1 The Birth of Science - MSRI

10.14 The First Few Definitions in Euclid’s Elements 287
instead that “a point is a unit in a location”: an ancient definition, Pythagorean
in origin, that appears in the Definitions but not in Euclid’s book. 226c
An even more significant passage, for chronological reasons if nothing
else, is found in Philo of Alexandria (first century B.C.). He makes a distinction
among geometric concepts, placing on the one hand circles and
isosceles triangles and polygons, for example, and on the other concepts
like points and lines, which can be defined only in philosophical terms.
He wonders:
How could [geometry], giving definitions, say that a point is that
which has no part, a line is breadthless length, a surface only has
length and breadth, and a solid is three-dimensional, because it has
length, breadth and depth? This is the stuff of philosophy. . . 227
Clearly, in first century B.C. Alexandria, the type of Platonist-essentialist
definitions that now head the Elements could not be found in geometric
works.
There remains to explain what sources were used in compiling the Definitions.
Consider the first entry: after the beginning quoted on page 283,
it goes on with illustrative properties of a point and other divagations,
some of which are found in Aristotle. The second entry says among other
things that a line is the extremity of a surface; this characterization, interpolated
as Definition 6 into our Elements, goes back to Plato and had already
been criticized by Aristotle. 228 It is clear, then, that far from reflecting
the method of Euclid’s Elements, the Definitions draws heavily from pre-
Hellenistic sources, though of course it also contains properly geometric
material.
The thesis espoused in this section openly challenges the traditional
view that Euclid was a Platonist. I think the idea of a Platonist Euclid has
three chief causes: the lasting influence of the only commentary to Euclid
preserved in Greek, by the neo-Platonist philosopher Proclus; the presence
in the Elements of the definitions that we have been discussing; and the ascendance
of Platonizing interpretations of Euclid, arising from the vigor
that Platonist views have enjoyed in schools of mathematical thought ever
since the imperial age.
Karl Popper seems to have implicitly reached the conclusion we have
articulated in this section. For if we apply his already quoted thoughts
226c
Plutarch, Platonicae quaestiones, 1003F; Heronis Definitiones, 14, 15, (ed. Heiberg), where the point
is said to be “like a unit having a location”.
227
Philo of Alexandria, De congressu eruditionis gratia, III, 102, 15–25 = [SVF], II, text 99.
228
Some relevant Aristotelian passages: Physica, IV, 11, 220a, 15 ff. (the point as the extremity of a
line); Metaphysica, V, 6, 1016b, 24–30 (indivisibility as a characteristic feature of points); De caelo, III,
1, 300a; Topica, VI, 6, 143b, 11 (line as extremity of a surface). See also footnote 226c above.
Revision: 1.11 Date: 2003/01/06 02:20:46