Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, computer (...) scientists, and anyone interested in the use of diagrams in geometry. (shrink)

In several articles, Mumma has presented a formal diagrammatic system Eu meant to give an account of one way in which Euclid's use of diagrams in the Elements could be formalized. However, largely because of the way in which it tries to limit case analysis, this system ends up being inconsistent, as shown here. Eu also suffers from several other problems: it is unable to prove several wide classes of correct geometric claims and contains a construction rule that is probably (...) computationally intractable and that may not even be decidable. (shrink)

In an article in the Journal of Philosophical Logic in 1996, "Towards a Model Theory of Venn Diagrams," (Vol. 25, No. 5, pp. 463-482), Hammer and Danner proved the full completeness of Shin's formal system for reasoning with Venn Diagrams. Their proof is eight pages long. This note gives a brief five line proof of this same result, using connections between diagrammatic and sentential representations.

In an article in the Journal of Philosophical Logic in 1996, "Towards a Model Theory of Venn Diagrams,", Hammer and Danner proved the full completeness of Shin's formal system for reasoning with Venn Diagrams. Their proof is eight pages long. This note gives a brief five line proof of this same result, using connections between diagrammatic and sentential representations.