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.
non-objective or arbitrary) aspects of the process leading to scientific results. They do so in order to relativise science, to make it seem less propositional, or less ideologyfree, or less objective. But I ask: what sort of a process is it, which makes possible a positive achievement such as deduction? And by asking such a question, I am E.g. (to continue with the distinguished name required by the pun) in Shapin (). Introduction led to look at aspects of the practice which the
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.
is possible. While there are no general, universal rules concerning, for example, reasoning, such rules do exist historically, in specific contexts. Reasoning, in general, can be done in an open way, appealing to whatever tools suggest themselves – linguistic, visual, for example – using those tools in any order, moving freely from one to the other. In Greek mathematics, however, reasoning is done in a very specific way. There is a method in its use of cognitive resources. And it must be so – had
absent object, for the diagrams of antiquity are not extant, and the medieval diagrams have never been studied as such. However, not all hope is lost. The texts – whose transmission is relatively well understood – refer to diagrams in various ways. On the basis of these references, observations concerning the practices of diagrams can be made. I thus start from the text, and from that base study the diagrams. The critical edition most useful from the point of view of the ancient diagrams is
performer; they are helped by a written background. There is a limit to how formulaic a totally oral work can be (without any help from writing, there is a limit on the emergence of a rigid, repetitive system). There is a limit on how developed a non-formulaic work can be, in a context which is to some extent oral (without formulae, it is difﬁcult to create a developed work). I will therefore say this. Without writing, the formulaic system of Greek mathematics could not have been conserved and