What is a proof for? What is the characteristic use of a proof as a computation, as opposed to its use as an experiment? What is the relationship between mathematical proceduresand natural processes?The essays collected in this volume address such questions from different points of view and will interest students and scholars in several branches of scientific knowledge. Someessays deal with the logical skeleton of deduction, others examine the interplay between natural systems and models of computation, yet others use significant results from thenatural sciences to illustrate the character of procedures in applied mathematics. Focusing on relevant conceptual and logical issues underlying the overall quest for proving, the volume seeks to cast light on what the effectiveness of proof rests on.