The Lemma Dilemma

Various proofs in the history of mathematics have necessitated an incredible number of steps. Some, because of step similarity, have been abbreviated by use of the words "and in like manner," an inclusion definitely in order and one much appreciated by the reader.

Others, due to the solver's desire for conciseness, suggest carelessness with details or risky assumptions. Because of this, famous "proofs" have been known to collapse. Consider, for example, Kempe's fallacious "proof" of the Four Color Map Theorem or Kummer's erroneous disposition of Fermat's Theorem.

Yet the desire for elegance in solving, proving, and deriving is a strong one-and thus leads the mathematician to varying techniques in attacking famous problems. One such technique is the quest for powerful and far-reaching lemmas that can reduce a critical problem to a direct corollary.

Table of Contents:

CHANGING POINTS OF VIEW

OVERCLAIMING, OMITTING, OBSCURING

ASSUMPTIONS

THE QUEST FOR LEMMAS

REFERENCES