Extended Abstract of Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry

CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION " We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic musical. 1 Background In 1879, the English mathematician Charles Dodgson, better known to the world under his pen name of Lewis Carroll, published a little book entitled Euclid and His Modern Rivals. Dodgson was concerned by the fact that quite a number of different nineteenth century authors had written their own treatments of planar geometry, most claiming to improve on Euclid, and each one slightly different in the order of its theorems, in which theorems it chose to include, in the proofs given of these theorems, in its treatment of the theory of parallel lines, and in other aspects. Dodgson's book was written " [i]n furtherance of the great cause which I have at heart—the vindication of Euclid's masterpiece. .. . " 1 It is written mostly in the form of a dream dialogue between a nineteenth century mathematician, Minos, and the ghost of Euclid. In it, they consider each of the modern rivals in turn, and conclude in each case that, while many of the rivals have interesting things to say, none of them are a more appropriate basis for the study of a beginning geometry student than Euclid's Elements. At the time at which Dodgson wrote his book, the subjects of geometry and logic were both entering a period of rapid change after having remained relatively constant for two thousand years. There had been enough change already to make Dodgson feel that Euclid needed defending. In the hundred and twenty-five years since then, however, there have been much larger changes in these fields, and, as a result, rather than just undergoing some small changes, Euclidean geometry in general, and Euclid's proofs in particular, have mostly fallen out of the standard mathematics curriculum. This is at least in part because Euclid's Elements, which was viewed for most of its existence as being the gold standard of careful reasoning and mathematical rigor, has come to be viewed as being inherently and unsalvageably informal and 1 Dodgson (1885) 1 2 / Euclid and His Twentieth Century Rivals …