Axiomatic Systems and Finite Geometries

• Judith N. Cederberg
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Finite geometries were developed in the late 19th 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.

Keywords

Projective Plane Distinct Point Projective Geometry Axiomatic System Code Word
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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1. Albert, A.A., and Sandler, R. (1968). An Introduction to Finite Projective Planes. New York: Holt, Rinehart and Winston. (Contains a thorough group theoretic treatment of finite projective planes. )Google Scholar
2. Anderson, I. (1974). A First Course in Combinatorial Mathematics. Oxford, England: Clarendon Press. (Chapter 6 discusses block designs and error-correcting codes.)Google Scholar
3. 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
4. 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
5. Gensler, H.J. (1984). Gödel’s Theorem Simplified. Lanham, MD: University Press of America.Google Scholar
6. Hofstadter, D.R. (1984). Analogies and metaphors to explain Gödel’s theorem. In: D.M. Campbell and J.C. Higgins (Eds.), Mathematics: People, Problems, Results, Vol. 2, pp. 262–275. Belmont, CA: Wadsworth.Google Scholar
7. Kolata, G. (1982). Does Gödel’s theorem matter to mathematics? Science 218: 779–780.
8. Lockwood, J.R., and Runion, G.E. (1978). Deductive Systems: Finite and non-Euclidean Geometries. Reston, VA: N.C.T.M. (Chapter 1 contains an elementary discussion of axiomatic systems.)Google Scholar
9. Nagel, E., and Newman, J.R. (1956). Gödel’s proof. In: J.R. Newman (Ed.), The World of Mathematics, Vol. 3, pp. 1668–1695. New York: Simon and Schuster.Google Scholar
10. Pless, V. (1982). Introduction to the Theory of Error-Correcting Codes. New York: Wiley. (A well-written explanation of this new discipline and the mathematics involved. )
11. Smart, J.R. (1978). Modern Geometries,2nd ed. Belmont, CA: Wadsworth. (Chapter 1 contains an easily readable discussion of axiomatic systems and several finite geometries.)Google Scholar
12. Thompson, T.M. (1983). From Error-Correcting Codes Through Sphere Packings to Simple Groups. The Carus Mathematical Monographs, No. 21. Ithaca, NY: M.A.A. (Incorporates numerous historical antecdotes while tracing 20th century mathematical developments involved in these topics.)Google Scholar

1. Beck, A., Bleicher, M.N., and Crowe, D.W. (1972). Excursions into Mathematics, pp. 262–279. New York: Worth.Google Scholar
2. Crowe, D.W., and Thompson, T.M. (1987). Some modern uses of geometry. In: M.M. Lindquist and A.P. Schulte (Eds.). Learning and Teaching Geometry, K-12, 1987 Yearbook, pp. 101-112. Reston, VA: N.C.T.M.Google Scholar
3. Gardner, M. (1959). Euler’s spoilers: The discovery of an order-10 Graeco-Latin square. Scientific American 201: 181–188.
4. Sawyer, W.W. (1971). Finite arithmetics and geometries. In: Prelude to Mathematics, Chap. 13. New York: Pengu in Books.Google Scholar