Motivated proofs: What they are, why they matter and how to write them

Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is motivated if and only if mathematicians can identify (i) the tasks each step is intended to perform; and (ii) where each step could have reasonably come from. I argue that motivated proofs promote understanding, convey new mathematical resources and stimulate new discoveries. They thus have significant epistemic benefits and directly contribute to the efficient dissemination and advancement of mathematical knowledge. Given their benefits, I also discuss the more practical matter of how we can produce motivated proofs. Finally I consider the relationship between motivated proofs and proofs which are explanatory, beautiful and fitting.