When does a 'visual proof by induction' serve a proof-like function in mathematics?

Abstract

A proof by mathematical induction demonstrates that a general theorem is necessarily true for all natural numbers. It has been suggested that some theorems may also be proven by a ‘visual proof by induction’ (Brown, 2010), despite the fact that the image only displays particular cases of the general theorem. In this study we examine the nature of the conclusions drawn from a visual proof by induction. We find that, while most university-educated viewers demonstrate a willingness to generalize the statement to nearby cases not depicted in the image, only viewers who have been trained in formal proof strategies show significantly higher resistance to the suggestion of large-magnitude counterexamples to the theorem. We conclude that for most university-educated adults without proof-training the image serves as the basis of a standard inductive generalization and does not provide the degree of certainty required for mathematical proof.


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