Appropriate Steps A Theory of Motivated Proofs

While philosophers have long been interested in proofs, their primary focus has traditionally been on the justificational role that they play, and particularly on the certainty that they provide. Mathematicians throughout history, however, have expected proofs to do more than “just” establish a certain result. For example, mathematicians have especially valued proofs which are fruitful, obtained in a pure manner, or explanatory. Philosophers of mathematical practice have recently started to investigate such non-justificational aspects of proofs. In this dissertation, I explore another desideratum taken from the practice of mathematics: that proofs be motivated. The topic of motivated proofs has received some discussion by mathematicians, most notably George Pólya, who emphasized that the steps in a motivated proof must not only be correct, but also appropriate, and recognized as such by the reader. However, the topic of motivated proofs has received almost no philosophical attention. It might be thought that this is because such a topic is not philosophically interesting and is better investigated by other disciplines such as psychology or pedagogy. However, while I encourage a highly interdisciplinary approach, I argue that the topic is philosophically interesting and requires philosophical work, even for it to be fruitfully studied from within a different discipline. Throughout the rest of this dissertation, I undertake such philosophical work. More precisely, I address philosophical questions such as “What criteria should a proof meet to be motivated?”, and “Should obtaining motivated proofs be a mathematical goal?” In addition I tackle the more practical question, “How can a poorly motivated proof be transformed into a better motivated one?”