1 The Birth of Science - MSRI

6.4 Definitions, Scientific Terms and Theoretical Entities 157
the notion in Aristotle were of this type. 32 Euclid avoids the word
in the Elements, using instead ¨, which originally meant “sign”. 33
This replacement suggests that Euclid may have wished to cut himself
off from the tradition of Platonic speculations on the true nature of the
point, sticking to a conventionalist notion of language and a new notion of
mathematics. This new way of seeing mathematical entities is in evidence page 202
in some of Euclid’s definitions, in particular the definition of proportion.
If one thinks of the “ratio between magnitudes” as something that exists
in and of itself, the equality of two ratios seems like an obvious notion (as
it did to Galileo 34 ), whereas Euclid’s definition, as we saw in Section 2.5,
is tantamount to a subtle and complex implicit definition of the notion of
the ratio between magnitudes.
Nominalist definitions are certainly very valuable for enriching scientific
terminology, but they cannot create it from scratch. Any definition of
this type can only reduce the meaning of a new term to that of terms already
assumed known. Just as the hypothetico-deductive method requires
demonstrationless statements on which to build, so the nominalist definition
procedure requires definitionless terms from which to start.
The awareness of the need to avoid infinite regress, which must be clear
to a person who shares a nominalist view of definitions, is documented
already in pre-Hellenistic philosophy. Thus, Aristotle reports that, according
to the school of Antisthenes, since every definition requires a reference
to something else, it is only possible to define what is composite (whether
materially or conceptually), not what is simple. 35 This observation of Antisthenes
is not an isolated one, because around 200 A.D. Sextus Empiricus
wrote:
And given that, if we want to define everything we define nothing at
all, because of regression to infinity, whereas if we admit that some
things can be understood without definition we are declaring that
definitions are not necessary for understanding, . . . we must either
define nothing at all or declare that definitions are not necessary. 36
The conclusion drawn by Sextus Empiricus reflects his own Skeptic
ideas. What concerns us is that the possibility of “admitting that some
32 Sample passages are listed in note 228 on page 287; see also the surrounding discussion.
33 The occasional presence in some works of the Aristotelian corpus of this word ¨ in the
sense of “point” is neither here nor there, because these works were certainly revised in post-
Euclidean times. The term recurs, for example, in some geometric constructions contained in the
Meteorologica (see especially 373a and 375b–377a), whose redaction into the form that has come
down to us is probably due to a student of Theophrastus.
34 We will return to this on page 307.
35 Aristotle, Metaphysica, VIII, 3, 1043b, 23–32.
36 Sextus Empiricus, Pyrrhoneae hypotyposes, II, xvi, 207–208.
Revision: 1.7 Date: 2002/09/14 23:17:37