The Concept of Existence and the Role of Constructions in Euclid's Elements |
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Authors: | ORNA HARARI |
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Institution: | (1) Dibner Institute for the History of Science and Technology MIT E56-100 38 Memorial Drive Cambridge, MA 02139, USA oharari-eshel@dibinst.mit.edu, US |
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Abstract: | This paper examines the widely accepted contention that geometrical constructions serve in Greek mathematics as proofs of
the existence of the constructed figures. In particular, I consider the following two questions: first, whether the evidence
taken from Aristotle's philosophy does support the modern existential interpretation of geometrical constructions; and second,
whether Euclid's Elements presupposes Aristotle's concept of being. With regard to the first question, I argue that Aristotle's ontology cannot serve
as evidence to support the existential interpretation, since Aristotle's ontological discussions address the question of the
relation between the whole and its parts, while the modern discussions of mathematical existence consider the question of
the validity of a concept. In considering the second question, I analyze two syllogistic reformulations of Euclidean proofs.
This analysis leads to two conclusions: first, it discloses the discrepancy between Aristotle's view of mathematical objects
and Euclid's practice, whereby it will cast doubt on the historical and theoretical adequacy of the existential interpretation.
Second, it sets the conceptual background for an alternative interpretation of geometrical constructions. I argue, on the
basis of this analysis that geometrical constructions do not serve in the Elements as a means of ascertaining the existence of geometrical objects, but rather as a means of exhibiting spatial relations between
geometrical figures.
(Received January 1, 2002)
Communicated by A. JONES
Dedicated to the memory of Yonathan Begin |
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