1 The Birth of Science - MSRI

6.6 Postulates and the Meaning of “Mathematics” and “Physics” 165
astronomy, it reigned uniformly. The modern distinction between physical
and mathematical sciences was alien to Hellenistic science, which was
unitary. This point cannot be stressed too often, given our well-nigh unavoidable
tendency to think in terms of modern categories: the theories
developed in Archimedes’ On floating bodies and On the equilibrium of plane
figures or in Euclid’s Optics are homogeneous with Euclid’s more famous
work, the Elements, not merely in their instrumental use of geometric notions
and results, but in that they are made of theorems based on postulates
of hydrostatics, statics and optics — just as the Elements is made
of theorems based on geometric postulates. Conversely, just as works on
statics and optics bear a clear relation to concrete activities such as the use
of balances and optical instruments (dioptra, astrolabe, and so on), the exact
same relation, as we have seen, obtains between Euclidean geometry
and drawing with ruler and compass.
It is essentially correct to say that the original name of the unitary science
that we have been discussing was mathematike ( , also
in the neuter plural, ). Substituting “mathematics” for the
Greek term requires the use of quotation marks and the awareness that
the meaning of the word changed profoundly in the modern age. In fact
it changed twice: the original meaning was “all that is studied”, coming
from the verb (learn) and the noun (object of study, subject
of learning): Plato uses the term in this sense at least twice, 56a and in
the Pythagorean school “mathematicians” were the disciples who shared page 212
in the more profound teachings. Later came the sense, attested in Aristotle
and systematically in Hellenistic authors, of a body of knowledge characterized
by a certain methodological outlook: much broader than what
we understand today as mathematics, but narrower than the etymological
meaning.
Anatolius, in the late third century A.D., recorded an explanation for the
origin of the Hellenistic use of the word:
Why does mathematics have this name? The Peripatetics say that,
whereas rhetoric, poetry and popular music can be practiced even
without being studied, no one can understand what’s called mathematics
without first having studied it. Thus they explain why the
theory of these things is called mathematics. 57
56a In the neuter singular: (Sophista, 219c, 2; Timaeus, 88c, 1). Plato did use, in a
sense close to the modern one, narrower terms such as “geometry”.
57 Anatolius, in [Heron: OO], vol. IV, 160, 17–24.
Revision: 1.7 Date: 2002/09/14 23:17:37