A Formal System for Euclid's Elements
THE REVIEW OF SYMBOLIC LOGIC Volume 2, Number 4, December 2009 A FORMAL SYSTEM FOR EUCLID’S ELEMENTS JEREMY AVIGAD Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University EDWARD DEAN Department of Philosophy, Carnegie Mellon University JOHN MUMMA Division of Logic, Methodology, and Philosophy of Science, Suppes Center for History and Philosophy of Science Abstract. We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. §1. Introduction. For more than two millennia, Euclid’s Elements was viewed by mathematicians and philosophers alike as a paradigm of rigorous argumentation. But the work lost some of its lofty status in the nineteenth century, amidst concerns related to the use of diagrams in its proofs. Recognizing the correctness of Euclid’s inferences was thought to require an “intuitive” use of these diagrams, whereas, in a proper mathemat- ical argument, every assumption should be spelled out explicitly. Moreover, there is the question as to how an argument that relies on a single diagram can serve to justify a general mathematical claim: any triangle one draws will, for example, be either acute, right, or obtuse, leaving the same intuitive faculty burdened with the task of ensuring that the argument is equally valid for all triangles.1 Such a reliance on intuition was therefore felt to fall short of delivering mathematical certainty. Without denying the importance of the Elements, by the end of the nineteenth century the common attitude among mathematicians and philosophers was that the appropriate logical analysis of geometric inference should be cast in terms of axioms and rules of inference.
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