Abbreviations

96 The mathematical lexicon
definitions may be a focus of interest in themselves. They answer such
questions as the ‘what is . . . ?’ question. Only such an interest can explain
such notorious definitions as Elements VII.1: ‘A unit is that by virtue of
which each of the things that exist is called one.’ No use can be made
of such definitions in the course of the first-order, demonstrative discourse.
Such definitions belong to the second-order discourse alone.
In general, we should view the Greek definition enterprise as belonging
to the discourse about mathematics – the discourse where mathematicians
meet with non-mathematicians – precisely the discourse
least important for the mathematical demonstration.
1.3 What is not defined?
My survey is far from exhaustive, but it is fair to estimate the number
of extant definitions as a few hundred; and definitions are very well
represented in the manuscript tradition. I suspect the total number
of definitions offered in antiquity was in the hundreds.
These definitions cover a smaller number of word-types. Even
ignoring the rare cases of double definition (e.g. ‘solid angle’, defined
in two different ways in Elements XI. Def. 11), definitions often return
to the same word-type, though in different combinations (for instance,
to pick up ‘solid angle’ again: this is a combination of ‘angle’, defined
in Elements I. Def. 8, and ‘solid’, XI. Def. 1). It is thus to be expected that
some, perhaps most, of the words used in Greek mathematics will be
undefined. This is the case, for two separate reasons. First, the role of
formulae. Return to the first definition: ‘A point is that which has no
part.’ The definiendum is the Greek word sBmeion. As I repeatedly
explained above, this is not what a Greek mathematician would normally
use when discussing points. Much more often, he would use an
expression such as τ� Α, ‘the Α’ (the gender of ‘point’ supplied by the
article). This is a very short formula indeed – the minimum formula –
yet a formula. But – and here is the crucial point – τ� Α is nowhere
defined. It was only sBmeion which was defined. The concept was defined,
conceptually, but the really functional unit was left undefined. The
same may be said of the most important words (often, the defined
nouns): words such as ‘line’, ‘triangle’, ‘rectangle’, ‘circle’. Beneath the
process of defining such concepts explicitly, there runs a much more
powerful silent current, establishing the real semantic usage through
formulae. I shall return to this subject in the next chapter.