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.
Rulers and compasses may have been used. Generally speaking, a Greek viewer would have read into them, directly, the objects depicted, though this would have required some imagination (and, probably, what was seen then was just the schematic configuration); but then, any viewing demands imagination. . The mutual dependence of text and diagram There are several ways in which diagram and text are interdependent. The most important is what I call ‘fixation of
when the genders of the relative pronoun and the signified object clash. But there are other cases, where the gender, or more often the number of the relative pronoun do change according to the signified object. The most consistent feature of this Aristotelian usage is its inconsistency – not a paradox, but a helpful hint on the nature of the usage. Aristotle, I suggest, uses language in a strange, forced way. That his usage of letters is borrowed from mathematics is extremely likely. That in
a point is specified in the following way: στω κωνικ πιφáνεια, κορυφ τ Α σηµε ον ‘Let there be a conic surface, whose vertex is the point Α’. The point Α has been defined as a vertex, and it will function in the proposition qua vertex, not qua point. Yet it will always be called, as in the specification itself, τ Α, in the neuter (‘point’ in Greek is neuter, while ‘vertex’ is feminine). This is the general rule: points, even when acquiring a special significance, are always called simply
as if he never deﬁned the one by means of the other. Clearly the tangle of the haptesthai family was inextricable, and post-Euclidean mathematicians evaded the tangle by using (as a rule) a third, unrelated verb, epipsauein. This verb originally meant ‘to touch lightly’. One wonders why Euclid did not choose it himself. At any rate, a regular expression for tangents in post-Euclidean mathematics was a non-deﬁned term, whose reference was derived from its connotations in ordinary language.
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