Uses of construction in problems and theorems in Euclid’s Elements I–VI

Nathan Sidoli*

*この研究の対応する著者

研究成果: Article査読

4 被引用数 (Scopus)

抄録

In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

本文言語English
ページ(範囲)403-452
ページ数50
ジャーナルArchive for History of Exact Sciences
72
4
DOI
出版ステータスPublished - 2018 7 1

ASJC Scopus subject areas

  • 数学(その他)
  • 科学史および科学哲学

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