Carroll, Lewis [addendum]
The success of the "Alice" books established Charles L. Dodgson's reputation as a gifted writer of children's literature. His admirers expected humor in everything he wrote from then on, an attitude that affected the reception of his serious pieces and prevented his work from contributing to the development of their subjects. For example, the more amusing Euclid and His Modern Rivals (1879) overshadowed his more important book, Curiosa Mathematica. Part 1. A New Theory of Parallels (1888).
Dodgson made significant contributions to linear algebra in An Elementary Treatise on Determinants (1867), a book that though marred by odd notation and unusual terminology, contains the first written proof of a standard theorem connecting the rank of a matrix with the existence of solutions to certain linear systems (chapter 4, proposition II). One of his techniques, condensation, was used in an early step of the solution to the alternating sign matrix problem (Bressoud 1999).
In the field of cryptology his five cipher systems, based on the three cipher paradigms of his time (Vige-nère, Beaufort, Variant Beaufort) are not well known. These were: Key-Vowel, Matrix, Alphabet, Telegraph, and Memoria Technica. The first two (1858, unpublished) were unbreakable from a practical point of view. The third and fourth (1868) were secure by the standards of his time for ordinary telegrams and mailed postcards. The last (1875), directly tied to word games, was the most literary.
His publications on the theory of voting consisted of four pamphlets, three written between 1873 and 1876, and The Principles of Parliamentary Representation (1884). The pamphlets of the 1870s, an outcome of Dodgson's involvement with college and university affairs, reflect his independent rediscovery of Condorcet's cyclical majorities and include the first application of game theory to sophisticated voting. The argument of the 1884 pamphlet, written to influence the outcome of two electoral reforms, a goal it did not accomplish, is based on the zero-sum game. Dodgson was the first to treat formally apportionment (allocating seats to districts) and proportional representation (assigning seats to political parties) together.
Dodgson's contributions to logic have been widely recognized since William Warren Bartley, III's edition of Lewis Carroll's Symbolic Logic (1977) which includes the unpublished manuscript of part 2 of Dodgson's Symbolic Logic. Dodgson developed a formal logic where he set down intuitively valid rules for making inferences. A comparison of the two parts reveals the progress he made toward an automated approach to the solution of multiple connected syllogistic problems, many being humorous puzzle problems. The most important of his techniques, the method of trees, foreshadowed modern concepts and techniques in automated reasoning that were developed from the 1950s. Dodgson's use of existential import, abandoned in modern logical usage, marred the reception of part 1 of his book. He developed a method of diagrams as a visual proof system for syllogisms that he introduced in The Game of Logic (1887). Like his tree test, which is a proof system for soriteses, it is sound and complete. His self similar diagrams (invariant under a change of scale) are capable of handling existential statements and are easily extended to any number of sets using a linear iterative process. In this regard, they are superior to the diagrams described by John Venn in 1880.
Logic Diagrams; Logic, History Of; Logic, Traditional; Venn, John.
Bibliography
Abeles, Francine F. "Lewis Carroll's Ciphers: The Literary Connections." Advances in Applied Mathematics 34(2005): 697–708.
Abeles, Francine F., ed. The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces. New York: Lewis Carroll Society of North America; Charlottesville: Distributed by the University Press of Virginia, 1994.
Abeles, Francine F., ed. The Political Pamphlets and Letters of Charles L. Dodgson and Related Pieces. New York: Lewis Carroll Society of North America; Charlottesville: Distributed by the University Press of Virginia, 2001.
Bressoud, David M. Proofs and Confirmations. The Story of the Alternating Sign Matrix Conjecture. Washington, D.C.: The Mathematical Association of America; Cambridge, U.K.: Cambridge University Press, 1999.
Cohen, Morton N. Lewis Carroll. A Biography. New York: Knopf, 1995.
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