Proofs, pictures, and Euclid

Synthese 175 (2):255 - 287 (2010)
Abstract
Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received view, this essay provides a contrary analysis by introducing a formal account of Euclid's proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid's arguments. It specifies what diagrams Euclid's diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored
Keywords Proof  Diagrams  Logic  Geometry
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References found in this work BETA
Gottfried Wilhelm Leibniz (2007). New Essays Concerning Human Understanding. In Elizabeth Schmidt Radcliffe, Richard McCarty, Fritz Allhoff & Anand Vaidya (eds.), Philosophical Review. Blackwell Pub. Ltd. 293-297.
Kenneth Manders (2008). The Euclidean Diagram. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press 80--133.
Y. Rav (1999). Why Do We Prove Theorems? Philosophia Mathematica 7 (1):5-41.

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Citations of this work BETA
Lorenzo Magnani (2015). The Eco-Cognitive Model of Abduction. Journal of Applied Logic 13 (3):285-315.
Catherine Legg & James Franklin (2015). Perceiving Necessity. Pacific Philosophical Quarterly 97 (2):n/a-n/a.

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