The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Ideas in Context)
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This book provides a way to understand a momentous development in human intellectual history: the phenomenon of deductive argument in classical Greek mathematics. The argument rests on a close description of the practices of Greek mathematics, principally the use of lettered diagrams and the regulated, formulaic use of language.
one of the correlations, and should probably be assumed to govern all the rest. A typical example is a: φ δ τ Γ ëνθρωπο ‘And [if that] on which Γ [is] man’ / ‘and [if that] which Γ stands for [is] man’. I have offered two alternative translations, but the second should probably be preferred, for after all Γ does not, spatially speaking, stand on the class of all human beings. It’s true that the antecedent of the relative clause need not be taken here to be ‘man’. Indeed, often it cannot,
through deﬁnitions. I have concentrated on the nature of Greek deﬁnitions. Now is the time to say that the emphasis on deﬁnitions is fundamentally misplaced, regardless of what deﬁnitions may do. This is because the emphasis on deﬁnitions implies an emphasis on words, piecemeal, rather than on the lexicon as a The mathematical lexicon whole. It is the lexicon as a whole which is the subject of the following discussion. . Description As already mentioned above, I have made a census of
word-types in the mathematical text versus % in the philosophical, clearly a statistically meaningful difference. The mathematical text is strongly repetitive: there are no dead ends which are entered once but never followed up later. Further, the relative paucity of hapax legomena results also from another, more purely lexical feature: the text includes no synonyms or near-synonyms (i.e. words expressing close shades of meanings). I shall return to this later. Ignoring hapax legomena, one is
convey its meaning. What we see is that the logic of self-regulating conventions is a constraint on the development of the lexicon. A small, well-deﬁned lexicon is thus a self-perpetuating mechanism. In the discussion above I have fastened upon the process of disambiguation in, as it were, the ontogenetic plane. It will be seen that my hypothesis concerning the phylogenetic plane is exactly similar. Once the lexicon began to take the shape described in chapter , The mathematical lexicon
not oral performers – they were not performers at all (see chapter below). But then, imagine a Greek mathematician during the moment of creation. He has got a diagram in front of him, no doubt. But what else? What can he rely upon, what is available in his arsenal? There are no written symbols there, no shortcuts for the representation of mathematical relations, nothing besides language itself. He may jot down his thoughts, but if so, this is precisely what he does: write down, in full, Greek