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Diagrammatic reasoning in Euclid’s Elements. (English) Zbl 1219.00013

Van Kerkhove, Bert (ed.) et al., Philosophical perspectives on mathematical practice. Papers from the 2nd perspectives on mathematical practices conference (PMP2007), Free University of Brussels, Brussels, March 26–28, 2007. London: College Publications (ISBN 978-1-904987-59-8/hbk). Texts in Philosophy 12, 235-267 (2010).
Starting with K. Manders [“Diagram-based geometric practice” and “The Euclidean diagram”, in: P. Mancosu (ed.), The philosophy of mathematical practice, Oxford: Oxford University Press, 65–79, 80–133 (2008; Zbl 1163.03001)], the interest in diagrammatic reasoning in Euclid’s Elements has led to several formal systems (such as the one in [J. Avigad, E. Dean and J. Mumma, “A formal system for Euclid’s Elements”, Rev. Symb. Log. 2, No. 4, 700–768 (2009; Zbl 1188.03008)]) that mimic diagrammatic reasoning, as well as to interest among philosophers for this hitherto unexplored form of reasoning.
The paper under review is concerned with the questions whether an axiomatic system and deductions from axioms are present in Euclid’s Elements, and which aspects of the diagram are used in diagram-based reasoning as evident in Books I, II, and III of the Elements. The first question is answered in the negative, given that the system presented in the Elements functions “more like a system of natural deduction; its Common Notions, Postulates, and Definitions function not as premises from which to reason but instead as rules or principles according which to reason.” Regarding the role and nature of diagrams and the reasoning based on them, the author finds that they are neither “merely images” nor “instances of geometrical figures”, but rather are icons with what H. P. Grice [Philosophical Review 66, 377–388 (1957)] called “non-natural meaning”. Reasoning in Euclid is not “merely diagram-based”, but rather “properly diagrammatic”, as “one reasons in the diagram”, by using “pop-up objects” and by “actualizing [\(\ldots\)] some potential of the diagram”. Proofs are then not about a sequence of sentences starting with axioms, but rather an “activity [\(\ldots\)] focused directly on the diagram”, its modus operandi being a constantly shifting way of conceiving the diagram, the different manners in which it is construed leading to different conclusions, which together make up the proof.
For the entire collection see [Zbl 1200.03011].

MSC:

00A30 Philosophy of mathematics
01A20 History of Greek and Roman mathematics
03A05 Philosophical and critical aspects of logic and foundations
03B30 Foundations of classical theories (including reverse mathematics)
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