Proofs, pictures, and Euclid

Synthese 175 (2):255 - 287 (2010)

John Mumma
California State University, San Bernardino
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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DOI 10.1007/s11229-009-9509-9
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References found in this work BETA

The Euclidean Diagram.Kenneth Manders - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 80--133.
A Formal System for Euclid’s Elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
Why Do We Prove Theorems?Yehuda Rav - 1999 - Philosophia Mathematica 7 (1):5-41.
The Derivation-Indicator View of Mathematical Practice.Jody Azzouni - 2004 - Philosophia Mathematica 12 (2):81-106.
Logical Reasoning with Diagrams.Gerard Allwein & Jon Barwise (eds.) - 1996 - Oxford University Press.

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Citations of this work BETA

A Formal System for Euclid’s Elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
The Eco-Cognitive Model of Abduction.Lorenzo Magnani - 2015 - Journal of Applied Logic 13 (3):285-315.
Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).

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