We Need a Better Style of Proof William M. Farmer 2016 Abstract Proofs serve several diverse purposes in mathematics. They are used to communicate mathematical ideas, certify that mathematical results are correct, discover new mathematical facts, learn mathematics, establish the interconnections between mathematical ideas, show the universality of mathematical results, and create mathematical beauty. Traditional proofs and (computer-supported) formal proofs do not fulfill these purposes equally well. In fact, traditional proofs serve some purposes much better than formal proofs, and vice versa. For example, traditional proofs are usually better for communication, while formal proofs are usually better for certification. We will compare both traditional and formal proofs with respect to these seven purposes and show that both styles of proof have serious shortcomings. We will offer a new style of proof in which (1) informal and formal proof components are combined (in accordance with Michael Kohlhase's notion of flexiformality), (2) results are proved at the optimal level of abstraction (in accordance with the little theories method), and (3) cross-checks are employed systematically. We will argue that this style of proof fulfills the purposes of mathematical proofs much better than both traditional and formal proofs.