Abbreviations

120 The mathematical lexicon
about mathematics; the latter is mathematics. We have glimpsed this
distinction often before: we now confront it directly.
It should be stressed that Archimedes does not use the letter form
to commend the attractions of Syracuse or to complain about his
health. The introductions are mathematical texts, as dry as any. The difference
of register could not be explained by a difference of subject
matter.
Or perhaps, as ever in the lexicon, a more structural approach
should be taken? The important thing is not how the second-order
lexicon is different, but that it is different. The two are separated.
Indeed, they are sealed off from each other, literally. Second-order
interludes between proofs, not to mention within proofs, are remarkably
rare. 84 The two are set as opposites. And it is of course the firstorder
discourse which is marked by this, since the second-order discourse
is simply a continuation of normal Greek prose.
I have said already that Greek mathematics focuses directly on objects.
The compartmentalisation of lexica is one of the more radical
ways in which this is done. When it is talking about objects, the lexicon
is reduced to the barest minimum, so that any wider considerations
are ruled out – because there are no words to speak with, as it were.
And, in this, the main body of Greek mathematics is marked off from
ordinary Greek, as no other Greek subject ever was. 85
2.6 Summary
Some of the results we have obtained are significant for the question of
the shaping of the lexicon. These are the small role of definitions, the
persistence of exceptions to any rule, the role of the structure of the
lexicon as an entirety, and the great divide between first- and secondorder
discourse. I shall return to such points later on.
More positively for the lexicon itself, we saw that it is dramatically
small – not only in specifically mathematical words, but in any words,
including the most common Greek grammatical words. It is strongly
repetitive, within authors and between authors. And it follows, on the
whole, a principle of one-concept-one-word.
84 One thinks of CF 374.16, coming in the middle of proposition 8: το�το δ � �ν ε�χρηστον ποτ�
τ� δε�ξαι – ‘now that was useful for the proof’. I do not think there are more than a handful
of such remarks in the whole of Greek mathematics (and, needless to say, ε�χρηστος is a hapax
legomenon in the Archimedean corpus).
85 I shall return to show this in section 3 below. Blomquist (1969) also shows that no other set of
authors had similar peculiarities in their use of particles.