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Definitions and Nondefinability in Geometry

by James T. Smith (San Francisco State University (Emeritus))

Award: Lester R. Ford

Year of Award: 2011

Publication Information: American Mathematical Monthly, vol. 117, June, 2010, pp. 475-489.


In the two millennia following Euclid, mathematicians wrestled with the choices, consequences, and relationships between various possible axioms for Euclidean geometry. Around the beginning of the last century, many mathematicians turned their attention to a more basic question: Which geometric concepts are the most elementary?

In this Smith‟s story begins with a discussion of the “Axiomatic Method” dating back to Aristotle and Euclid, and describes various attempts in the early nineteen hundreds to develop a presentation of geometric theory based on a minimum of undefined terms. The story Smith tells describes a path that led from an imprecise and somewhat intuitive approach to geometry to Pasch‟s first completely rigorous axiomatic presentation of a geometric theory. This path eventually led to Tarski‟s formal, rigorous (and opaque) use of logical symbols that minimized the use of set-theoretic notions and “has become a standard of comparison for work in the foundations of geometry”. Along the way we meet principle players in this development: Pasch, Peano, Pieri, Hilbert, Veblen, and finally, Tarski. The story is presented within a rich framework of brief biographies and discussion of the influence of their work upon one another.

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About the Author (From the MathFest 2011 Prizes and Awards Booklet)

James Smith, the first in his Ohio family to attend college, earned the A.B. from Harvard in 1961, then moved to San Francisco. For several years he worked with computers for the Navy, and attended San Francisco State (SFSU) and Stanford Universities. He earned a Ph.D. from the University of Saskatchewan, Regina in 1970, with a dissertation in foundations of geometry, supervised by H. N. Gupta. Smith continued that research, joined the SFSU faculty, became department chair in 1975, and helped start the SFSU computer science program. From 1982 he worked for several organizations and wrote books on software engineering and geometry. After 2003 he gradually retired from SFSU, switching to full-time preoccupation with history, particularly the careers of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983); he has books underway on each. Smith continues to serve the MAA's Golden Section; he divides his time between San Francisco and Siskiyou County.

Subject classification(s): Geometry and Topology | Geometric Proof
Publication Date: 
Monday, August 22, 2011