1 The Birth of Science - MSRI

6.4 Definitions, Scientific Terms and Theoretical Entities 155
definition is to identify the essence of the thing defined. 24 Thus, for example,
the many attempts to define “good” and “justice” in Plato’s Socratic
dialogues. In the Platonist view, essentialist definitions apply just as well
to mathematical entities, which are regarded as having an objective reality,
the mathematician’s function being solely to describe and use them. 25 This
view prevailed in the imperial age, in the Middle Ages and in the early
modern age.
Karl Popper wrote:
The development of thought since Aristotle could, I think, be summed
up by saying that every discipline, as long as it used the Aristotelian
method of definition, has remained arrested in a state of empty verbiage
and barren scholasticism, and that the degree to which the var- page 200
ious sciences have been able to make any progress depended on the
degree to which they have been able to get rid of this essentialist
method. 26
One may take issue with Popper’s opinion about the “empty verbiage”
of the Aristotelian method. In fact, the Aristotelian method of definition,
which consists in pinpointing the essence of what is being defined through
a series of dichotomies, is useful and applicable for singling out an existing
object among a finite set of possibilities; this is the case of animal species,
which Aristotle was particularly interested in. One can, for instance, define
the swallow by saying that it is a bird and by listing enough features
to allow it to be distinguished from all other known bird species. But it
was not this method that led to the creation of the scientific terminology
that concerns us. In exact science, indeed, a definition is not meant to identify
a concrete object among a finite set of possibilities, but to characterize
uniquely a theoretical entity among infinitely many possibilities. 27
24 For Aristotle’s opinion that defining something means to identify its essence, see for example
Topica, I, 5, 101b, 36; Metaphysica, VII, 5, 1031a, 13; VIII, 1, 1042a, 17. For the substantial agreement
on this subjection between Aristotle and Plato see [Popper: OSE], Chapter 11, §2, following note
31. 25Plato’s conception of mathematical entities (put forth, for example, in the Republic, VI, 509c–
511a) was, it is true, criticized at length by Aristotle (see in particular Metaphysica, XI, 4: XIII; XIV),
who maintained that these entities were not immanent in objects and did not possess a separate
reality. Aristotle’s position can be summarized approximately by saying that mathematical beings
have a particular type of existence: they exist only as properties of perceivable objects. But although
his view has different philosophical bases than Plato’s, the difference is not such that the attitude
toward the mathematician’s work is significantly changed. In this respect the essential point is that,
for Aristotle as for Plato, humans do not construct mathematical entities: they somehow preexist.
26 [Popper: OSE], Chapter 11, §2, following note 26.
27 If in order to define an object one must take into account its differences vis-à-vis all others,
any definition has as a prerequisite the knowledge of all reality. This difficulty (raised, according
to scholiasts, by Speusippus) had already been faced by Aristotle in other contexts (Analytica posteriora,
II, 13, 97a, 6–10), but it becomes insurmountable in the case of mathematics.
Revision: 1.7 Date: 2002/09/14 23:17:37