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 procedures and 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. Some essays deal with the logical skeleton of deduction, others examine the interplay between natural systems and models of computation, yet others use significant results from the natural 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.

Zusammenfassung
This volume is located in a cross-disciplinary ?eld bringing together mat- matics, logic, natural science and philosophy. Re?ection on the e?ectiveness of proof brings out a number of questions that have always been latent in the informal understanding of the subject. What makes a symbolic constr- tion signi?cant? What makes an assumption reasonable? What makes a proof reliable? G¨ odel, Church and Turing, in di?erent ways, achieve a deep und- standing of the notion of e?ective calculability involved in the nature of proof. Turing's work in particular provides a precise and unquestionably adequate de?nition of the general notion of a formal system in terms of a machine with a ?nite number of parts. On the other hand, Eugene Wigner refers to the - reasonable e?ectiveness of mathematics in the natural sciences as a miracle. Where should the boundary be traced between mathematical procedures and physical processes? What is the characteristic use of a proof as a com- tation, as opposed to its use as an experiment? What does natural science tell us about the e?ectiveness of proof? What is the role of mathematical proofs in the discovery and validation of empirical theories? The papers collected in this book are intended to search for some answers, to discuss conceptual and logical issues underlying such questions and, perhaps, to call attention to other relevant questions.

Inhalt
Why Proof? What is a Proof?.- On Formal Proofs.- Toy Models in Physics and the Reasonable Effectiveness of Mathematics.- Experimental Methods in Proofs.- Proofs Verifying Programs and Programs Producing Proofs: A Conceptual Analysis.- The Logic of the Weak Excluded Middle: A Case Study of Proof-Search.- Automated Search for Gödel's Proofs.- Proofs as Efficient Programs.- Quantum Combing.- Proofs instead of Meaning Explanations: Understanding Classical vs Intuitionistic Mathematics from the Outside.- Proof as a Path of Light.- Computability and Incomputability of Differential Equations.- Phenomenology of Incompleteness: From Formal Deductions to Mathematics and Physics.