# Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries

by Marvin Jay Greenberg (University of California at Santa Cruz -- Emeritus Professor)

Award: Lester R. Ford

Year of Award: 2011

Publication Information: The American Mathematical Monthly, vol. 117, no. 3, March 2010, pp. 198-219.

Summary

For two thousand years, Euclid's Elements reigned supreme as the paradigm of axiomatic reasoning. By the late 1800s, however, questions on the foundations of mathematics led to reconsideration of certain mathematical systems, including the axioms of geometry. Hilbert's Grundlagen der Geometrie (2nd ed., 1903) put both elementary plane Euclidean and hyperbolic geometries on a rigorous axiomatic foundation, revealing the impossibility of proving certain geometric statements from restricted axioms. Hilbert emphasized purity of method. This author shows how the elementary (i.e., lines and circles) part of those two geometries has been cleverly purged by Hilbert and the tiny community of foundations of geometry researchers of unnecessary reliance on real numbers, remaining faithful to the synthetic method of Euclid. The author deftly introduces relevant ideas of Archimedes, Eudoxus, Proclus, and Aristotle, thus justifying the title "old and new results". He details his own discovery that Aristotle's Axiom is a missing link in foundations of both Euclidean and hyperbolic plane geometries.

This broad overview of the foundations of plane geometry closes with a discussion of incompleteness, undecidability and consistency in geometry. Many mathematics majors are intrigued by Gödel and his famous incompleteness and undecidability results in arithmetic. The author shows that the theory of elementary Euclidean geometry is also incomplete and undecidable. However, elementary geometry has finitary consistency proofs, whereas Gödel showed that such proof is impossible for arithmetic; in this sense, elementary geometry is simpler than arithmetic.

About the Author: (From the Prizes and Awards booklet, MathFest 2011)

Marvin Jay Greenberg won a Ford scholarship at age 15 to attend Columbia College, after convincing the Dean that because he played golf he was not just a bookworm. At Princeton, he solved the Ph.D. problem suggested by Serge Lang after being yelled at by Serge in the Fine Hall common room. In algebraic geometry he discovered the functor Jean-Pierre Serre named after him. Later, he was the translator of Serre's Corps Locaux.

He taught at UC Berkeley from 1959 to 1964, excluding a year off to study with A. Grothendieck. In 1965, he discovered the approximation theorem in arithmetical algebraic geometry named after him. His first book, Lectures on Algebraic Topology, appeared in 1967, his second, Lectures on Forms in Many Variables, in 1969.

He lives in the hills of beautiful Berkeley, CA, where he wrote the recent fourth edition of his Euclidean and Non-Euclidean Geometries: Development and History. He is a founding member of the Shivas Irons golf society (www.shivas.org).

Subject classification(s): Geometry and Topology | Differential Geometry | Plane Geometry
Publication Date:
Sunday, August 21, 2011