Proofs, pictures, and Euclid

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. The prevailing conception of mathematical proof, or at least the conception which has been developed most thoroughly, is logical. A proof, accordingly, is a sequence of sentences. Each sentence is either an assumption of the proof, or is derived via sound inference rules from sentences preceding it. The sentence appearing at the end of the sequence is what has been proved. This conception has been enormously fruitful and illuminating. Yet its great success in giving a precise account of mathematical reasoning does not imply that all mathematical proofs are, in essence, a sequence of sentences. My aim in this paper is to consider data which do not sit comfortably with the standard logical conception: proofs in which pictures seem to be instrumental in establishing a result. I focus, in particular, on a famous collection of picture proofs— Euclid’s diagrammatic arguments in the early books of the Elements. The familiar sentential model of proof portrays inferences as transitions between sentences. And so, by the familiar model, Euclid’s diagrams would at best serve as a heuristic, illustrative device. They could not be part of the rigorous proof itself. In direct opposition to this, I introduce the proof system Eu, which accounts for the role of the diagram within Euclid’s mathematical arguments. It possesses a diagrammatic symbol type, and specifies rules of proof for these symbols. It thus provides a formal model where Euclid’s diagrams are part of the rigorous proof. Though Eu has been designed specifically to formalize these arguments, we can subsequently look to it to understand what is distinctive about proving with pictures. Eu represents a species of rigorous mathematical c © 2010 Kluwer Academic Publishers. Printed in the Netherlands. Euclid-pictures6.tex; 7/06/2010; 17:39; p.1