TAILIEUCHUNG - Aesthetic Analysis of Proofs of the Binomial Theorem

The anthropology of aesthetics as I see it, then, consists in the comparative study of valued perceptual experience in different societies. While our common human physiology no doubt results in our having universal, generalized responses to certain stimuli, perception is an active and cognitive process in which cultural factors play a dominant role. Perceptions are cultural phenomena. Forge touched on this some twenty years ago when he wrote (1970: 282) concerning the visual art of the Abelam of New Guinea: What do the Abelam see? Quite obviously there can be no absolute answer to this question:. | Aesthetic Analysis of Proofs of the Binomial Theorem Lawrence Neff Stout Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington IL 61702-2900 lstout@ August 16 1999 This paper explores aesthetics of mathematical proof. Certain important aspects of proofs are not relevant to aesthetics validity utility exposition but others are immediacy enlightenment economy of means establishment of connections opening of mathematical vistas . Three different proofs of the binomial theorem are used as illustrations. 1 Introduction Proof in mathematics has two central roles it provides the definitive criterion for truth in the subject an epistemological role and it is the canvas for part of the aesthetic of mathematics. In order to meet the demands of the epistemological role a proof must follow the rules of deductive logic. Each statement in the proof must either be an axiom or definition or be known to be correct from a previous proof or it must follow from earlier statements in the proof. Proofs are usually informal in that they do not fill in all of the steps but rather depend on the mathematical knowledge of the reader to provide if desired all of the connections. Thus a proof depends on an intellectual tradition and a social context for satisfaction of its epistemological role. 1 That context will provide for an agreed upon notion of number specification of the logical constructs allowed in the proof usually classical predicate calculus unless an explicit constructivist or intuitionist viewpoint is taken notational conventions and familiarity with other theorems which may be brought to bear. In particular there is a need for knowledge of the proofs of those other results so that hidden circularity is avoided. But satisfaction with and appreciation of a proof does not end with determination of its validity. We ask for insight. A proof should not only tell us that a mathematical statement is true but why it is true. We will find a proof .

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