Axiomatic Systems and Finite Geometries
Finite geometries were developed in the late nineteenth century, in part to demonstrate and test the axiomatic properties of completeness, consistency, and independence. They are introduced in this chapter to fulfill this historical role and to develop both an appreciation for and an understanding of the revolution in mathematical and philosophical thought brought about by the development of non-Euclidean geometry. In addition, finite geometries provide relatively simple axiomatic systems in which we can begin to develop the skills and techniques of geometric reasoning. The finite geometries introduced in Sections 1.3 and 1.5 also illustrate some of the fundamental properties of non-Euclidean and projective geometry.
KeywordsProjective Plane Distinct Point Projective Geometry Axiomatic System Code Word
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Suggestions for Further Reading
- Beck, A., Bleicher, M. N., and Crowe, D. W. (1972). Excursions into Mathematics. New York: Worth. (Sections 4.9–4.15 give a very readable discussion of finite planes, including the development of analytic models.)Google Scholar
- Benedicty, M., and Sledge, F. R. (1987). Discrete Mathematical Structures. Orlando, FL: Harcourt Brace Jovanovich. (Chapter 13 gives an elementary presentation of coding theory.)Google Scholar
- Gensler, H. J. (1984). Gödel’s Theorem Simplified. Lanham, MD: University Press of America.Google Scholar
- Hofstadter, D. R. (1984). Analogies and metaphors to explain Gödel’s theorem. In Mathematics: People, Problems, Results. D. M. Campbell and J. C. Higgins (Eds.), Vol. 2, pp. 262–275. Belmont, CA: Wadsworth.Google Scholar
- Lockwood, J. R., and Runion, G. E. (1978). Deductive Systems: Finite and Non-Euclidean Geometries. Reston,VA: NCTM (Chapter 1 contains an elementary discussion of axiomatic systems.)Google Scholar
- Nagel, E. and Newman, J. R. (1956). Gödel’s proof. In The World of Mathematics. J. R. Newman (Ed.), Vol. 3, pp. 1668–1695. New York: Simon and Schuster.Google Scholar
- Smart, J. R. (1998). Modern Geometries, 5th ed. Pacific Grove, CA: Brooks/Cole. (Chapter 1 contains an easily readable discussion of axiomatic systems and several finite geometries.)Google Scholar
Readings on Latin Squares
- Beck, A., Bleicher, M. N. and Crowe, D. W. (1972). Exccursions into Mathematics, pp. 262–279. New York: Worth.Google Scholar
- Crowe, D. W., and Thompson, T. M. (1987). Some modern uses of geometry. In Learning and Teaching Geometry, iC-12, 1987 Yearbook, M. M. Lindquist and A. P. Schulte (Eds.), pp. 101–112. Reston, VA: NCTM.Google Scholar
- Sawyer, W. W. (1971). Finite arithmetics and geometries. In Prelude to Mathematics, Chap. 13. New York: Penguin Books.Google Scholar