanding. 31
The notion of a point () was analyzed at length in pre-Hellenistic
times in the framework that we have called Platonist. The discussions of
28 Archimedes, On conoids and spheroids, 155, 4–13 (ed. Mugler, vol. I).
28a By contrast, for Plato the same term really meant round like a ball (Plato, Timaeus, 33b). The
Greek ¦ is of course the etymon of our “spheroid”, the word traditionally used to translate
this term in Archimedes; the technical expression used nowadays for the same notion is ellipsoid
of revolution (prolate for lengthened and oblate for flattened).
29 Like the one just quoted, many definitions, particularly in Archimedes and Apollonius, consisted
of a long expression that was identified with a new term by means of some form of the verb
(to call). In Euclid’s Elements there are also some definitions of Platonist type, for instance
for point, line and plane. We will return in Section 10.14 to the problems offered by these latter
definitions.
30 See page 130 and especially footnote 30 thereon. A hint of nominalist definitions in mathematics
is perhaps present already in Plato (Theaetetus, 184a–b). The passages in the Theaetetus dealing
with the practices of the mathematician Theodorus and his school seem to look forward to later
scientific elements that are not otherwise present in Plato (whose notion of language, as laid down
in the Cratylus, is miles away from conventionalism).
31 [Proclus/Friedlein], 100.
Revision: 1.7 Date: 2002/09/14 23:17:37